/[escript]/trunk/escript/py_src/pdetools.py
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revision 1956 by gross, Mon Nov 3 05:08:42 2008 UTC revision 4071 by gross, Tue Nov 13 21:43:11 2012 UTC
# Line 1  Line 1 
1    
2  ########################################################  ##############################################################################
3  #  #
4  # Copyright (c) 2003-2008 by University of Queensland  # Copyright (c) 2003-2012 by University of Queensland
5  # Earth Systems Science Computational Center (ESSCC)  # http://www.uq.edu.au
 # http://www.uq.edu.au/esscc  
6  #  #
7  # Primary Business: Queensland, Australia  # Primary Business: Queensland, Australia
8  # Licensed under the Open Software License version 3.0  # Licensed under the Open Software License version 3.0
9  # http://www.opensource.org/licenses/osl-3.0.php  # http://www.opensource.org/licenses/osl-3.0.php
10  #  #
11  ########################################################  # Development until 2012 by Earth Systems Science Computational Center (ESSCC)
12    # Development since 2012 by School of Earth Sciences
13    #
14    ##############################################################################
15    
16  __copyright__="""Copyright (c) 2003-2008 by University of Queensland  __copyright__="""Copyright (c) 2003-2012 by University of Queensland
17  Earth Systems Science Computational Center (ESSCC)  http://www.uq.edu.au
 http://www.uq.edu.au/esscc  
18  Primary Business: Queensland, Australia"""  Primary Business: Queensland, Australia"""
19  __license__="""Licensed under the Open Software License version 3.0  __license__="""Licensed under the Open Software License version 3.0
20  http://www.opensource.org/licenses/osl-3.0.php"""  http://www.opensource.org/licenses/osl-3.0.php"""
21  __url__="http://www.uq.edu.au/esscc/escript-finley"  __url__="https://launchpad.net/escript-finley"
22    
23  """  """
24  Provides some tools related to PDEs.  Provides some tools related to PDEs.
25    
26  Currently includes:  Currently includes:
27      - Projector - to project a discontinuous      - Projector - to project a discontinuous function onto a continuous function
28      - Locator - to trace values in data objects at a certain location      - Locator - to trace values in data objects at a certain location
29      - TimeIntegrationManager - to handel extraplotion in time      - TimeIntegrationManager - to handle extrapolation in time
30      - SaddlePointProblem - solver for Saddle point problems using the inexact uszawa scheme      - SaddlePointProblem - solver for Saddle point problems using the inexact uszawa scheme
31    
32  @var __author__: name of author  :var __author__: name of author
33  @var __copyright__: copyrights  :var __copyright__: copyrights
34  @var __license__: licence agreement  :var __license__: licence agreement
35  @var __url__: url entry point on documentation  :var __url__: url entry point on documentation
36  @var __version__: version  :var __version__: version
37  @var __date__: date of the version  :var __date__: date of the version
38  """  """
39    
40  __author__="Lutz Gross, l.gross@uq.edu.au"  __author__="Lutz Gross, l.gross@uq.edu.au"
41    
42    
43  import escript  #from . import escript
44  import linearPDEs  from . import escriptcpp
45  import numarray  escore=escriptcpp
46  import util  from . import linearPDEs
47    import numpy
48    from . import util
49  import math  import math
50    
 ##### Added by Artak  
 # from Numeric import zeros,Int,Float64  
 ###################################  
   
   
51  class TimeIntegrationManager:  class TimeIntegrationManager:
52    """    """
53    a simple mechanism to manage time dependend values.    A simple mechanism to manage time dependend values.
54    
55    typical usage is::    Typical usage is::
56    
57       dt=0.1 # time increment       dt=0.1 # time increment
58       tm=TimeIntegrationManager(inital_value,p=1)       tm=TimeIntegrationManager(inital_value,p=1)
# Line 64  class TimeIntegrationManager: Line 62  class TimeIntegrationManager:
62           tm.checkin(dt,v)           tm.checkin(dt,v)
63           t+=dt           t+=dt
64    
65    @note: currently only p=1 is supported.    :note: currently only p=1 is supported.
66    """    """
67    def __init__(self,*inital_values,**kwargs):    def __init__(self,*inital_values,**kwargs):
68       """       """
69       sets up the value manager where inital_value is the initial value and p is order used for extrapolation       Sets up the value manager where ``inital_values`` are the initial values
70         and p is the order used for extrapolation.
71       """       """
72       if kwargs.has_key("p"):       if "p" in kwargs:
73              self.__p=kwargs["p"]              self.__p=kwargs["p"]
74       else:       else:
75              self.__p=1              self.__p=1
76       if kwargs.has_key("time"):       if "time" in kwargs:
77              self.__t=kwargs["time"]              self.__t=kwargs["time"]
78       else:       else:
79              self.__t=0.              self.__t=0.
# Line 85  class TimeIntegrationManager: Line 84  class TimeIntegrationManager:
84    
85    def getTime(self):    def getTime(self):
86        return self.__t        return self.__t
87    
88    def getValue(self):    def getValue(self):
89        out=self.__v_mem[0]        out=self.__v_mem[0]
90        if len(out)==1:        if len(out)==1:
# Line 94  class TimeIntegrationManager: Line 94  class TimeIntegrationManager:
94    
95    def checkin(self,dt,*values):    def checkin(self,dt,*values):
96        """        """
97        adds new values to the manager. the p+1 last value get lost        Adds new values to the manager. The p+1 last values are lost.
98        """        """
99        o=min(self.__order+1,self.__p)        o=min(self.__order+1,self.__p)
100        self.__order=min(self.__order+1,self.__p)        self.__order=min(self.__order+1,self.__p)
# Line 111  class TimeIntegrationManager: Line 111  class TimeIntegrationManager:
111    
112    def extrapolate(self,dt):    def extrapolate(self,dt):
113        """        """
114        extrapolates to dt forward in time.        Extrapolates to ``dt`` forward in time.
115        """        """
116        if self.__order==0:        if self.__order==0:
117           out=self.__v_mem[0]           out=self.__v_mem[0]
# Line 126  class TimeIntegrationManager: Line 126  class TimeIntegrationManager:
126           return out[0]           return out[0]
127        else:        else:
128           return out           return out
129    
130    
131  class Projector:  class Projector:
132    """    """
133    The Projector is a factory which projects a discontiuous function onto a    The Projector is a factory which projects a discontinuous function onto a
134    continuous function on the a given domain.    continuous function on a given domain.
135    """    """
136    def __init__(self, domain, reduce = True, fast=True):    def __init__(self, domain, reduce=True, fast=True):
137      """      """
138      Create a continuous function space projector for a domain.      Creates a continuous function space projector for a domain.
139    
140      @param domain: Domain of the projection.      :param domain: Domain of the projection.
141      @param reduce: Flag to reduce projection order (default is True)      :param reduce: Flag to reduce projection order
142      @param fast: Flag to use a fast method based on matrix lumping (default is true)      :param fast: Flag to use a fast method based on matrix lumping
143      """      """
144      self.__pde = linearPDEs.LinearPDE(domain)      self.__pde = linearPDEs.LinearPDE(domain)
145      if fast:      if fast:
146        self.__pde.setSolverMethod(linearPDEs.LinearPDE.LUMPING)          self.__pde.getSolverOptions().setSolverMethod(linearPDEs.SolverOptions.LUMPING)
147      self.__pde.setSymmetryOn()      self.__pde.setSymmetryOn()
148      self.__pde.setReducedOrderTo(reduce)      self.__pde.setReducedOrderTo(reduce)
149      self.__pde.setValue(D = 1.)      self.__pde.setValue(D = 1.)
150      return      return
151      def getSolverOptions(self):
152        """
153        Returns the solver options of the PDE solver.
154        
155        :rtype: `linearPDEs.SolverOptions`
156        """
157        return self.__pde.getSolverOptions()
158    
159      def getValue(self, input_data):
160        """
161        Projects ``input_data`` onto a continuous function.
162    
163        :param input_data: the data to be projected
164        """
165        return self(input_data)
166    
167    def __call__(self, input_data):    def __call__(self, input_data):
168      """      """
169      Projects input_data onto a continuous function      Projects ``input_data`` onto a continuous function.
170    
171      @param input_data: The input_data to be projected.      :param input_data: the data to be projected
172      """      """
173      out=escript.Data(0.,input_data.getShape(),self.__pde.getFunctionSpaceForSolution())      out=escore.Data(0.,input_data.getShape(),self.__pde.getFunctionSpaceForSolution())
174      self.__pde.setValue(Y = escript.Data(), Y_reduced = escript.Data())      self.__pde.setValue(Y = escore.Data(), Y_reduced = escore.Data())
175      if input_data.getRank()==0:      if input_data.getRank()==0:
176          self.__pde.setValue(Y = input_data)          self.__pde.setValue(Y = input_data)
177          out=self.__pde.getSolution()          out=self.__pde.getSolution()
# Line 186  class Projector: Line 201  class Projector:
201    
202  class NoPDE:  class NoPDE:
203       """       """
204       solves the following problem for u:       Solves the following problem for u:
205    
206       M{kronecker[i,j]*D[j]*u[j]=Y[i]}       *kronecker[i,j]*D[j]*u[j]=Y[i]*
207    
208       with constraint       with constraint
209    
210       M{u[j]=r[j]}  where M{q[j]>0}       *u[j]=r[j]*  where *q[j]>0*
211    
212       where D, Y, r and q are given functions of rank 1.       where *D*, *Y*, *r* and *q* are given functions of rank 1.
213    
214       In the case of scalars this takes the form       In the case of scalars this takes the form
215    
216       M{D*u=Y}       *D*u=Y*
217    
218       with constraint       with constraint
219    
220       M{u=r}  where M{q>0}       *u=r* where *q>0*
221    
222       where D, Y, r and q are given scalar functions.       where *D*, *Y*, *r* and *q* are given scalar functions.
223    
224       The constraint is overwriting any other condition.       The constraint overwrites any other condition.
225    
226       @note: This class is similar to the L{linearPDEs.LinearPDE} class with A=B=C=X=0 but has the intention       :note: This class is similar to the `linearPDEs.LinearPDE` class with
227              that all input parameter are given in L{Solution} or L{ReducedSolution}. The whole              A=B=C=X=0 but has the intention that all input parameters are given
228              thing is a bit strange and I blame Robert.Woodcock@csiro.au for this.              in `Solution` or `ReducedSolution`.
229       """       """
230         # The whole thing is a bit strange and I blame Rob Woodcock (CSIRO) for
231         # this.
232       def __init__(self,domain,D=None,Y=None,q=None,r=None):       def __init__(self,domain,D=None,Y=None,q=None,r=None):
233           """           """
234           initialize the problem           Initializes the problem.
235    
236           @param domain: domain of the PDE.           :param domain: domain of the PDE
237           @type domain: L{Domain}           :type domain: `Domain`
238           @param D: coefficient of the solution.           :param D: coefficient of the solution
239           @type D: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type D: ``float``, ``int``, ``numpy.ndarray``, `Data`
240           @param Y: right hand side           :param Y: right hand side
241           @type Y: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type Y: ``float``, ``int``, ``numpy.ndarray``, `Data`
242           @param q: location of constraints           :param q: location of constraints
243           @type q: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type q: ``float``, ``int``, ``numpy.ndarray``, `Data`
244           @param r: value of solution at locations of constraints           :param r: value of solution at locations of constraints
245           @type r: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type r: ``float``, ``int``, ``numpy.ndarray``, `Data`
246           """           """
247           self.__domain=domain           self.__domain=domain
248           self.__D=D           self.__D=D
# Line 233  class NoPDE: Line 250  class NoPDE:
250           self.__q=q           self.__q=q
251           self.__r=r           self.__r=r
252           self.__u=None           self.__u=None
253           self.__function_space=escript.Solution(self.__domain)           self.__function_space=escore.Solution(self.__domain)
254    
255       def setReducedOn(self):       def setReducedOn(self):
256           """           """
257           sets the L{FunctionSpace} of the solution to L{ReducedSolution}           Sets the `FunctionSpace` of the solution to `ReducedSolution`.
258           """           """
259           self.__function_space=escript.ReducedSolution(self.__domain)           self.__function_space=escore.ReducedSolution(self.__domain)
260           self.__u=None           self.__u=None
261    
262       def setReducedOff(self):       def setReducedOff(self):
263           """           """
264           sets the L{FunctionSpace} of the solution to L{Solution}           Sets the `FunctionSpace` of the solution to `Solution`.
265           """           """
266           self.__function_space=escript.Solution(self.__domain)           self.__function_space=escore.Solution(self.__domain)
267           self.__u=None           self.__u=None
268            
269       def setValue(self,D=None,Y=None,q=None,r=None):       def setValue(self,D=None,Y=None,q=None,r=None):
270           """           """
271           assigns values to the parameters.           Assigns values to the parameters.
272    
273           @param D: coefficient of the solution.           :param D: coefficient of the solution
274           @type D: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type D: ``float``, ``int``, ``numpy.ndarray``, `Data`
275           @param Y: right hand side           :param Y: right hand side
276           @type Y: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type Y: ``float``, ``int``, ``numpy.ndarray``, `Data`
277           @param q: location of constraints           :param q: location of constraints
278           @type q: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type q: ``float``, ``int``, ``numpy.ndarray``, `Data`
279           @param r: value of solution at locations of constraints           :param r: value of solution at locations of constraints
280           @type r: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type r: ``float``, ``int``, ``numpy.ndarray``, `Data`
281           """           """
282           if not D==None:           if not D==None:
283              self.__D=D              self.__D=D
# Line 276  class NoPDE: Line 294  class NoPDE:
294    
295       def getSolution(self):       def getSolution(self):
296           """           """
297           returns the solution           Returns the solution.
298            
299           @return: the solution of the problem           :return: the solution of the problem
300           @rtype: L{Data} object in the L{FunctionSpace} L{Solution} or L{ReducedSolution}.           :rtype: `Data` object in the `FunctionSpace` `Solution` or
301                     `ReducedSolution`
302           """           """
303           if self.__u==None:           if self.__u==None:
304              if self.__D==None:              if self.__D==None:
305                 raise ValueError,"coefficient D is undefined"                 raise ValueError("coefficient D is undefined")
306              D=escript.Data(self.__D,self.__function_space)              D=escore.Data(self.__D,self.__function_space)
307              if D.getRank()>1:              if D.getRank()>1:
308                 raise ValueError,"coefficient D must have rank 0 or 1"                 raise ValueError("coefficient D must have rank 0 or 1")
309              if self.__Y==None:              if self.__Y==None:
310                 self.__u=escript.Data(0.,D.getShape(),self.__function_space)                 self.__u=escore.Data(0.,D.getShape(),self.__function_space)
311              else:              else:
312                 self.__u=util.quotient(self.__Y,D)                 self.__u=1./D*self.__Y
313              if not self.__q==None:              if not self.__q==None:
314                  q=util.wherePositive(escript.Data(self.__q,self.__function_space))                  q=util.wherePositive(escore.Data(self.__q,self.__function_space))
315                  self.__u*=(1.-q)                  self.__u*=(1.-q)
316                  if not self.__r==None: self.__u+=q*self.__r                  if not self.__r==None: self.__u+=q*self.__r
317           return self.__u           return self.__u
318                
319  class Locator:  class Locator:
320       """       """
321       Locator provides access to the values of data objects at a given       Locator provides access to the values of data objects at a given spatial
322       spatial coordinate x.         coordinate x.
323        
324       In fact, a Locator object finds the sample in the set of samples of a       In fact, a Locator object finds the sample in the set of samples of a
325       given function space or domain where which is closest to the given       given function space or domain which is closest to the given point x.
      point x.  
326       """       """
327    
328       def __init__(self,where,x=numarray.zeros((3,))):       def __init__(self,where,x=numpy.zeros((3,))):
329         """         """
330         Initializes a Locator to access values in Data objects on the Doamin         Initializes a Locator to access values in Data objects on the Doamin
331         or FunctionSpace where for the sample point which         or FunctionSpace for the sample point which is closest to the given
332         closest to the given point x.         point x.
333    
334         @param where: function space         :param where: function space
335         @type where: L{escript.FunctionSpace}         :type where: `escript.FunctionSpace`
336         @param x: coefficient of the solution.         :param x: location(s) of the Locator
337         @type x: L{numarray.NumArray} or C{list} of L{numarray.NumArray}         :type x: ``numpy.ndarray`` or ``list`` of ``numpy.ndarray``
338         """         """
339         if isinstance(where,escript.FunctionSpace):         if isinstance(where,escore.FunctionSpace):
340            self.__function_space=where            self.__function_space=where
341         else:         else:
342            self.__function_space=escript.ContinuousFunction(where)            self.__function_space=escore.ContinuousFunction(where)
343           iterative=False
344         if isinstance(x, list):         if isinstance(x, list):
345               if len(x)==0:
346                  raise ValueError("At least one point must be given.")
347               try:
348                 iter(x[0])
349                 iterative=True
350               except TypeError:
351                 iterative=False
352           xxx=self.__function_space.getX()
353           if iterative:
354             self.__id=[]             self.__id=[]
355             for p in x:             for p in x:
356                self.__id.append(util.length(self.__function_space.getX()-p[:self.__function_space.getDim()]).minGlobalDataPoint())                self.__id.append(util.length(xxx-p[:self.__function_space.getDim()]).minGlobalDataPoint())
357         else:         else:
358             self.__id=util.length(self.__function_space.getX()-x[:self.__function_space.getDim()]).minGlobalDataPoint()             self.__id=util.length(xxx-x[:self.__function_space.getDim()]).minGlobalDataPoint()
359    
360       def __str__(self):       def __str__(self):
361         """         """
362         Returns the coordinates of the Locator as a string.         Returns the coordinates of the Locator as a string.
363         """         """
364         x=self.getX()         x=self.getX()
365         if instance(x,list):         if isinstance(x,list):
366            out="["            out="["
367            first=True            first=True
368            for xx in x:            for xx in x:
# Line 350  class Locator: Line 378  class Locator:
378    
379       def getX(self):       def getX(self):
380          """          """
381      Returns the exact coordinates of the Locator.          Returns the exact coordinates of the Locator.
382      """          """
383          return self(self.getFunctionSpace().getX())          return self(self.getFunctionSpace().getX())
384    
385       def getFunctionSpace(self):       def getFunctionSpace(self):
386          """          """
387      Returns the function space of the Locator.          Returns the function space of the Locator.
388      """          """
389          return self.__function_space          return self.__function_space
390    
391       def getId(self,item=None):       def getId(self,item=None):
392          """          """
393      Returns the identifier of the location.          Returns the identifier of the location.
394      """          """
395          if item == None:          if item == None:
396             return self.__id             return self.__id
397          else:          else:
# Line 375  class Locator: Line 403  class Locator:
403    
404       def __call__(self,data):       def __call__(self,data):
405          """          """
406      Returns the value of data at the Locator of a Data object otherwise          Returns the value of data at the Locator of a Data object.
407      the object is returned.          """
     """  
408          return self.getValue(data)          return self.getValue(data)
409    
410       def getValue(self,data):       def getValue(self,data):
411          """          """
412      Returns the value of data at the Locator if data is a Data object          Returns the value of ``data`` at the Locator if ``data`` is a `Data`
413      otherwise the object is returned.          object otherwise the object is returned.
414      """          """
415          if isinstance(data,escript.Data):          if isinstance(data,escore.Data):
416             if data.getFunctionSpace()==self.getFunctionSpace():             dat=util.interpolate(data,self.getFunctionSpace())
              dat=data  
            else:  
              dat=data.interpolate(self.getFunctionSpace())  
417             id=self.getId()             id=self.getId()
418             r=data.getRank()             r=data.getRank()
419             if isinstance(id,list):             if isinstance(id,list):
420                 out=[]                 out=[]
421                 for i in id:                 for i in id:
422                    o=data.getValueOfGlobalDataPoint(*i)                    o=numpy.array(dat.getTupleForGlobalDataPoint(*i))
423                    if data.getRank()==0:                    if data.getRank()==0:
424                       out.append(o[0])                       out.append(o[0])
425                    else:                    else:
426                       out.append(o)                       out.append(o)
427                 return out                 return out
428             else:             else:
429               out=data.getValueOfGlobalDataPoint(*id)               out=numpy.array(dat.getTupleForGlobalDataPoint(*id))
430               if data.getRank()==0:               if data.getRank()==0:
431                  return out[0]                  return out[0]
432               else:               else:
433                  return out                  return out
434          else:          else:
435             return data             return data
436              
437         def setValue(self, data, v):
438          """
439          Sets the value of the ``data`` at the Locator.
440          """
441          if isinstance(data, escore.Data):
442             if data.getFunctionSpace()!=self.getFunctionSpace():
443               raise TypeError("setValue: FunctionSpace of Locator and Data object must match.")
444             data.expand()  
445             id=self.getId()
446             if isinstance(id, list):
447              for i in id:
448               data._setTupleForGlobalDataPoint(i[1], i[0], v)
449             else:
450               data._setTupleForGlobalDataPoint(id[1], id[0], v)
451          else:
452               raise TypeError("setValue: Invalid argument type.")
453    
454    
455    def getInfLocator(arg):
456        """
457        Return a Locator for a point with the inf value over all arg.
458        """
459        if not isinstance(arg, escore.Data):
460           raise TypeError("getInfLocator: Unknown argument type.")
461        a_inf=util.inf(arg)
462        loc=util.length(arg-a_inf).minGlobalDataPoint() # This gives us the location but not coords
463        x=arg.getFunctionSpace().getX()
464        x_min=x.getTupleForGlobalDataPoint(*loc)
465        return Locator(arg.getFunctionSpace(),x_min)
466    
467    def getSupLocator(arg):
468        """
469        Return a Locator for a point with the sup value over all arg.
470        """
471        if not isinstance(arg, escore.Data):
472           raise TypeError("getInfLocator: Unknown argument type.")
473        a_inf=util.sup(arg)
474        loc=util.length(arg-a_inf).minGlobalDataPoint() # This gives us the location but not coords
475        x=arg.getFunctionSpace().getX()
476        x_min=x.getTupleForGlobalDataPoint(*loc)
477        return Locator(arg.getFunctionSpace(),x_min)
478        
479    
480  class SolverSchemeException(Exception):  class SolverSchemeException(Exception):
481     """     """
482     exceptions thrown by solvers     This is a generic exception thrown by solvers.
483     """     """
484     pass     pass
485    
486  class IndefinitePreconditioner(SolverSchemeException):  class IndefinitePreconditioner(SolverSchemeException):
487     """     """
488     the preconditioner is not positive definite.     Exception thrown if the preconditioner is not positive definite.
489     """     """
490     pass     pass
491    
492  class MaxIterReached(SolverSchemeException):  class MaxIterReached(SolverSchemeException):
493     """     """
494     maxium number of iteration steps is reached.     Exception thrown if the maximum number of iteration steps is reached.
495       """
496       pass
497    
498    class CorrectionFailed(SolverSchemeException):
499       """
500       Exception thrown if no convergence has been achieved in the solution
501       correction scheme.
502     """     """
503     pass     pass
504    
505  class IterationBreakDown(SolverSchemeException):  class IterationBreakDown(SolverSchemeException):
506     """     """
507     iteration scheme econouters an incurable breakdown.     Exception thrown if the iteration scheme encountered an incurable breakdown.
508     """     """
509     pass     pass
510    
511  class NegativeNorm(SolverSchemeException):  class NegativeNorm(SolverSchemeException):
512     """     """
513     a norm calculation returns a negative norm.     Exception thrown if a norm calculation returns a negative norm.
514     """     """
515     pass     pass
516    
517  class IterationHistory(object):  def PCG(r, Aprod, x, Msolve, bilinearform, atol=0, rtol=1.e-8, iter_max=100, initial_guess=True, verbose=False):
518     """     """
519     The IterationHistory class is used to define a stopping criterium. It keeps track of the     Solver for
    residual norms. The stoppingcriterium indicates termination if the residual norm has been reduced by  
    a given tolerance.  
    """  
    def __init__(self,tolerance=math.sqrt(util.EPSILON),verbose=False):  
       """  
       Initialization  
   
       @param tolerance: tolerance  
       @type tolerance: positive C{float}  
       @param verbose: switches on the printing out some information  
       @type verbose: C{bool}  
       """  
       if not tolerance>0.:  
           raise ValueError,"tolerance needs to be positive."  
       self.tolerance=tolerance  
       self.verbose=verbose  
       self.history=[]  
    def stoppingcriterium(self,norm_r,r,x):  
        """  
        returns True if the C{norm_r} is C{tolerance}*C{norm_r[0]} where C{norm_r[0]}  is the residual norm at the first call.  
   
         
        @param norm_r: current residual norm  
        @type norm_r: non-negative C{float}  
        @param r: current residual (not used)  
        @param x: current solution approximation (not used)  
        @return: C{True} is the stopping criterium is fullfilled. Otherwise C{False} is returned.  
        @rtype: C{bool}  
   
        """  
        self.history.append(norm_r)  
        if self.verbose: print "iter: %s:  inner(rhat,r) = %e"#(len(self.history)-1, self.history[-1])  
        return self.history[-1]<=self.tolerance * self.history[0]  
520    
521     def stoppingcriterium2(self,norm_r,norm_b,solver="GMRES",TOL=None):     *Ax=b*
        """  
        returns True if the C{norm_r} is C{tolerance}*C{norm_b}  
522    
523             with a symmetric and positive definite operator A (more details required!).
524         @param norm_r: current residual norm     It uses the conjugate gradient method with preconditioner M providing an
525         @type norm_r: non-negative C{float}     approximation of A.
        @param norm_b: norm of right hand side  
        @type norm_b: non-negative C{float}  
        @return: C{True} is the stopping criterium is fullfilled. Otherwise C{False} is returned.  
        @rtype: C{bool}  
526    
527         """     The iteration is terminated if
        if TOL==None:  
           TOL=self.tolerance  
        self.history.append(norm_r)  
        if self.verbose: print "iter: %s:  norm(r) = %e"#(len(self.history)-1, self.history[-1])  
        return self.history[-1]<=TOL * norm_b  
528    
529  def PCG(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100):     *|r| <= atol+rtol*|r0|*
    """  
    Solver for  
530    
531     M{Ax=b}     where *r0* is the initial residual and *|.|* is the energy norm. In fact
532    
533     with a symmetric and positive definite operator A (more details required!).     *|r| = sqrt( bilinearform(Msolve(r),r))*
    It uses the conjugate gradient method with preconditioner M providing an approximation of A.  
   
    The iteration is terminated if the C{stoppingcriterium} function return C{True}.  
534    
535     For details on the preconditioned conjugate gradient method see the book:     For details on the preconditioned conjugate gradient method see the book:
536    
537     Templates for the Solution of Linear Systems by R. Barrett, M. Berry,     I{Templates for the Solution of Linear Systems by R. Barrett, M. Berry,
538     T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo,     T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo,
539     C. Romine, and H. van der Vorst.     C. Romine, and H. van der Vorst}.
540    
541     @param b: the right hand side of the liner system. C{b} is altered.     :param r: initial residual *r=b-Ax*. ``r`` is altered.
542     @type b: any object supporting inplace add (x+=y) and scaling (x=scalar*y)     :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
543     @param Aprod: returns the value Ax     :param x: an initial guess for the solution
544     @type Aprod: function C{Aprod(x)} where C{x} is of the same object like argument C{x}. The returned object needs to be of the same type like argument C{b}.     :type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
545     @param Msolve: solves Mx=r     :param Aprod: returns the value Ax
546     @type Msolve: function C{Msolve(r)} where C{r} is of the same type like argument C{b}. The returned object needs to be of the same     :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
547  type like argument C{x}.                  argument ``x``. The returned object needs to be of the same type
548     @param bilinearform: inner product C{<x,r>}                  like argument ``r``.
549     @type bilinearform: function C{bilinearform(x,r)} where C{x} is of the same type like argument C{x} and C{r} is . The returned value is a C{float}.     :param Msolve: solves Mx=r
550     @param stoppingcriterium: function which returns True if a stopping criterium is meet. C{stoppingcriterium} has the arguments C{norm_r}, C{r} and C{x} giving the current norm of the residual (=C{sqrt(bilinearform(Msolve(r),r)}), the current residual and the current solution approximation. C{stoppingcriterium} is called in each iteration step.     :type Msolve: function ``Msolve(r)`` where ``r`` is of the same type like
551     @type stoppingcriterium: function that returns C{True} or C{False}                   argument ``r``. The returned object needs to be of the same
552     @param x: an initial guess for the solution. If no C{x} is given 0*b is used.                   type like argument ``x``.
553     @type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)     :param bilinearform: inner product ``<x,r>``
554     @param iter_max: maximum number of iteration steps.     :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
555     @type iter_max: C{int}                         type like argument ``x`` and ``r`` is. The returned value
556     @return: the solution approximation and the corresponding residual                         is a ``float``.
557     @rtype: C{tuple}     :param atol: absolute tolerance
558     @warning: C{b} and C{x} are altered.     :type atol: non-negative ``float``
559       :param rtol: relative tolerance
560       :type rtol: non-negative ``float``
561       :param iter_max: maximum number of iteration steps
562       :type iter_max: ``int``
563       :return: the solution approximation and the corresponding residual
564       :rtype: ``tuple``
565       :warning: ``r`` and ``x`` are altered.
566     """     """
567     iter=0     iter=0
    if x==None:  
       x=0*b  
    else:  
       b += (-1)*Aprod(x)  
    r=b  
568     rhat=Msolve(r)     rhat=Msolve(r)
569     d = rhat     d = rhat
570     rhat_dot_r = bilinearform(rhat, r)     rhat_dot_r = bilinearform(rhat, r)
571     if rhat_dot_r<0: raise NegativeNorm,"negative norm."     if rhat_dot_r<0: raise NegativeNorm("negative norm.")
572       norm_r0=math.sqrt(rhat_dot_r)
573       atol2=atol+rtol*norm_r0
574       if atol2<=0:
575          raise ValueError("Non-positive tolarance.")
576       atol2=max(atol2, 100. * util.EPSILON * norm_r0)
577    
578       if verbose: print(("PCG: initial residual norm = %e (absolute tolerance = %e)"%(norm_r0, atol2)))
579    
580    
581     while not stoppingcriterium(math.sqrt(rhat_dot_r),r,x):     while not math.sqrt(rhat_dot_r) <= atol2:
582         iter+=1         iter+=1
583         if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max         if iter  >= iter_max: raise MaxIterReached("maximum number of %s steps reached."%iter_max)
584    
585         q=Aprod(d)         q=Aprod(d)
586         alpha = rhat_dot_r / bilinearform(d, q)         alpha = rhat_dot_r / bilinearform(d, q)
587         x += alpha * d         x += alpha * d
588         r += (-alpha) * q         if isinstance(q,ArithmeticTuple):
589              r += q * (-alpha)      # Doing it the other way calls the float64.__mul__ not AT.__rmul__
590           else:
591               r += (-alpha) * q
592         rhat=Msolve(r)         rhat=Msolve(r)
593         rhat_dot_r_new = bilinearform(rhat, r)         rhat_dot_r_new = bilinearform(rhat, r)
594         beta = rhat_dot_r_new / rhat_dot_r         beta = rhat_dot_r_new / rhat_dot_r
# Line 556  type like argument C{x}. Line 596  type like argument C{x}.
596         d=rhat         d=rhat
597    
598         rhat_dot_r = rhat_dot_r_new         rhat_dot_r = rhat_dot_r_new
599         if rhat_dot_r<0: raise NegativeNorm,"negative norm."         if rhat_dot_r<0: raise NegativeNorm("negative norm.")
600           if verbose: print(("PCG: iteration step %s: residual norm = %e"%(iter, math.sqrt(rhat_dot_r))))
601     return x,r     if verbose: print(("PCG: tolerance reached after %s steps."%iter))
602       return x,r,math.sqrt(rhat_dot_r)
603    
604  class Defect(object):  class Defect(object):
605      """      """
606      defines a non-linear defect F(x) of a variable x      Defines a non-linear defect F(x) of a variable x.
607      """      """
608      def __init__(self):      def __init__(self):
609          """          """
610          initialize defect          Initializes defect.
611          """          """
612          self.setDerivativeIncrementLength()          self.setDerivativeIncrementLength()
613    
614      def bilinearform(self, x0, x1):      def bilinearform(self, x0, x1):
615          """          """
616          returns the inner product of x0 and x1          Returns the inner product of x0 and x1
617          @param x0: a value for x  
618          @param x1: a value for x          :param x0: value for x0
619          @return: the inner product of x0 and x1          :param x1: value for x1
620          @rtype: C{float}          :return: the inner product of x0 and x1
621            :rtype: ``float``
622          """          """
623          return 0          return 0
624          
625      def norm(self,x):      def norm(self,x):
626          """          """
627          the norm of argument C{x}          Returns the norm of argument ``x``.
628    
629          @param x: a value for x          :param x: a value
630          @return: norm of argument x          :return: norm of argument x
631          @rtype: C{float}          :rtype: ``float``
632          @note: by default C{sqrt(self.bilinearform(x,x)} is retrurned.          :note: by default ``sqrt(self.bilinearform(x,x)`` is returned.
633          """          """
634          s=self.bilinearform(x,x)          s=self.bilinearform(x,x)
635          if s<0: raise NegativeNorm,"negative norm."          if s<0: raise NegativeNorm("negative norm.")
636          return math.sqrt(s)          return math.sqrt(s)
637    
   
638      def eval(self,x):      def eval(self,x):
639          """          """
640          returns the value F of a given x          Returns the value F of a given ``x``.
641    
642          @param x: value for which the defect C{F} is evalulated.          :param x: value for which the defect ``F`` is evaluated
643          @return: value of the defect at C{x}          :return: value of the defect at ``x``
644          """          """
645          return 0          return 0
646    
647      def __call__(self,x):      def __call__(self,x):
648          return self.eval(x)          return self.eval(x)
649    
650      def setDerivativeIncrementLength(self,inc=math.sqrt(util.EPSILON)):      def setDerivativeIncrementLength(self,inc=1000.*math.sqrt(util.EPSILON)):
651          """          """
652          sets the relative length of the increment used to approximate the derivative of the defect          Sets the relative length of the increment used to approximate the
653          the increment is inc*norm(x)/norm(v)*v in the direction of v with x as a staring point.          derivative of the defect. The increment is inc*norm(x)/norm(v)*v in the
654            direction of v with x as a starting point.
655    
656          @param inc: relative increment length          :param inc: relative increment length
657          @type inc: positive C{float}          :type inc: positive ``float``
658          """          """
659          if inc<=0: raise ValueError,"positive increment required."          if inc<=0: raise ValueError("positive increment required.")
660          self.__inc=inc          self.__inc=inc
661    
662      def getDerivativeIncrementLength(self):      def getDerivativeIncrementLength(self):
663          """          """
664          returns the relative increment length used to approximate the derivative of the defect          Returns the relative increment length used to approximate the
665          @return: value of the defect at C{x}          derivative of the defect.
666          @rtype: positive C{float}          :return: value of the defect at ``x``
667            :rtype: positive ``float``
668          """          """
669          return self.__inc          return self.__inc
670    
671      def derivative(self, F0, x0, v, v_is_normalised=True):      def derivative(self, F0, x0, v, v_is_normalised=True):
672          """          """
673          returns the directional derivative at x0 in the direction of v          Returns the directional derivative at ``x0`` in the direction of ``v``.
674    
675          @param F0: value of this defect at x0          :param F0: value of this defect at x0
676          @param x0: value at which derivative is calculated.          :param x0: value at which derivative is calculated
677          @param v: direction          :param v: direction
678          @param v_is_normalised: is true to indicate that C{v} is nomalized (self.norm(v)=0)          :param v_is_normalised: True to indicate that ``v`` is nomalized
679          @return: derivative of this defect at x0 in the direction of C{v}                                  (self.norm(v)=0)
680          @note: by default numerical evaluation (self.eval(x0+eps*v)-F0)/eps is used but this method          :return: derivative of this defect at x0 in the direction of ``v``
681          maybe oepsnew verwritten to use exact evalution.          :note: by default numerical evaluation (self.eval(x0+eps*v)-F0)/eps is
682                   used but this method maybe overwritten to use exact evaluation.
683          """          """
684          normx=self.norm(x0)          normx=self.norm(x0)
685          if normx>0:          if normx>0:
# Line 651  class Defect(object): Line 695  class Defect(object):
695          F1=self.eval(x0 + epsnew * v)          F1=self.eval(x0 + epsnew * v)
696          return (F1-F0)/epsnew          return (F1-F0)/epsnew
697    
698  ######################################      ######################################
699  def NewtonGMRES(defect, x, iter_max=100, sub_iter_max=20, atol=0,rtol=1.e-4, sub_tol_max=0.5, gamma=0.9, verbose=False):  def NewtonGMRES(defect, x, iter_max=100, sub_iter_max=20, atol=0,rtol=1.e-4, subtol_max=0.5, gamma=0.9, verbose=False):
700     """     """
701     solves a non-linear problem M{F(x)=0} for unknown M{x} using the stopping criterion:     Solves a non-linear problem *F(x)=0* for unknown *x* using the stopping
702       criterion:
703    
704       *norm(F(x) <= atol + rtol * norm(F(x0)*
705    
706     M{norm(F(x) <= atol + rtol * norm(F(x0)}     where *x0* is the initial guess.
707      
708     where M{x0} is the initial guess.     :param defect: object defining the function *F*. ``defect.norm`` defines the
709                      *norm* used in the stopping criterion.
710     @param defect: object defining the the function M{F}, C{defect.norm} defines the M{norm} used in the stopping criterion.     :type defect: `Defect`
711     @type defect: L{Defect}     :param x: initial guess for the solution, ``x`` is altered.
712     @param x: initial guess for the solution, C{x} is altered.     :type x: any object type allowing basic operations such as
713     @type x: any object type allowing basic operations such as  L{numarray.NumArray}, L{Data}              ``numpy.ndarray``, `Data`
714     @param iter_max: maximum number of iteration steps     :param iter_max: maximum number of iteration steps
715     @type iter_max: positive C{int}     :type iter_max: positive ``int``
716     @param sub_iter_max:     :param sub_iter_max: maximum number of inner iteration steps
717     @type sub_iter_max:     :type sub_iter_max: positive ``int``
718     @param atol: absolute tolerance for the solution     :param atol: absolute tolerance for the solution
719     @type atol: positive C{float}     :type atol: positive ``float``
720     @param rtol: relative tolerance for the solution     :param rtol: relative tolerance for the solution
721     @type rtol: positive C{float}     :type rtol: positive ``float``
722     @param gamma: tolerance safety factor for inner iteration     :param gamma: tolerance safety factor for inner iteration
723     @type gamma: positive C{float}, less than 1     :type gamma: positive ``float``, less than 1
724     @param sub_tol_max: upper bound for inner tolerance.     :param subtol_max: upper bound for inner tolerance
725     @type sub_tol_max: positive C{float}, less than 1     :type subtol_max: positive ``float``, less than 1
726     @return: an approximation of the solution with the desired accuracy     :return: an approximation of the solution with the desired accuracy
727     @rtype: same type as the initial guess.     :rtype: same type as the initial guess
728     """     """
729     lmaxit=iter_max     lmaxit=iter_max
730     if atol<0: raise ValueError,"atol needs to be non-negative."     if atol<0: raise ValueError("atol needs to be non-negative.")
731     if rtol<0: raise ValueError,"rtol needs to be non-negative."     if rtol<0: raise ValueError("rtol needs to be non-negative.")
732     if rtol+atol<=0: raise ValueError,"rtol or atol needs to be non-negative."     if rtol+atol<=0: raise ValueError("rtol or atol needs to be non-negative.")
733     if gamma<=0 or gamma>=1: raise ValueError,"tolerance safety factor for inner iteration (gamma =%s) needs to be positive and less than 1."%gamma     if gamma<=0 or gamma>=1: raise ValueError("tolerance safety factor for inner iteration (gamma =%s) needs to be positive and less than 1."%gamma)
734     if sub_tol_max<=0 or sub_tol_max>=1: raise ValueError,"upper bound for inner tolerance for inner iteration (sub_tol_max =%s) needs to be positive and less than 1."%sub_tol_max     if subtol_max<=0 or subtol_max>=1: raise ValueError("upper bound for inner tolerance for inner iteration (subtol_max =%s) needs to be positive and less than 1."%subtol_max)
735    
736     F=defect(x)     F=defect(x)
737     fnrm=defect.norm(F)     fnrm=defect.norm(F)
738     stop_tol=atol + rtol*fnrm     stop_tol=atol + rtol*fnrm
739     sub_tol=sub_tol_max     subtol=subtol_max
740     if verbose: print "NewtonGMRES: initial residual = %e."%fnrm     if verbose: print(("NewtonGMRES: initial residual = %e."%fnrm))
741     if verbose: print "             tolerance = %e."%sub_tol     if verbose: print(("             tolerance = %e."%subtol))
742     iter=1     iter=1
743     #     #
744     # main iteration loop     # main iteration loop
745     #     #
746     while not fnrm<=stop_tol:     while not fnrm<=stop_tol:
747              if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max              if iter  >= iter_max: raise MaxIterReached("maximum number of %s steps reached."%iter_max)
748              #              #
749          #   adjust sub_tol_          #   adjust subtol_
750          #          #
751              if iter > 1:              if iter > 1:
752             rat=fnrm/fnrmo                 rat=fnrm/fnrmo
753                 sub_tol_old=sub_tol                 subtol_old=subtol
754             sub_tol=gamma*rat**2                 subtol=gamma*rat**2
755             if gamma*sub_tol_old**2 > .1: sub_tol=max(sub_tol,gamma*sub_tol_old**2)                 if gamma*subtol_old**2 > .1: subtol=max(subtol,gamma*subtol_old**2)
756             sub_tol=max(min(sub_tol,sub_tol_max), .5*stop_tol/fnrm)                 subtol=max(min(subtol,subtol_max), .5*stop_tol/fnrm)
757          #          #
758          # calculate newton increment xc          # calculate newton increment xc
759              #     if iter_max in __FDGMRES is reached MaxIterReached is thrown              #     if iter_max in __FDGMRES is reached MaxIterReached is thrown
760              #     if iter_restart -1 is returned as sub_iter              #     if iter_restart -1 is returned as sub_iter
761              #     if  atol is reached sub_iter returns the numer of steps performed to get there              #     if  atol is reached sub_iter returns the numer of steps performed to get there
762              #              #
763              #                #
764              if verbose: print "             subiteration (GMRES) is called with relative tolerance %e."%sub_tol              if verbose: print(("             subiteration (GMRES) is called with relative tolerance %e."%subtol))
765              try:              try:
766                 xc, sub_iter=__FDGMRES(F, defect, x, sub_tol*fnrm, iter_max=iter_max-iter, iter_restart=sub_iter_max)                 xc, sub_iter=__FDGMRES(F, defect, x, subtol*fnrm, iter_max=iter_max-iter, iter_restart=sub_iter_max)
767              except MaxIterReached:              except MaxIterReached:
768                 raise MaxIterReached,"maximum number of %s steps reached."%iter_max                 raise MaxIterReached("maximum number of %s steps reached."%iter_max)
769              if sub_iter<0:              if sub_iter<0:
770                 iter+=sub_iter_max                 iter+=sub_iter_max
771              else:              else:
772                 iter+=sub_iter                 iter+=sub_iter
773              # ====              # ====
774          x+=xc              x+=xc
775              F=defect(x)              F=defect(x)
776          iter+=1              iter+=1
777              fnrmo, fnrm=fnrm, defect.norm(F)              fnrmo, fnrm=fnrm, defect.norm(F)
778              if verbose: print "             step %s: residual %e."%(iter,fnrm)              if verbose: print(("             step %s: residual %e."%(iter,fnrm)))
779     if verbose: print "NewtonGMRES: completed after %s steps."%iter     if verbose: print(("NewtonGMRES: completed after %s steps."%iter))
780     return x     return x
781    
782  def __givapp(c,s,vin):  def __givapp(c,s,vin):
783      """      """
784      apply a sequence of Givens rotations (c,s) to the recuirsively to the vector vin      Applies a sequence of Givens rotations (c,s) recursively to the vector
785      @warning: C{vin} is altered.      ``vin``
786    
787        :warning: ``vin`` is altered.
788      """      """
789      vrot=vin      vrot=vin
790      if isinstance(c,float):      if isinstance(c,float):
791          vrot=[c*vrot[0]-s*vrot[1],s*vrot[0]+c*vrot[1]]          vrot=[c*vrot[0]-s*vrot[1],s*vrot[0]+c*vrot[1]]
792      else:      else:
793          for i in range(len(c)):          for i in range(len(c)):
794              w1=c[i]*vrot[i]-s[i]*vrot[i+1]              w1=c[i]*vrot[i]-s[i]*vrot[i+1]
795          w2=s[i]*vrot[i]+c[i]*vrot[i+1]              w2=s[i]*vrot[i]+c[i]*vrot[i+1]
796              vrot[i:i+2]=w1,w2              vrot[i]=w1
797                vrot[i+1]=w2
798      return vrot      return vrot
799    
800  def __FDGMRES(F0, defect, x0, atol, iter_max=100, iter_restart=20):  def __FDGMRES(F0, defect, x0, atol, iter_max=100, iter_restart=20):
801     h=numarray.zeros((iter_restart,iter_restart),numarray.Float64)     h=numpy.zeros((iter_restart,iter_restart),numpy.float64)
802     c=numarray.zeros(iter_restart,numarray.Float64)     c=numpy.zeros(iter_restart,numpy.float64)
803     s=numarray.zeros(iter_restart,numarray.Float64)     s=numpy.zeros(iter_restart,numpy.float64)
804     g=numarray.zeros(iter_restart,numarray.Float64)     g=numpy.zeros(iter_restart,numpy.float64)
805     v=[]     v=[]
806    
807     rho=defect.norm(F0)     rho=defect.norm(F0)
808     if rho<=0.: return x0*0     if rho<=0.: return x0*0
809      
810     v.append(-F0/rho)     v.append(-F0/rho)
811     g[0]=rho     g[0]=rho
812     iter=0     iter=0
813     while rho > atol and iter<iter_restart-1:     while rho > atol and iter<iter_restart-1:
814            if iter  >= iter_max:
815      if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max              raise MaxIterReached("maximum number of %s steps reached."%iter_max)
816    
817          p=defect.derivative(F0,x0,v[iter], v_is_normalised=True)          p=defect.derivative(F0,x0,v[iter], v_is_normalised=True)
818      v.append(p)          v.append(p)
819    
820            v_norm1=defect.norm(v[iter+1])
821    
822      v_norm1=defect.norm(v[iter+1])          # Modified Gram-Schmidt
823            for j in range(iter+1):
824                h[j,iter]=defect.bilinearform(v[j],v[iter+1])
825                v[iter+1]-=h[j,iter]*v[j]
826    
827          # Modified Gram-Schmidt          h[iter+1,iter]=defect.norm(v[iter+1])
828      for j in range(iter+1):          v_norm2=h[iter+1,iter]
          h[j][iter]=defect.bilinearform(v[j],v[iter+1])    
          v[iter+1]-=h[j][iter]*v[j]  
         
     h[iter+1][iter]=defect.norm(v[iter+1])  
     v_norm2=h[iter+1][iter]  
829    
830          # Reorthogonalize if needed          # Reorthogonalize if needed
831      if v_norm1 + 0.001*v_norm2 == v_norm1:   #Brown/Hindmarsh condition (default)          if v_norm1 + 0.001*v_norm2 == v_norm1:   #Brown/Hindmarsh condition (default)
832          for j in range(iter+1):                for j in range(iter+1):
833             hr=defect.bilinearform(v[j],v[iter+1])                  hr=defect.bilinearform(v[j],v[iter+1])
834                 h[j][iter]=h[j][iter]+hr                  h[j,iter]=h[j,iter]+hr
835                 v[iter+1] -= hr*v[j]                  v[iter+1] -= hr*v[j]
836    
837          v_norm2=defect.norm(v[iter+1])              v_norm2=defect.norm(v[iter+1])
838          h[iter+1][iter]=v_norm2              h[iter+1,iter]=v_norm2
839          #   watch out for happy breakdown          #   watch out for happy breakdown
840          if not v_norm2 == 0:          if not v_norm2 == 0:
841                  v[iter+1]=v[iter+1]/h[iter+1][iter]              v[iter+1]=v[iter+1]/h[iter+1,iter]
842    
843          #   Form and store the information for the new Givens rotation          #   Form and store the information for the new Givens rotation
844      if iter > 0 :          if iter > 0 :
845          hhat=numarray.zeros(iter+1,numarray.Float64)              hhat=numpy.zeros(iter+1,numpy.float64)
846          for i in range(iter+1) : hhat[i]=h[i][iter]              for i in range(iter+1) : hhat[i]=h[i,iter]
847          hhat=__givapp(c[0:iter],s[0:iter],hhat);              hhat=__givapp(c[0:iter],s[0:iter],hhat);
848              for i in range(iter+1) : h[i][iter]=hhat[i]              for i in range(iter+1) : h[i,iter]=hhat[i]
849    
850      mu=math.sqrt(h[iter][iter]*h[iter][iter]+h[iter+1][iter]*h[iter+1][iter])          mu=math.sqrt(h[iter,iter]*h[iter,iter]+h[iter+1,iter]*h[iter+1,iter])
851    
852      if mu!=0 :          if mu!=0 :
853          c[iter]=h[iter][iter]/mu              c[iter]=h[iter,iter]/mu
854          s[iter]=-h[iter+1][iter]/mu              s[iter]=-h[iter+1,iter]/mu
855          h[iter][iter]=c[iter]*h[iter][iter]-s[iter]*h[iter+1][iter]              h[iter,iter]=c[iter]*h[iter,iter]-s[iter]*h[iter+1,iter]
856          h[iter+1][iter]=0.0              h[iter+1,iter]=0.0
857          g[iter:iter+2]=__givapp(c[iter],s[iter],g[iter:iter+2])              gg=__givapp(c[iter],s[iter],[g[iter],g[iter+1]])
858                g[iter]=gg[0]
859                g[iter+1]=gg[1]
860    
861          # Update the residual norm          # Update the residual norm
862          rho=abs(g[iter+1])          rho=abs(g[iter+1])
863      iter+=1          iter+=1
864    
865     # At this point either iter > iter_max or rho < tol.     # At this point either iter > iter_max or rho < tol.
866     # It's time to compute x and leave.             # It's time to compute x and leave.
867     if iter > 0 :     if iter > 0 :
868       y=numarray.zeros(iter,numarray.Float64)           y=numpy.zeros(iter,numpy.float64)
869       y[iter-1] = g[iter-1] / h[iter-1][iter-1]       y[iter-1] = g[iter-1] / h[iter-1,iter-1]
870       if iter > 1 :         if iter > 1 :
871          i=iter-2            i=iter-2
872          while i>=0 :          while i>=0 :
873            y[i] = ( g[i] - numarray.dot(h[i][i+1:iter], y[i+1:iter])) / h[i][i]            y[i] = ( g[i] - numpy.dot(h[i,i+1:iter], y[i+1:iter])) / h[i,i]
874            i=i-1            i=i-1
875       xhat=v[iter-1]*y[iter-1]       xhat=v[iter-1]*y[iter-1]
876       for i in range(iter-1):       for i in range(iter-1):
877      xhat += v[i]*y[i]         xhat += v[i]*y[i]
878     else :     else :
879        xhat=v[0] * 0        xhat=v[0] * 0
880    
881     if iter<iter_restart-1:     if iter<iter_restart-1:
882        stopped=iter        stopped=iter
883     else:     else:
884        stopped=-1        stopped=-1
885    
886     return xhat,stopped     return xhat,stopped
887    
888  ##############################################  def GMRES(r, Aprod, x, bilinearform, atol=0, rtol=1.e-8, iter_max=100, iter_restart=20, verbose=False,P_R=None):
889  def GMRES(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100, iter_restart=20):     """
890  ################################################     Solver for
891    
892       *Ax=b*
893    
894       with a general operator A (more details required!).
895       It uses the generalized minimum residual method (GMRES).
896    
897       The iteration is terminated if
898    
899       *|r| <= atol+rtol*|r0|*
900    
901       where *r0* is the initial residual and *|.|* is the energy norm. In fact
902    
903       *|r| = sqrt( bilinearform(r,r))*
904    
905       :param r: initial residual *r=b-Ax*. ``r`` is altered.
906       :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
907       :param x: an initial guess for the solution
908       :type x: same like ``r``
909       :param Aprod: returns the value Ax
910       :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
911                    argument ``x``. The returned object needs to be of the same
912                    type like argument ``r``.
913       :param bilinearform: inner product ``<x,r>``
914       :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
915                           type like argument ``x`` and ``r``. The returned value is
916                           a ``float``.
917       :param atol: absolute tolerance
918       :type atol: non-negative ``float``
919       :param rtol: relative tolerance
920       :type rtol: non-negative ``float``
921       :param iter_max: maximum number of iteration steps
922       :type iter_max: ``int``
923       :param iter_restart: in order to save memory the orthogonalization process
924                            is terminated after ``iter_restart`` steps and the
925                            iteration is restarted.
926       :type iter_restart: ``int``
927       :return: the solution approximation and the corresponding residual
928       :rtype: ``tuple``
929       :warning: ``r`` and ``x`` are altered.
930       """
931     m=iter_restart     m=iter_restart
932       restarted=False
933     iter=0     iter=0
934     xc=x     if rtol>0:
935          r_dot_r = bilinearform(r, r)
936          if r_dot_r<0: raise NegativeNorm("negative norm.")
937          atol2=atol+rtol*math.sqrt(r_dot_r)
938          if verbose: print(("GMRES: norm of right hand side = %e (absolute tolerance = %e)"%(math.sqrt(r_dot_r), atol2)))
939       else:
940          atol2=atol
941          if verbose: print(("GMRES: absolute tolerance = %e"%atol2))
942       if atol2<=0:
943          raise ValueError("Non-positive tolarance.")
944    
945     while True:     while True:
946        if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached"%iter_max        if iter  >= iter_max: raise MaxIterReached("maximum number of %s steps reached"%iter_max)
947        xc,stopped=__GMRESm(b*1, Aprod, Msolve, bilinearform, stoppingcriterium, x=xc*1, iter_max=iter_max-iter, iter_restart=m)        if restarted:
948             r2 = r-Aprod(x-x2)
949          else:
950             r2=1*r
951          x2=x*1.
952          x,stopped=_GMRESm(r2, Aprod, x, bilinearform, atol2, iter_max=iter_max-iter, iter_restart=m, verbose=verbose,P_R=P_R)
953          iter+=iter_restart
954        if stopped: break        if stopped: break
955        iter+=iter_restart            if verbose: print("GMRES: restart.")
956     return xc        restarted=True
957       if verbose: print("GMRES: tolerance has been reached.")
958       return x
959    
960  def __GMRESm(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100, iter_restart=20):  def _GMRESm(r, Aprod, x, bilinearform, atol, iter_max=100, iter_restart=20, verbose=False, P_R=None):
961     iter=0     iter=0
    r=Msolve(b)  
    r_dot_r = bilinearform(r, r)  
    if r_dot_r<0: raise NegativeNorm,"negative norm."  
    norm_b=math.sqrt(r_dot_r)  
962    
963     if x==None:     h=numpy.zeros((iter_restart+1,iter_restart),numpy.float64)
964        x=0*b     c=numpy.zeros(iter_restart,numpy.float64)
965     else:     s=numpy.zeros(iter_restart,numpy.float64)
966        r=Msolve(b-Aprod(x))     g=numpy.zeros(iter_restart+1,numpy.float64)
       r_dot_r = bilinearform(r, r)  
       if r_dot_r<0: raise NegativeNorm,"negative norm."  
     
    h=numarray.zeros((iter_restart,iter_restart),numarray.Float64)  
    c=numarray.zeros(iter_restart,numarray.Float64)  
    s=numarray.zeros(iter_restart,numarray.Float64)  
    g=numarray.zeros(iter_restart,numarray.Float64)  
967     v=[]     v=[]
968    
969       r_dot_r = bilinearform(r, r)
970       if r_dot_r<0: raise NegativeNorm("negative norm.")
971     rho=math.sqrt(r_dot_r)     rho=math.sqrt(r_dot_r)
972      
973     v.append(r/rho)     v.append(r/rho)
974     g[0]=rho     g[0]=rho
975    
976     while not (stoppingcriterium(rho,norm_b) or iter==iter_restart-1):     if verbose: print(("GMRES: initial residual %e (absolute tolerance = %e)"%(rho,atol)))
977       while not (rho<=atol or iter==iter_restart):
978    
979      if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max          if iter  >= iter_max: raise MaxIterReached("maximum number of %s steps reached."%iter_max)
980    
981      p=Msolve(Aprod(v[iter]))          if P_R!=None:
982                p=Aprod(P_R(v[iter]))
983            else:
984                p=Aprod(v[iter])
985            v.append(p)
986    
987      v.append(p)          v_norm1=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
988    
989      v_norm1=math.sqrt(bilinearform(v[iter+1], v[iter+1]))    # Modified Gram-Schmidt
990            for j in range(iter+1):
991              h[j,iter]=bilinearform(v[j],v[iter+1])
992              v[iter+1]-=h[j,iter]*v[j]
993    
994  # Modified Gram-Schmidt          h[iter+1,iter]=math.sqrt(bilinearform(v[iter+1],v[iter+1]))
995      for j in range(iter+1):          v_norm2=h[iter+1,iter]
       h[j][iter]=bilinearform(v[j],v[iter+1])    
       v[iter+1]-=h[j][iter]*v[j]  
         
     h[iter+1][iter]=math.sqrt(bilinearform(v[iter+1],v[iter+1]))  
     v_norm2=h[iter+1][iter]  
996    
997  # Reorthogonalize if needed  # Reorthogonalize if needed
998      if v_norm1 + 0.001*v_norm2 == v_norm1:   #Brown/Hindmarsh condition (default)          if v_norm1 + 0.001*v_norm2 == v_norm1:   #Brown/Hindmarsh condition (default)
999       for j in range(iter+1):             for j in range(iter+1):
1000          hr=bilinearform(v[j],v[iter+1])              hr=bilinearform(v[j],v[iter+1])
1001              h[j][iter]=h[j][iter]+hr              h[j,iter]=h[j,iter]+hr
1002              v[iter+1] -= hr*v[j]              v[iter+1] -= hr*v[j]
1003    
1004       v_norm2=math.sqrt(bilinearform(v[iter+1], v[iter+1]))             v_norm2=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
1005       h[iter+1][iter]=v_norm2           h[iter+1,iter]=v_norm2
1006    
1007  #   watch out for happy breakdown  #   watch out for happy breakdown
1008          if not v_norm2 == 0:          if not v_norm2 == 0:
1009           v[iter+1]=v[iter+1]/h[iter+1][iter]           v[iter+1]=v[iter+1]/h[iter+1,iter]
1010    
1011  #   Form and store the information for the new Givens rotation  #   Form and store the information for the new Givens rotation
1012      if iter > 0 :          if iter > 0: h[:iter+1,iter]=__givapp(c[:iter],s[:iter],h[:iter+1,iter])
1013          hhat=numarray.zeros(iter+1,numarray.Float64)          mu=math.sqrt(h[iter,iter]*h[iter,iter]+h[iter+1,iter]*h[iter+1,iter])
         for i in range(iter+1) : hhat[i]=h[i][iter]  
         hhat=__givapp(c[0:iter],s[0:iter],hhat);  
             for i in range(iter+1) : h[i][iter]=hhat[i]  
   
     mu=math.sqrt(h[iter][iter]*h[iter][iter]+h[iter+1][iter]*h[iter+1][iter])  
   
     if mu!=0 :  
         c[iter]=h[iter][iter]/mu  
         s[iter]=-h[iter+1][iter]/mu  
         h[iter][iter]=c[iter]*h[iter][iter]-s[iter]*h[iter+1][iter]  
         h[iter+1][iter]=0.0  
         g[iter:iter+2]=__givapp(c[iter],s[iter],g[iter:iter+2])  
1014    
1015            if mu!=0 :
1016                    c[iter]=h[iter,iter]/mu
1017                    s[iter]=-h[iter+1,iter]/mu
1018                    h[iter,iter]=c[iter]*h[iter,iter]-s[iter]*h[iter+1,iter]
1019                    h[iter+1,iter]=0.0
1020                    gg=__givapp(c[iter],s[iter],[g[iter],g[iter+1]])
1021                    g[iter]=gg[0]
1022                    g[iter+1]=gg[1]
1023  # Update the residual norm  # Update the residual norm
1024                  
1025          rho=abs(g[iter+1])          rho=abs(g[iter+1])
1026      iter+=1          if verbose: print(("GMRES: iteration step %s: residual %e"%(iter,rho)))
1027            iter+=1
1028    
1029  # At this point either iter > iter_max or rho < tol.  # At this point either iter > iter_max or rho < tol.
1030  # It's time to compute x and leave.          # It's time to compute x and leave.
1031    
1032     if iter > 0 :     if verbose: print(("GMRES: iteration stopped after %s step."%iter))
1033       y=numarray.zeros(iter,numarray.Float64)         if iter > 0 :
1034       y[iter-1] = g[iter-1] / h[iter-1][iter-1]       y=numpy.zeros(iter,numpy.float64)
1035       if iter > 1 :         y[iter-1] = g[iter-1] / h[iter-1,iter-1]
1036          i=iter-2         if iter > 1 :
1037            i=iter-2
1038          while i>=0 :          while i>=0 :
1039            y[i] = ( g[i] - numarray.dot(h[i][i+1:iter], y[i+1:iter])) / h[i][i]            y[i] = ( g[i] - numpy.dot(h[i,i+1:iter], y[i+1:iter])) / h[i,i]
1040            i=i-1            i=i-1
1041       xhat=v[iter-1]*y[iter-1]       xhat=v[iter-1]*y[iter-1]
1042       for i in range(iter-1):       for i in range(iter-1):
1043      xhat += v[i]*y[i]         xhat += v[i]*y[i]
1044     else : xhat=v[0]     else:
1045         xhat=v[0] * 0
1046     x += xhat     if P_R!=None:
1047     if iter<iter_restart-1:        x += P_R(xhat)
1048        stopped=True     else:
1049     else:        x += xhat
1050       if iter<iter_restart-1:
1051          stopped=True
1052       else:
1053        stopped=False        stopped=False
1054    
1055     return x,stopped     return x,stopped
1056    
1057  #################################################  def MINRES(r, Aprod, x, Msolve, bilinearform, atol=0, rtol=1.e-8, iter_max=100):
1058  def MINRES(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100):      """
1059  #################################################      Solver for
1060      #  
1061      #  minres solves the system of linear equations Ax = b      *Ax=b*
1062      #  where A is a symmetric matrix (possibly indefinite or singular)  
1063      #  and b is a given vector.      with a symmetric and positive definite operator A (more details required!).
1064      #        It uses the minimum residual method (MINRES) with preconditioner M
1065      #  "A" may be a dense or sparse matrix (preferably sparse!)      providing an approximation of A.
1066      #  or the name of a function such that  
1067      #               y = A(x)      The iteration is terminated if
1068      #  returns the product y = Ax for any given vector x.  
1069      #      *|r| <= atol+rtol*|r0|*
1070      #  "M" defines a positive-definite preconditioner M = C C'.  
1071      #  "M" may be a dense or sparse matrix (preferably sparse!)      where *r0* is the initial residual and *|.|* is the energy norm. In fact
1072      #  or the name of a function such that  
1073      #  solves the system My = x for any given vector x.      *|r| = sqrt( bilinearform(Msolve(r),r))*
1074      #  
1075      #      For details on the preconditioned conjugate gradient method see the book:
1076        
1077        I{Templates for the Solution of Linear Systems by R. Barrett, M. Berry,
1078        T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo,
1079        C. Romine, and H. van der Vorst}.
1080    
1081        :param r: initial residual *r=b-Ax*. ``r`` is altered.
1082        :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1083        :param x: an initial guess for the solution
1084        :type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1085        :param Aprod: returns the value Ax
1086        :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
1087                     argument ``x``. The returned object needs to be of the same
1088                     type like argument ``r``.
1089        :param Msolve: solves Mx=r
1090        :type Msolve: function ``Msolve(r)`` where ``r`` is of the same type like
1091                      argument ``r``. The returned object needs to be of the same
1092                      type like argument ``x``.
1093        :param bilinearform: inner product ``<x,r>``
1094        :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
1095                            type like argument ``x`` and ``r`` is. The returned value
1096                            is a ``float``.
1097        :param atol: absolute tolerance
1098        :type atol: non-negative ``float``
1099        :param rtol: relative tolerance
1100        :type rtol: non-negative ``float``
1101        :param iter_max: maximum number of iteration steps
1102        :type iter_max: ``int``
1103        :return: the solution approximation and the corresponding residual
1104        :rtype: ``tuple``
1105        :warning: ``r`` and ``x`` are altered.
1106        """
1107      #------------------------------------------------------------------      #------------------------------------------------------------------
1108      # Set up y and v for the first Lanczos vector v1.      # Set up y and v for the first Lanczos vector v1.
1109      # y  =  beta1 P' v1,  where  P = C**(-1).      # y  =  beta1 P' v1,  where  P = C**(-1).
1110      # v is really P' v1.      # v is really P' v1.
1111      #------------------------------------------------------------------      #------------------------------------------------------------------
1112      if x==None:      r1    = r
1113        x=0*b      y = Msolve(r)
1114      else:      beta1 = bilinearform(y,r)
       b += (-1)*Aprod(x)  
1115    
1116      r1    = b      if beta1< 0: raise NegativeNorm("negative norm.")
     y = Msolve(b)  
     beta1 = bilinearform(y,b)  
   
     if beta1< 0: raise NegativeNorm,"negative norm."  
1117    
1118      #  If b = 0 exactly, stop with x = 0.      #  If r = 0 exactly, stop with x
1119      if beta1==0: return x*0.      if beta1==0: return x
1120    
1121      if beta1> 0:      if beta1> 0: beta1  = math.sqrt(beta1)
       beta1  = math.sqrt(beta1)        
1122    
1123      #------------------------------------------------------------------      #------------------------------------------------------------------
1124      # Initialize quantities.      # Initialize quantities.
# Line 1008  def MINRES(b, Aprod, Msolve, bilinearfor Line 1138  def MINRES(b, Aprod, Msolve, bilinearfor
1138      ynorm2 = 0      ynorm2 = 0
1139      cs     = -1      cs     = -1
1140      sn     = 0      sn     = 0
1141      w      = b*0.      w      = r*0.
1142      w2     = b*0.      w2     = r*0.
1143      r2     = r1      r2     = r1
1144      eps    = 0.0001      eps    = 0.0001
1145    
1146      #---------------------------------------------------------------------      #---------------------------------------------------------------------
1147      # Main iteration loop.      # Main iteration loop.
1148      # --------------------------------------------------------------------      # --------------------------------------------------------------------
1149      while not stoppingcriterium(rnorm,Anorm*ynorm,'MINRES'):    #  checks ||r|| < (||A|| ||x||) * TOL      while not rnorm<=atol+rtol*Anorm*ynorm:    #  checks ||r|| < (||A|| ||x||) * TOL
1150    
1151      if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max          if iter  >= iter_max: raise MaxIterReached("maximum number of %s steps reached."%iter_max)
1152          iter    = iter  +  1          iter    = iter  +  1
1153    
1154          #-----------------------------------------------------------------          #-----------------------------------------------------------------
# Line 1035  def MINRES(b, Aprod, Msolve, bilinearfor Line 1165  def MINRES(b, Aprod, Msolve, bilinearfor
1165          #-----------------------------------------------------------------          #-----------------------------------------------------------------
1166          s = 1/beta                 # Normalize previous vector (in y).          s = 1/beta                 # Normalize previous vector (in y).
1167          v = s*y                    # v = vk if P = I          v = s*y                    # v = vk if P = I
1168        
1169          y      = Aprod(v)          y      = Aprod(v)
1170        
1171          if iter >= 2:          if iter >= 2:
1172            y = y - (beta/oldb)*r1            y = y - (beta/oldb)*r1
1173    
1174          alfa   = bilinearform(v,y)              # alphak          alfa   = bilinearform(v,y)              # alphak
1175          y      += (- alfa/beta)*r2          y      += (- alfa/beta)*r2
1176          r1     = r2          r1     = r2
1177          r2     = y          r2     = y
1178          y = Msolve(r2)          y = Msolve(r2)
1179          oldb   = beta                           # oldb = betak          oldb   = beta                           # oldb = betak
1180          beta   = bilinearform(y,r2)             # beta = betak+1^2          beta   = bilinearform(y,r2)             # beta = betak+1^2
1181          if beta < 0: raise NegativeNorm,"negative norm."          if beta < 0: raise NegativeNorm("negative norm.")
1182    
1183          beta   = math.sqrt( beta )          beta   = math.sqrt( beta )
1184          tnorm2 = tnorm2 + alfa*alfa + oldb*oldb + beta*beta          tnorm2 = tnorm2 + alfa*alfa + oldb*oldb + beta*beta
1185            
1186          if iter==1:                 # Initialize a few things.          if iter==1:                 # Initialize a few things.
1187            gmax   = abs( alfa )      # alpha1            gmax   = abs( alfa )      # alpha1
1188            gmin   = gmax             # alpha1            gmin   = gmax             # alpha1
# Line 1060  def MINRES(b, Aprod, Msolve, bilinearfor Line 1190  def MINRES(b, Aprod, Msolve, bilinearfor
1190          # Apply previous rotation Qk-1 to get          # Apply previous rotation Qk-1 to get
1191          #   [deltak epslnk+1] = [cs  sn][dbark    0   ]          #   [deltak epslnk+1] = [cs  sn][dbark    0   ]
1192          #   [gbar k dbar k+1]   [sn -cs][alfak betak+1].          #   [gbar k dbar k+1]   [sn -cs][alfak betak+1].
1193        
1194          oldeps = epsln          oldeps = epsln
1195          delta  = cs * dbar  +  sn * alfa  # delta1 = 0         deltak          delta  = cs * dbar  +  sn * alfa  # delta1 = 0         deltak
1196          gbar   = sn * dbar  -  cs * alfa  # gbar 1 = alfa1     gbar k          gbar   = sn * dbar  -  cs * alfa  # gbar 1 = alfa1     gbar k
# Line 1070  def MINRES(b, Aprod, Msolve, bilinearfor Line 1200  def MINRES(b, Aprod, Msolve, bilinearfor
1200          # Compute the next plane rotation Qk          # Compute the next plane rotation Qk
1201    
1202          gamma  = math.sqrt(gbar*gbar+beta*beta)  # gammak          gamma  = math.sqrt(gbar*gbar+beta*beta)  # gammak
1203          gamma  = max(gamma,eps)          gamma  = max(gamma,eps)
1204          cs     = gbar / gamma             # ck          cs     = gbar / gamma             # ck
1205          sn     = beta / gamma             # sk          sn     = beta / gamma             # sk
1206          phi    = cs * phibar              # phik          phi    = cs * phibar              # phik
# Line 1078  def MINRES(b, Aprod, Msolve, bilinearfor Line 1208  def MINRES(b, Aprod, Msolve, bilinearfor
1208    
1209          # Update  x.          # Update  x.
1210    
1211          denom = 1/gamma          denom = 1/gamma
1212          w1    = w2          w1    = w2
1213          w2    = w          w2    = w
1214          w     = (v - oldeps*w1 - delta*w2) * denom          w     = (v - oldeps*w1 - delta*w2) * denom
1215          x     +=  phi*w          x     +=  phi*w
1216    
# Line 1095  def MINRES(b, Aprod, Msolve, bilinearfor Line 1225  def MINRES(b, Aprod, Msolve, bilinearfor
1225    
1226          # Estimate various norms and test for convergence.          # Estimate various norms and test for convergence.
1227    
1228          Anorm  = math.sqrt( tnorm2 )          Anorm  = math.sqrt( tnorm2 )
1229          ynorm  = math.sqrt( ynorm2 )          ynorm  = math.sqrt( ynorm2 )
1230    
1231          rnorm  = phibar          rnorm  = phibar
1232    
1233      return x      return x
1234    
1235  def TFQMR(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100):  def TFQMR(r, Aprod, x, bilinearform, atol=0, rtol=1.e-8, iter_max=100):
1236      """
1237      Solver for
1238    
1239  # TFQMR solver for linear systems    *Ax=b*
 #  
 #  
 # initialization  
 #  
   errtol = math.sqrt(bilinearform(b,b))  
   norm_b=errtol  
   kmax  = iter_max  
   error = []  
   
   if math.sqrt(bilinearform(x,x)) != 0.0:  
     r = b - Aprod(x)  
   else:  
     r = b  
1240    
1241    r=Msolve(r)    with a general operator A (more details required!).
1242      It uses the Transpose-Free Quasi-Minimal Residual method (TFQMR).
1243    
1244      The iteration is terminated if
1245    
1246      *|r| <= atol+rtol*|r0|*
1247    
1248      where *r0* is the initial residual and *|.|* is the energy norm. In fact
1249    
1250      *|r| = sqrt( bilinearform(r,r))*
1251    
1252      :param r: initial residual *r=b-Ax*. ``r`` is altered.
1253      :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1254      :param x: an initial guess for the solution
1255      :type x: same like ``r``
1256      :param Aprod: returns the value Ax
1257      :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
1258                   argument ``x``. The returned object needs to be of the same type
1259                   like argument ``r``.
1260      :param bilinearform: inner product ``<x,r>``
1261      :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
1262                          type like argument ``x`` and ``r``. The returned value is
1263                          a ``float``.
1264      :param atol: absolute tolerance
1265      :type atol: non-negative ``float``
1266      :param rtol: relative tolerance
1267      :type rtol: non-negative ``float``
1268      :param iter_max: maximum number of iteration steps
1269      :type iter_max: ``int``
1270      :rtype: ``tuple``
1271      :warning: ``r`` and ``x`` are altered.
1272      """
1273    u1=0    u1=0
1274    u2=0    u2=0
1275    y1=0    y1=0
1276    y2=0    y2=0
1277    
1278    w = r    w = r
1279    y1 = r    y1 = r
1280    iter = 0    iter = 0
1281    d = 0    d = 0
1282        v = Aprod(y1)
   v = Msolve(Aprod(y1))  
1283    u1 = v    u1 = v
1284      
1285    theta = 0.0;    theta = 0.0;
1286    eta = 0.0;    eta = 0.0;
1287    tau = math.sqrt(bilinearform(r,r))    rho=bilinearform(r,r)
1288    error = [ error, tau ]    if rho < 0: raise NegativeNorm("negative norm.")
1289    rho = tau * tau    tau = math.sqrt(rho)
1290    m=1    norm_r0=tau
1291  #    while tau>atol+rtol*norm_r0:
1292  #  TFQMR iteration      if iter  >= iter_max: raise MaxIterReached("maximum number of %s steps reached."%iter_max)
 #  
 #  while ( iter < kmax-1 ):  
     
   while not stoppingcriterium(tau*math.sqrt ( m + 1 ),norm_b,'TFQMR'):  
     if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max  
1293    
1294      sigma = bilinearform(r,v)      sigma = bilinearform(r,v)
1295        if sigma == 0.0: raise IterationBreakDown('TFQMR breakdown, sigma=0')
     if ( sigma == 0.0 ):  
       raise 'TFQMR breakdown, sigma=0'  
       
1296    
1297      alpha = rho / sigma      alpha = rho / sigma
1298    
# Line 1162  def TFQMR(b, Aprod, Msolve, bilinearform Line 1302  def TFQMR(b, Aprod, Msolve, bilinearform
1302  #  #
1303        if ( j == 1 ):        if ( j == 1 ):
1304          y2 = y1 - alpha * v          y2 = y1 - alpha * v
1305          u2 = Msolve(Aprod(y2))          u2 = Aprod(y2)
1306    
1307        m = 2 * (iter+1) - 2 + (j+1)        m = 2 * (iter+1) - 2 + (j+1)
1308        if j==0:        if j==0:
1309           w = w - alpha * u1           w = w - alpha * u1
1310           d = y1 + ( theta * theta * eta / alpha ) * d           d = y1 + ( theta * theta * eta / alpha ) * d
1311        if j==1:        if j==1:
# Line 1180  def TFQMR(b, Aprod, Msolve, bilinearform Line 1320  def TFQMR(b, Aprod, Msolve, bilinearform
1320  #  #
1321  #  Try to terminate the iteration at each pass through the loop  #  Try to terminate the iteration at each pass through the loop
1322  #  #
1323       # if ( tau * math.sqrt ( m + 1 ) <= errtol ):      if rho == 0.0: raise IterationBreakDown('TFQMR breakdown, rho=0')
      #   error = [ error, tau ]  
      #   total_iters = iter  
      #   break  
         
   
     if ( rho == 0.0 ):  
       raise 'TFQMR breakdown, rho=0'  
       
1324    
1325      rhon = bilinearform(r,w)      rhon = bilinearform(r,w)
1326      beta = rhon / rho;      beta = rhon / rho;
1327      rho = rhon;      rho = rhon;
1328      y1 = w + beta * y2;      y1 = w + beta * y2;
1329      u1 = Msolve(Aprod(y1))      u1 = Aprod(y1)
1330      v = u1 + beta * ( u2 + beta * v )      v = u1 + beta * ( u2 + beta * v )
1331      error = [ error, tau ]  
1332      total_iters = iter      iter += 1
       
     iter = iter + 1  
1333    
1334    return x    return x
1335    
# Line 1208  def TFQMR(b, Aprod, Msolve, bilinearform Line 1338  def TFQMR(b, Aprod, Msolve, bilinearform
1338    
1339  class ArithmeticTuple(object):  class ArithmeticTuple(object):
1340     """     """
1341     tuple supporting inplace update x+=y and scaling x=a*y where x,y is an ArithmeticTuple and a is a float.     Tuple supporting inplace update x+=y and scaling x=a*y where ``x,y`` is an
1342       ArithmeticTuple and ``a`` is a float.
1343    
1344     example of usage:     Example of usage::
1345    
1346     from esys.escript import Data         from esys.escript import Data
1347     from numarray import array         from numpy import array
1348     a=Data(...)         a=eData(...)
1349     b=array([1.,4.])         b=array([1.,4.])
1350     x=ArithmeticTuple(a,b)         x=ArithmeticTuple(a,b)
1351     y=5.*x         y=5.*x
1352    
1353     """     """
1354     def __init__(self,*args):     def __init__(self,*args):
1355         """         """
1356         initialize object with elements args.         Initializes object with elements ``args``.
1357    
1358         @param args: tuple of object that support implace add (x+=y) and scaling (x=a*y)         :param args: tuple of objects that support inplace add (x+=y) and
1359                        scaling (x=a*y)
1360         """         """
1361         self.__items=list(args)         self.__items=list(args)
1362    
1363     def __len__(self):     def __len__(self):
1364         """         """
1365         number of items         Returns the number of items.
1366    
1367         @return: number of items         :return: number of items
1368         @rtype: C{int}         :rtype: ``int``
1369         """         """
1370         return len(self.__items)         return len(self.__items)
1371    
1372     def __getitem__(self,index):     def __getitem__(self,index):
1373         """         """
1374         get an item         Returns item at specified position.
1375    
1376         @param index: item to be returned         :param index: index of item to be returned
1377         @type index: C{int}         :type index: ``int``
1378         @return: item with index C{index}         :return: item with index ``index``
1379         """         """
1380         return self.__items.__getitem__(index)         return self.__items.__getitem__(index)
1381    
1382     def __mul__(self,other):     def __mul__(self,other):
1383         """         """
1384         scaling from the right         Scales by ``other`` from the right.
1385    
1386         @param other: scaling factor         :param other: scaling factor
1387         @type other: C{float}         :type other: ``float``
1388         @return: itemwise self*other         :return: itemwise self*other
1389         @rtype: L{ArithmeticTuple}         :rtype: `ArithmeticTuple`
1390         """         """
1391         out=[]         out=[]
1392         try:           try:
1393             l=len(other)             l=len(other)
1394             if l!=len(self):             if l!=len(self):
1395                 raise ValueError,"length of of arguments don't match."                 raise ValueError("length of arguments don't match.")
1396             for i in range(l): out.append(self[i]*other[i])             for i in range(l):
1397            if self.__isEmpty(self[i]) or self.__isEmpty(other[i]):
1398                out.append(escore.Data())
1399            else:
1400                out.append(self[i]*other[i])
1401         except TypeError:         except TypeError:
1402         for i in range(len(self)): out.append(self[i]*other)          for i in range(len(self)):  
1403            if self.__isEmpty(self[i]) or self.__isEmpty(other):
1404                out.append(escore.Data())
1405            else:
1406                out.append(self[i]*other)
1407         return ArithmeticTuple(*tuple(out))         return ArithmeticTuple(*tuple(out))
1408    
1409     def __rmul__(self,other):     def __rmul__(self,other):
1410         """        """
1411         scaling from the left        Scales by ``other`` from the left.
1412    
1413         @param other: scaling factor        :param other: scaling factor
1414         @type other: C{float}        :type other: ``float``
1415         @return: itemwise other*self        :return: itemwise other*self
1416         @rtype: L{ArithmeticTuple}        :rtype: `ArithmeticTuple`
1417         """        """
1418         out=[]        out=[]
1419         try:          try:
1420             l=len(other)        l=len(other)
1421             if l!=len(self):        if l!=len(self):
1422                 raise ValueError,"length of of arguments don't match."            raise ValueError("length of arguments don't match.")
1423             for i in range(l): out.append(other[i]*self[i])        for i in range(l):
1424         except TypeError:          if self.__isEmpty(self[i]) or self.__isEmpty(other[i]):
1425         for i in range(len(self)): out.append(other*self[i])              out.append(escore.Data())
1426         return ArithmeticTuple(*tuple(out))          else:
1427                out.append(other[i]*self[i])
1428          except TypeError:
1429          for i in range(len(self)):  
1430            if self.__isEmpty(self[i]) or self.__isEmpty(other):
1431                out.append(escore.Data())
1432            else:
1433                out.append(other*self[i])
1434          return ArithmeticTuple(*tuple(out))
1435    
1436     def __div__(self,other):     def __div__(self,other):
1437         """         """
1438         dividing from the right         Scales by (1/``other``) from the right.
1439    
1440         @param other: scaling factor         :param other: scaling factor
1441         @type other: C{float}         :type other: ``float``
1442         @return: itemwise self/other         :return: itemwise self/other
1443         @rtype: L{ArithmeticTuple}         :rtype: `ArithmeticTuple`
1444         """         """
1445         return self*(1/other)         return self*(1/other)
1446    
1447     def __rdiv__(self,other):     def __rdiv__(self,other):
1448         """        """
1449         dividing from the left        Scales by (1/``other``) from the left.
1450    
1451          :param other: scaling factor
1452          :type other: ``float``
1453          :return: itemwise other/self
1454          :rtype: `ArithmeticTuple`
1455          """
1456          out=[]
1457          try:
1458          l=len(other)
1459          if l!=len(self):
1460              raise ValueError("length of arguments don't match.")
1461          
1462          for i in range(l):
1463            if self.__isEmpty(self[i]):
1464                raise ZeroDivisionError("in component %s"%i)
1465            else:
1466                if self.__isEmpty(other[i]):
1467                out.append(escore.Data())
1468                else:
1469                out.append(other[i]/self[i])
1470          except TypeError:
1471          for i in range(len(self)):
1472            if self.__isEmpty(self[i]):
1473                raise ZeroDivisionError("in component %s"%i)
1474            else:
1475                if self.__isEmpty(other):
1476                out.append(escore.Data())
1477                else:
1478                out.append(other/self[i])
1479          return ArithmeticTuple(*tuple(out))
1480    
        @param other: scaling factor  
        @type other: C{float}  
        @return: itemwise other/self  
        @rtype: L{ArithmeticTuple}  
        """  
        out=[]  
        try:    
            l=len(other)  
            if l!=len(self):  
                raise ValueError,"length of of arguments don't match."  
            for i in range(l): out.append(other[i]/self[i])  
        except TypeError:  
        for i in range(len(self)): out.append(other/self[i])  
        return ArithmeticTuple(*tuple(out))  
     
1481     def __iadd__(self,other):     def __iadd__(self,other):
1482         """        """
1483         in-place add of other to self        Inplace addition of ``other`` to self.
1484    
1485         @param other: increment        :param other: increment
1486         @type other: C{ArithmeticTuple}        :type other: ``ArithmeticTuple``
1487         """        """
1488         if len(self) != len(other):        if len(self) != len(other):
1489             raise ValueError,"tuple length must match."        raise ValueError("tuple lengths must match.")
1490         for i in range(len(self)):        for i in range(len(self)):
1491             self.__items[i]+=other[i]        if self.__isEmpty(self.__items[i]):
1492         return self            self.__items[i]=other[i]
1493          else:
1494              self.__items[i]+=other[i]
1495              
1496          return self
1497    
1498     def __add__(self,other):     def __add__(self,other):
1499         """        """
1500         add other to self        Adds ``other`` to self.
1501    
1502         @param other: increment        :param other: increment
1503         @type other: C{ArithmeticTuple}        :type other: ``ArithmeticTuple``
1504         """        """
1505         out=[]        out=[]
1506         try:          try:
1507             l=len(other)        l=len(other)
1508             if l!=len(self):        if l!=len(self):
1509                 raise ValueError,"length of of arguments don't match."            raise ValueError("length of arguments don't match.")
1510             for i in range(l): out.append(self[i]+other[i])        for i in range(l):
1511         except TypeError:          if self.__isEmpty(self[i]):
1512         for i in range(len(self)): out.append(self[i]+other)              out.append(other[i])
1513         return ArithmeticTuple(*tuple(out))          elif self.__isEmpty(other[i]):
1514                out.append(self[i])
1515            else:
1516                out.append(self[i]+other[i])
1517          except TypeError:
1518            for i in range(len(self)):    
1519            if self.__isEmpty(self[i]):
1520                out.append(other)
1521            elif self.__isEmpty(other):
1522                out.append(self[i])
1523            else:
1524                out.append(self[i]+other)
1525          return ArithmeticTuple(*tuple(out))
1526    
1527     def __sub__(self,other):     def __sub__(self,other):
1528         """        """
1529         subtract other from self        Subtracts ``other`` from self.
1530    
1531          :param other: decrement
1532          :type other: ``ArithmeticTuple``
1533          """
1534          out=[]
1535          try:
1536          l=len(other)
1537          if l!=len(self):
1538              raise ValueError("length of arguments don't match.")
1539          for i in range(l):
1540            if self.__isEmpty(other[i]):
1541                out.append(self[i])
1542            elif self.__isEmpty(self[i]):
1543                out.append(-other[i])
1544            else:
1545                out.append(self[i]-other[i])
1546          except TypeError:
1547            for i in range(len(self)):    
1548            if  self.__isEmpty(other):
1549                out.append(self[i])
1550            elif self.__isEmpty(self[i]):
1551                out.append(-other)
1552            else:
1553                out.append(self[i]-other)
1554                
1555          return ArithmeticTuple(*tuple(out))
1556    
        @param other: increment  
        @type other: C{ArithmeticTuple}  
        """  
        out=[]  
        try:    
            l=len(other)  
            if l!=len(self):  
                raise ValueError,"length of of arguments don't match."  
            for i in range(l): out.append(self[i]-other[i])  
        except TypeError:  
        for i in range(len(self)): out.append(self[i]-other)  
        return ArithmeticTuple(*tuple(out))  
     
1557     def __isub__(self,other):     def __isub__(self,other):
1558         """        """
1559         subtract other from self        Inplace subtraction of ``other`` from self.
1560    
1561         @param other: increment        :param other: decrement
1562         @type other: C{ArithmeticTuple}        :type other: ``ArithmeticTuple``
1563         """        """
1564         if len(self) != len(other):        if len(self) != len(other):
1565             raise ValueError,"tuple length must match."        raise ValueError("tuple length must match.")
1566         for i in range(len(self)):        for i in range(len(self)):
1567             self.__items[i]-=other[i]        if not self.__isEmpty(other[i]):
1568         return self            if self.__isEmpty(self.__items[i]):
1569              self.__items[i]=-other[i]
1570              else:
1571              self.__items[i]=other[i]
1572          return self
1573    
1574     def __neg__(self):     def __neg__(self):
1575         """        """
1576         negate        Negates values.
1577          """
1578         """        out=[]
1579         out=[]        for i in range(len(self)):
1580         for i in range(len(self)):        if self.__isEmpty(self[i]):
1581             out.append(-self[i])            out.append(escore.Data())
1582         return ArithmeticTuple(*tuple(out))        else:
1583              out.append(-self[i])
1584          
1585          return ArithmeticTuple(*tuple(out))
1586       def __isEmpty(self, d):
1587        if isinstance(d, escore.Data):
1588        return d.isEmpty()
1589        else:
1590        return False
1591    
1592    
1593  class HomogeneousSaddlePointProblem(object):  class HomogeneousSaddlePointProblem(object):
1594        """        """
1595        This provides a framwork for solving linear homogeneous saddle point problem of the form        This class provides a framework for solving linear homogeneous saddle
1596          point problems of the form::
              Av+B^*p=f  
              Bv    =0  
1597    
1598        for the unknowns v and p and given operators A and B and given right hand side f.            *Av+B^*p=f*
1599        B^* is the adjoint operator of B is the given inner product.            *Bv     =0*
1600    
1601          for the unknowns *v* and *p* and given operators *A* and *B* and
1602          given right hand side *f*. *B^** is the adjoint operator of *B*.
1603          *A* may depend weakly on *v* and *p*.
1604        """        """
1605        def __init__(self,**kwargs):        def __init__(self, **kwargs):
         self.setTolerance()  
         self.setToleranceReductionFactor()  
   
       def initialize(self):  
1606          """          """
1607          initialize the problem (overwrite)          initializes the saddle point problem
1608          """          """
1609          pass          self.resetControlParameters()
1610        def B(self,v):          self.setTolerance()
1611            self.setAbsoluteTolerance()
1612          def resetControlParameters(self, K_p=1., K_v=1., rtol_max=0.01, rtol_min = 1.e-7, chi_max=0.5, reduction_factor=0.3, theta = 0.1):
1613           """           """
1614           returns Bv (overwrite)           sets a control parameter
          @rtype: equal to the type of p  
1615    
1616           @note: boundary conditions on p should be zero!           :param K_p: initial value for constant to adjust pressure tolerance
1617             :type K_p: ``float``
1618             :param K_v: initial value for constant to adjust velocity tolerance
1619             :type K_v: ``float``
1620             :param rtol_max: maximuim relative tolerance used to calculate presssure and velocity increment.
1621             :type rtol_max: ``float``
1622             :param chi_max: maximum tolerable converegence rate.
1623             :type chi_max: ``float``
1624             :param reduction_factor: reduction factor for adjustment factors.
1625             :type reduction_factor: ``float``
1626           """           """
1627           pass           self.setControlParameter(K_p, K_v, rtol_max, rtol_min, chi_max, reduction_factor, theta)
1628    
1629        def inner(self,p0,p1):        def setControlParameter(self,K_p=None, K_v=None, rtol_max=None, rtol_min=None, chi_max=None, reduction_factor=None, theta=None):
1630           """           """
1631           returns inner product of two element p0 and p1  (overwrite)           sets a control parameter
           
          @type p0: equal to the type of p  
          @type p1: equal to the type of p  
          @rtype: C{float}  
1632    
          @rtype: equal to the type of p  
          """  
          pass  
1633    
1634        def solve_A(self,u,p):           :param K_p: initial value for constant to adjust pressure tolerance
1635             :type K_p: ``float``
1636             :param K_v: initial value for constant to adjust velocity tolerance
1637             :type K_v: ``float``
1638             :param rtol_max: maximuim relative tolerance used to calculate presssure and velocity increment.
1639             :type rtol_max: ``float``
1640             :param chi_max: maximum tolerable converegence rate.
1641             :type chi_max: ``float``
1642             :type reduction_factor: ``float``
1643           """           """
1644           solves Av=f-Au-B^*p with accuracy self.getReducedTolerance() (overwrite)           if not K_p == None:
1645                if K_p<1:
1646                   raise ValueError("K_p need to be greater or equal to 1.")
1647             else:
1648                K_p=self.__K_p
1649    
1650             if not K_v == None:
1651                if K_v<1:
1652                   raise ValueError("K_v need to be greater or equal to 1.")
1653             else:
1654                K_v=self.__K_v
1655    
1656             if not rtol_max == None:
1657                if rtol_max<=0 or rtol_max>=1:
1658                   raise ValueError("rtol_max needs to be positive and less than 1.")
1659             else:
1660                rtol_max=self.__rtol_max
1661    
1662             if not rtol_min == None:
1663                if rtol_min<=0 or rtol_min>=1:
1664                   raise ValueError("rtol_min needs to be positive and less than 1.")
1665             else:
1666                rtol_min=self.__rtol_min
1667    
1668             if not chi_max == None:
1669                if chi_max<=0 or chi_max>=1:
1670                   raise ValueError("chi_max needs to be positive and less than 1.")
1671             else:
1672                chi_max = self.__chi_max
1673    
1674             if not reduction_factor == None:
1675                if reduction_factor<=0 or reduction_factor>1:
1676                   raise ValueError("reduction_factor need to be between zero and one.")
1677             else:
1678                reduction_factor=self.__reduction_factor
1679    
1680             if not theta == None:
1681                if theta<=0 or theta>1:
1682                   raise ValueError("theta need to be between zero and one.")
1683             else:
1684                theta=self.__theta
1685    
1686             if rtol_min>=rtol_max:
1687                 raise ValueError("rtol_max = %e needs to be greater than rtol_min = %e"%(rtol_max,rtol_min))
1688             self.__chi_max = chi_max
1689             self.__rtol_max = rtol_max
1690             self.__K_p = K_p
1691             self.__K_v = K_v
1692             self.__reduction_factor = reduction_factor
1693             self.__theta = theta
1694             self.__rtol_min=rtol_min
1695    
1696           @rtype: equal to the type of v        #=============================================================
1697           @note: boundary conditions on v should be zero!        def inner_pBv(self,p,Bv):
1698           """           """
1699           pass           Returns inner product of element p and Bv (overwrite).
1700    
1701        def solve_prec(self,p):           :param p: a pressure increment
1702             :param Bv: a residual
1703             :return: inner product of element p and Bv
1704             :rtype: ``float``
1705             :note: used if PCG is applied.
1706           """           """
1707           provides a preconditioner for BA^{-1}B^* with accuracy self.getReducedTolerance() (overwrite)           raise NotImplementedError("no inner product for p and Bv implemented.")
1708    
1709           @rtype: equal to the type of p        def inner_p(self,p0,p1):
1710           """           """
1711           pass           Returns inner product of p0 and p1 (overwrite).
1712    
1713        def stoppingcriterium(self,Bv,v,p):           :param p0: a pressure
1714             :param p1: a pressure
1715             :return: inner product of p0 and p1
1716             :rtype: ``float``
1717             """
1718             raise NotImplementedError("no inner product for p implemented.")
1719      
1720          def norm_v(self,v):
1721           """           """
1722           returns a True if iteration is terminated. (overwrite)           Returns the norm of v (overwrite).
1723    
1724           @rtype: C{bool}           :param v: a velovity
1725             :return: norm of v
1726             :rtype: non-negative ``float``
1727           """           """
1728           pass           raise NotImplementedError("no norm of v implemented.")
1729                      def getDV(self, p, v, tol):
1730        def __inner(self,p,r):           """
1731           return self.inner(p,r[1])           return a correction to the value for a given v and a given p with accuracy `tol` (overwrite)
1732    
1733        def __inner_p(self,p1,p2):           :param p: pressure
1734           return self.inner(p1,p2)           :param v: pressure
1735                   :return: dv given as *dv= A^{-1} (f-A v-B^*p)*
1736        def __inner_a(self,a1,a2):           :note: Only *A* may depend on *v* and *p*
1737           return self.inner_a(a1,a2)           """
1738             raise NotImplementedError("no dv calculation implemented.")
1739    
1740        def __inner_a1(self,a1,a2):          
1741           return self.inner(a1[1],a2[1])        def Bv(self,v, tol):
1742            """
1743            Returns Bv with accuracy `tol` (overwrite)
1744    
1745        def __stoppingcriterium(self,norm_r,r,p):          :rtype: equal to the type of p
1746            return self.stoppingcriterium(r[1],r[0],p)          :note: boundary conditions on p should be zero!
1747            """
1748            raise NotImplementedError("no operator B implemented.")
1749    
1750        def __stoppingcriterium2(self,norm_r,norm_b,solver='GMRES',TOL=None):        def norm_Bv(self,Bv):
1751            return self.stoppingcriterium2(norm_r,norm_b,solver,TOL)          """
1752            Returns the norm of Bv (overwrite).
1753    
1754        def setTolerance(self,tolerance=1.e-8):          :rtype: equal to the type of p
1755                self.__tol=tolerance          :note: boundary conditions on p should be zero!
1756        def getTolerance(self):          """
1757                return self.__tol          raise NotImplementedError("no norm of Bv implemented.")
       def setToleranceReductionFactor(self,reduction=0.01):  
               self.__reduction=reduction  
       def getSubProblemTolerance(self):  
               return self.__reduction*self.getTolerance()  
   
       def solve(self,v,p,max_iter=20, verbose=False, show_details=False, solver='PCG',iter_restart=20):  
               """  
               solves the saddle point problem using initial guesses v and p.  
   
               @param max_iter: maximum number of iteration steps.  
               """  
               self.verbose=verbose  
               self.show_details=show_details and self.verbose  
   
               # assume p is known: then v=A^-1(f-B^*p)  
               # which leads to BA^-1B^*p = BA^-1f    
   
           # Az=f is solved as A(z-v)=f-Av (z-v = 0 on fixed_u_mask)        
           self.__z=v+self.solve_A(v,p*0)  
               Bz=self.B(self.__z)  
               #  
           #   solve BA^-1B^*p = Bz  
               #  
               #  
               #  
               self.iter=0  
           if solver=='GMRES':        
                 if self.verbose: print "enter GMRES method (iter_max=%s)"%max_iter  
                 p=GMRES(Bz,self.__Aprod2,self.__Msolve2,self.__inner_p,self.__stoppingcriterium2,iter_max=max_iter, x=p*1.,iter_restart=iter_restart)  
                 # solve Au=f-B^*p  
                 #       A(u-v)=f-B^*p-Av  
                 #       u=v+(u-v)  
         u=v+self.solve_A(v,p)  
   
           if solver=='TFQMR':        
                 if self.verbose: print "enter TFQMR method (iter_max=%s)"%max_iter  
                 p=TFQMR(Bz,self.__Aprod2,self.__Msolve2,self.__inner_p,self.__stoppingcriterium2,iter_max=max_iter, x=p*1.)  
                 # solve Au=f-B^*p  
                 #       A(u-v)=f-B^*p-Av  
                 #       u=v+(u-v)  
         u=v+self.solve_A(v,p)  
   
           if solver=='MINRES':        
                 if self.verbose: print "enter MINRES method (iter_max=%s)"%max_iter  
                 p=MINRES(Bz,self.__Aprod2,self.__Msolve2,self.__inner_p,self.__stoppingcriterium2,iter_max=max_iter, x=p*1.)  
                 # solve Au=f-B^*p  
                 #       A(u-v)=f-B^*p-Av  
                 #       u=v+(u-v)  
         u=v+self.solve_A(v,p)  
                 
           if solver=='GMRESC':        
                 if self.verbose: print "enter GMRES coupled method (iter_max=%s)"%max_iter  
                 p0=self.solve_prec1(Bz)  
             #alfa=(1/self.vol)*util.integrate(util.interpolate(p,escript.Function(self.domain)))  
                 #p-=alfa  
                 x=GMRES(ArithmeticTuple(self.__z*1.,p0*1),self.__Anext,self.__Mempty,self.__inner_a,self.__stoppingcriterium2,iter_max=max_iter, x=ArithmeticTuple(v*1,p*1),iter_restart=20)  
                 #x=NewtonGMRES(ArithmeticTuple(self.__z*1.,p0*1),self.__Aprod_Newton2,self.__Mempty,self.__inner_a,self.__stoppingcriterium2,iter_max=max_iter, x=ArithmeticTuple(v*1,p*1),atol=0,rtol=self.getTolerance())  
   
                 # solve Au=f-B^*p  
                 #       A(u-v)=f-B^*p-Av  
                 #       u=v+(u-v)  
             p=x[1]  
         u=v+self.solve_A(v,p)        
         #p=x[1]  
         #u=x[0]  
   
               if solver=='PCG':  
                 #   note that the residual r=Bz-BA^-1B^*p = B(z-A^-1B^*p) = Bv  
                 #  
                 #   with                    Av=Az-B^*p = f - B^*p (v=z on fixed_u_mask)  
                 #                           A(v-z)= f -Az - B^*p (v-z=0 on fixed_u_mask)  
                 if self.verbose: print "enter PCG method (iter_max=%s)"%max_iter  
                 p,r=PCG(ArithmeticTuple(self.__z*1.,Bz),self.__Aprod,self.__Msolve,self.__inner,self.__stoppingcriterium,iter_max=max_iter, x=p)  
             u=r[0]    
                 # print "DIFF=",util.integrate(p)  
   
               # print "RESULT div(u)=",util.Lsup(self.B(u)),util.Lsup(u)  
   
           return u,p  
   
       def __Msolve(self,r):  
           return self.solve_prec(r[1])  
   
       def __Msolve2(self,r):  
           return self.solve_prec(r*1)  
   
       def __Mempty(self,r):  
           return r  
   
   
       def __Aprod(self,p):  
           # return BA^-1B*p  
           #solve Av =B^*p as Av =f-Az-B^*(-p)  
           v=self.solve_A(self.__z,-p)  
           return ArithmeticTuple(v, self.B(v))  
   
       def __Anext(self,x):  
           # return next v,p  
           #solve Av =-B^*p as Av =f-Az-B^*p  
   
       pc=x[1]  
           v=self.solve_A(self.__z,-pc)  
       p=self.solve_prec1(self.B(v))  
   
           return ArithmeticTuple(v,p)  
   
   
       def __Aprod2(self,p):  
           # return BA^-1B*p  
           #solve Av =B^*p as Av =f-Az-B^*(-p)  
       v=self.solve_A(self.__z,-p)  
           return self.B(v)  
   
       def __Aprod_Newton(self,p):  
           # return BA^-1B*p - Bz  
           #solve Av =-B^*p as Av =f-Az-B^*p  
       v=self.solve_A(self.__z,-p)  
           return self.B(v-self.__z)  
   
       def __Aprod_Newton2(self,x):  
           # return BA^-1B*p - Bz  
           #solve Av =-B^*p as Av =f-Az-B^*p  
           pc=x[1]  
       v=self.solve_A(self.__z,-pc)  
           p=self.solve_prec1(self.B(v-self.__z))  
           return ArithmeticTuple(v,p)  
1758    
1759          def solve_AinvBt(self,dp, tol):
1760             """
1761             Solves *A dv=B^*dp* with accuracy `tol`
1762    
1763  def MaskFromBoundaryTag(domain,*tags):           :param dp: a pressure increment
1764     """           :return: the solution of *A dv=B^*dp*
1765     creates a mask on the Solution(domain) function space which one for samples           :note: boundary conditions on dv should be zero! *A* is the operator used in ``getDV`` and must not be altered.
1766     that touch the boundary tagged by tags.           """
1767             raise NotImplementedError("no operator A implemented.")
1768    
1769     usage: m=MaskFromBoundaryTag(domain,"left", "right")        def solve_prec(self,Bv, tol):
1770             """
1771             Provides a preconditioner for *(BA^{-1}B^ * )* applied to Bv with accuracy `tol`
1772    
1773     @param domain: a given domain           :rtype: equal to the type of p
1774     @type domain: L{escript.Domain}           :note: boundary conditions on p should be zero!
1775     @param tags: boundray tags           """
1776     @type tags: C{str}           raise NotImplementedError("no preconditioner for Schur complement implemented.")
1777     @return: a mask which marks samples that are touching the boundary tagged by any of the given tags.        #=============================================================
1778     @rtype: L{escript.Data} of rank 0        def __Aprod_PCG(self,dp):
1779     """            dv=self.solve_AinvBt(dp, self.__subtol)
1780     pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)            return ArithmeticTuple(dv,self.Bv(dv, self.__subtol))
1781     d=escript.Scalar(0.,escript.FunctionOnBoundary(domain))  
1782     for t in tags: d.setTaggedValue(t,1.)        def __inner_PCG(self,p,r):
1783     pde.setValue(y=d)           return self.inner_pBv(p,r[1])
1784     return util.whereNonZero(pde.getRightHandSide())  
1785  #============================================================================================================================================        def __Msolve_PCG(self,r):
1786  class SaddlePointProblem(object):            return self.solve_prec(r[1], self.__subtol)
1787     """        #=============================================================
1788     This implements a solver for a saddlepoint problem        def __Aprod_GMRES(self,p):
1789              return self.solve_prec(self.Bv(self.solve_AinvBt(p, self.__subtol), self.__subtol), self.__subtol)
1790          def __inner_GMRES(self,p0,p1):
1791             return self.inner_p(p0,p1)
1792    
1793          #=============================================================
1794          def norm_p(self,p):
1795              """
1796              calculates the norm of ``p``
1797              
1798              :param p: a pressure
1799              :return: the norm of ``p`` using the inner product for pressure
1800              :rtype: ``float``
1801              """
1802              f=self.inner_p(p,p)
1803              if f<0: raise ValueError("negative pressure norm.")
1804              return math.sqrt(f)
1805          
1806          def solve(self,v,p,max_iter=20, verbose=False, usePCG=True, iter_restart=20, max_correction_steps=10):
1807             """
1808             Solves the saddle point problem using initial guesses v and p.
1809    
1810     M{f(u,p)=0}           :param v: initial guess for velocity
1811     M{g(u)=0}           :param p: initial guess for pressure
1812             :type v: `Data`
1813             :type p: `Data`
1814             :param usePCG: indicates the usage of the PCG rather than GMRES scheme.
1815             :param max_iter: maximum number of iteration steps per correction
1816                              attempt
1817             :param verbose: if True, shows information on the progress of the
1818                             saddlepoint problem solver.
1819             :param iter_restart: restart the iteration after ``iter_restart`` steps
1820                                  (only used if useUzaw=False)
1821             :type usePCG: ``bool``
1822             :type max_iter: ``int``
1823             :type verbose: ``bool``
1824             :type iter_restart: ``int``
1825             :rtype: ``tuple`` of `Data` objects
1826             :note: typically this method is overwritten by a subclass. It provides a wrapper for the ``_solve`` method.
1827             """
1828             return self._solve(v=v,p=p,max_iter=max_iter,verbose=verbose, usePCG=usePCG, iter_restart=iter_restart, max_correction_steps=max_correction_steps)
1829    
1830     for u and p. The problem is solved with an inexact Uszawa scheme for p:        def _solve(self,v,p,max_iter=20, verbose=False, usePCG=True, iter_restart=20, max_correction_steps=10):
1831             """
1832             see `_solve` method.
1833             """
1834             self.verbose=verbose
1835             rtol=self.getTolerance()
1836             atol=self.getAbsoluteTolerance()
1837    
1838             K_p=self.__K_p
1839             K_v=self.__K_v
1840             correction_step=0
1841             converged=False
1842             chi=None
1843             eps=None
1844    
1845             if self.verbose: print(("HomogeneousSaddlePointProblem: start iteration: rtol= %e, atol=%e"%(rtol, atol)))
1846             while not converged:
1847    
1848                 # get tolerance for velecity increment:
1849                 if chi == None:
1850                    rtol_v=self.__rtol_max
1851                 else:
1852                    rtol_v=min(chi/K_v,self.__rtol_max)
1853                 rtol_v=max(rtol_v, self.__rtol_min)
1854                 if self.verbose: print(("HomogeneousSaddlePointProblem: step %s: rtol_v= %e"%(correction_step,rtol_v)))
1855                 # get velocity increment:
1856                 dv1=self.getDV(p,v,rtol_v)
1857                 v1=v+dv1
1858                 Bv1=self.Bv(v1, rtol_v)
1859                 norm_Bv1=self.norm_Bv(Bv1)
1860                 norm_dv1=self.norm_v(dv1)
1861                 if self.verbose: print(("HomogeneousSaddlePointProblem: step %s: norm_Bv1 = %e, norm_dv1 = %e"%(correction_step, norm_Bv1, norm_dv1)))
1862                 if norm_dv1*self.__theta < norm_Bv1:
1863                    # get tolerance for pressure increment:
1864                    large_Bv1=True
1865                    if chi == None or eps == None:
1866                       rtol_p=self.__rtol_max
1867                    else:
1868                       rtol_p=min(chi**2*eps/K_p/norm_Bv1, self.__rtol_max)
1869                    self.__subtol=max(rtol_p**2, self.__rtol_min)
1870                    if self.verbose: print(("HomogeneousSaddlePointProblem: step %s: rtol_p= %e"%(correction_step,rtol_p)))
1871                    # now we solve for the pressure increment dp from B*A^{-1}B^* dp = Bv1
1872                    if usePCG:
1873                        dp,r,a_norm=PCG(ArithmeticTuple(v1,Bv1),self.__Aprod_PCG,0*p,self.__Msolve_PCG,self.__inner_PCG,atol=0, rtol=rtol_p,iter_max=max_iter, verbose=self.verbose)
1874                        v2=r[0]
1875                        Bv2=r[1]
1876                    else:
1877                        # don't use!!!!
1878                        dp=GMRES(self.solve_prec(Bv1,self.__subtol),self.__Aprod_GMRES, 0*p, self.__inner_GMRES,atol=0, rtol=rtol_p,iter_max=max_iter, iter_restart=iter_restart, verbose=self.verbose)
1879                        dv2=self.solve_AinvBt(dp, self.__subtol)
1880                        v2=v1-dv2
1881                        Bv2=self.Bv(v2, self.__subtol)
1882                    p2=p+dp
1883                 else:
1884                    large_Bv1=False
1885                    v2=v1
1886                    p2=p
1887                 # update business:
1888                 norm_dv2=self.norm_v(v2-v)
1889                 norm_v2=self.norm_v(v2)
1890                 if self.verbose: print(("HomogeneousSaddlePointProblem: step %s: v2 = %e, norm_dv2 = %e"%(correction_step, norm_v2, self.norm_v(v2-v))))
1891                 eps, eps_old = max(norm_Bv1, norm_dv2), eps
1892                 if eps_old == None:
1893                      chi, chi_old = None, chi
1894                 else:
1895                      chi, chi_old = min(eps/ eps_old, self.__chi_max), chi
1896                 if eps != None:
1897                     if chi !=None:
1898                        if self.verbose: print(("HomogeneousSaddlePointProblem: step %s: convergence rate = %e, correction = %e"%(correction_step,chi, eps)))
1899                     else:
1900                        if self.verbose: print(("HomogeneousSaddlePointProblem: step %s: correction = %e"%(correction_step, eps)))
1901                 if eps <= rtol*norm_v2+atol :
1902                     converged = True
1903                 else:
1904                     if correction_step>=max_correction_steps:
1905                          raise CorrectionFailed("Given up after %d correction steps."%correction_step)
1906                     if chi_old!=None:
1907                        K_p=max(1,self.__reduction_factor*K_p,(chi-chi_old)/chi_old**2*K_p)
1908                        K_v=max(1,self.__reduction_factor*K_v,(chi-chi_old)/chi_old**2*K_p)
1909                        if self.verbose: print(("HomogeneousSaddlePointProblem: step %s: new adjustment factor K = %e"%(correction_step,K_p)))
1910                 correction_step+=1
1911                 v,p =v2, p2
1912             if self.verbose: print(("HomogeneousSaddlePointProblem: tolerance reached after %s steps."%correction_step))
1913             return v,p
1914          #========================================================================
1915          def setTolerance(self,tolerance=1.e-4):
1916             """
1917             Sets the relative tolerance for (v,p).
1918    
1919     M{Q_f (u^{k+1}-u^{k}) = - f(u^{k},p^{k})}           :param tolerance: tolerance to be used
1920     M{Q_g (p^{k+1}-p^{k}) =   g(u^{k+1})}           :type tolerance: non-negative ``float``
1921             """
1922             if tolerance<0:
1923                 raise ValueError("tolerance must be positive.")
1924             self.__rtol=tolerance
1925    
1926     where Q_f is an approximation of the Jacobiean A_f of f with respect to u  and Q_f is an approximation of        def getTolerance(self):
1927     A_g A_f^{-1} A_g with A_g is the jacobiean of g with respect to p. As a the construction of a 'proper'           """
1928     Q_g can be difficult, non-linear conjugate gradient method is applied to solve for p, so Q_g plays           Returns the relative tolerance.
    in fact the role of a preconditioner.  
    """  
    def __init__(self,verbose=False,*args):  
        """  
        initializes the problem  
1929    
1930         @param verbose: switches on the printing out some information           :return: relative tolerance
1931         @type verbose: C{bool}           :rtype: ``float``
1932         @note: this method may be overwritten by a particular saddle point problem           """
1933         """           return self.__rtol
        print "SaddlePointProblem should not be used anymore!"  
        if not isinstance(verbose,bool):  
             raise TypeError("verbose needs to be of type bool.")  
        self.__verbose=verbose  
        self.relaxation=1.  
        DeprecationWarning("SaddlePointProblem should not be used anymore.")  
1934    
1935     def trace(self,text):        def setAbsoluteTolerance(self,tolerance=0.):
1936         """           """
1937         prints text if verbose has been set           Sets the absolute tolerance.
1938    
1939         @param text: a text message           :param tolerance: tolerance to be used
1940         @type text: C{str}           :type tolerance: non-negative ``float``
1941         """           """
1942         if self.__verbose: print "%s: %s"%(str(self),text)           if tolerance<0:
1943                 raise ValueError("tolerance must be non-negative.")
1944             self.__atol=tolerance
1945    
1946     def solve_f(self,u,p,tol=1.e-8):        def getAbsoluteTolerance(self):
1947         """           """
1948         solves           Returns the absolute tolerance.
1949    
1950         A_f du = f(u,p)           :return: absolute tolerance
1951             :rtype: ``float``
1952             """
1953             return self.__atol
1954    
        with tolerance C{tol} and return du. A_f is Jacobiean of f with respect to u.  
1955    
1956         @param u: current approximation of u  def MaskFromBoundaryTag(domain,*tags):
1957         @type u: L{escript.Data}     """
1958         @param p: current approximation of p     Creates a mask on the Solution(domain) function space where the value is
1959         @type p: L{escript.Data}     one for samples that touch the boundary tagged by tags.
        @param tol: tolerance expected for du  
        @type tol: C{float}  
        @return: increment du  
        @rtype: L{escript.Data}  
        @note: this method has to be overwritten by a particular saddle point problem  
        """  
        pass  
1960    
1961     def solve_g(self,u,tol=1.e-8):     Usage: m=MaskFromBoundaryTag(domain, "left", "right")
        """  
        solves  
1962    
1963         Q_g dp = g(u)     :param domain: domain to be used
1964       :type domain: `escript.Domain`
1965       :param tags: boundary tags
1966       :type tags: ``str``
1967       :return: a mask which marks samples that are touching the boundary tagged
1968                by any of the given tags
1969       :rtype: `escript.Data` of rank 0
1970       """
1971       pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)
1972       d=escore.Scalar(0.,escore.FunctionOnBoundary(domain))
1973       for t in tags: d.setTaggedValue(t,1.)
1974       pde.setValue(y=d)
1975       return util.whereNonZero(pde.getRightHandSide())
1976    
1977         with Q_g is a preconditioner for A_g A_f^{-1} A_g with  A_g is the jacobiean of g with respect to p.  def MaskFromTag(domain,*tags):
1978       """
1979       Creates a mask on the Solution(domain) function space where the value is
1980       one for samples that touch regions tagged by tags.
1981    
1982         @param u: current approximation of u     Usage: m=MaskFromTag(domain, "ham")
        @type u: L{escript.Data}  
        @param tol: tolerance expected for dp  
        @type tol: C{float}  
        @return: increment dp  
        @rtype: L{escript.Data}  
        @note: this method has to be overwritten by a particular saddle point problem  
        """  
        pass  
1983    
1984     def inner(self,p0,p1):     :param domain: domain to be used
1985         """     :type domain: `escript.Domain`
1986         inner product of p0 and p1 approximating p. Typically this returns integrate(p0*p1)     :param tags: boundary tags
1987         @return: inner product of p0 and p1     :type tags: ``str``
1988         @rtype: C{float}     :return: a mask which marks samples that are touching the boundary tagged
1989         """              by any of the given tags
1990         pass     :rtype: `escript.Data` of rank 0
1991       """
1992       pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)
1993       d=escore.Scalar(0.,escore.Function(domain))
1994       for t in tags: d.setTaggedValue(t,1.)
1995       pde.setValue(Y=d)
1996       return util.whereNonZero(pde.getRightHandSide())
1997    
    subiter_max=3  
    def solve(self,u0,p0,tolerance=1.e-6,tolerance_u=None,iter_max=100,accepted_reduction=0.995,relaxation=None):  
         """  
         runs the solver  
1998    
         @param u0: initial guess for C{u}  
         @type u0: L{esys.escript.Data}  
         @param p0: initial guess for C{p}  
         @type p0: L{esys.escript.Data}  
         @param tolerance: tolerance for relative error in C{u} and C{p}  
         @type tolerance: positive C{float}  
         @param tolerance_u: tolerance for relative error in C{u} if different from C{tolerance}  
         @type tolerance_u: positive C{float}  
         @param iter_max: maximum number of iteration steps.  
         @type iter_max: C{int}  
         @param accepted_reduction: if the norm  g cannot be reduced by C{accepted_reduction} backtracking to adapt the  
                                    relaxation factor. If C{accepted_reduction=None} no backtracking is used.  
         @type accepted_reduction: positive C{float} or C{None}  
         @param relaxation: initial relaxation factor. If C{relaxation==None}, the last relaxation factor is used.  
         @type relaxation: C{float} or C{None}  
         """  
         tol=1.e-2  
         if tolerance_u==None: tolerance_u=tolerance  
         if not relaxation==None: self.relaxation=relaxation  
         if accepted_reduction ==None:  
               angle_limit=0.  
         elif accepted_reduction>=1.:  
               angle_limit=0.  
         else:  
               angle_limit=util.sqrt(1-accepted_reduction**2)  
         self.iter=0  
         u=u0  
         p=p0  
         #  
         #   initialize things:  
         #  
         converged=False  
         #  
         #  start loop:  
         #  
         #  initial search direction is g  
         #  
         while not converged :  
             if self.iter>iter_max:  
                 raise ArithmeticError("no convergence after %s steps."%self.iter)  
             f_new=self.solve_f(u,p,tol)  
             norm_f_new = util.Lsup(f_new)  
             u_new=u-f_new  
             g_new=self.solve_g(u_new,tol)  
             self.iter+=1  
             norm_g_new = util.sqrt(self.inner(g_new,g_new))  
             if norm_f_new==0. and norm_g_new==0.: return u, p  
             if self.iter>1 and not accepted_reduction==None:  
                #  
                #   did we manage to reduce the norm of G? I  
                #   if not we start a backtracking procedure  
                #  
                # print "new/old norm = ",norm_g_new, norm_g, norm_g_new/norm_g  
                if norm_g_new > accepted_reduction * norm_g:  
                   sub_iter=0  
                   s=self.relaxation  
                   d=g  
                   g_last=g  
                   self.trace("    start substepping: f = %s, g = %s, relaxation = %s."%(norm_f_new, norm_g_new, s))  
                   while sub_iter < self.subiter_max and  norm_g_new > accepted_reduction * norm_g:  
                      dg= g_new-g_last  
                      norm_dg=abs(util.sqrt(self.inner(dg,dg))/self.relaxation)  
                      rad=self.inner(g_new,dg)/self.relaxation  
                      # print "   ",sub_iter,": rad, norm_dg:",abs(rad), norm_dg*norm_g_new * angle_limit  
                      # print "   ",sub_iter,": rad, norm_dg:",rad, norm_dg, norm_g_new, norm_g  
                      if abs(rad) < norm_dg*norm_g_new * angle_limit:  
                          if sub_iter>0: self.trace("    no further improvements expected from backtracking.")  
                          break  
                      r=self.relaxation  
                      self.relaxation= - rad/norm_dg**2  
                      s+=self.relaxation  
                      #####  
                      # a=g_new+self.relaxation*dg/r  
                      # print "predicted new norm = ",util.sqrt(self.inner(a,a)),util.sqrt(self.inner(g_new,g_new)), self.relaxation  
                      #####  
                      g_last=g_new  
                      p+=self.relaxation*d  
                      f_new=self.solve_f(u,p,tol)  
                      u_new=u-f_new  
                      g_new=self.solve_g(u_new,tol)  
                      self.iter+=1  
                      norm_f_new = util.Lsup(f_new)  
                      norm_g_new = util.sqrt(self.inner(g_new,g_new))  
                      # print "   ",sub_iter," new g norm",norm_g_new  
                      self.trace("    %s th sub-step: f = %s, g = %s, relaxation = %s."%(sub_iter, norm_f_new, norm_g_new, s))  
                      #  
                      #   can we expect reduction of g?  
                      #  
                      # u_last=u_new  
                      sub_iter+=1  
                   self.relaxation=s  
             #  
             #  check for convergence:  
             #  
             norm_u_new = util.Lsup(u_new)  
             p_new=p+self.relaxation*g_new  
             norm_p_new = util.sqrt(self.inner(p_new,p_new))  
             self.trace("%s th step: f = %s, f/u = %s, g = %s, g/p = %s, relaxation = %s."%(self.iter, norm_f_new ,norm_f_new/norm_u_new, norm_g_new, norm_g_new/norm_p_new, self.relaxation))  
   
             if self.iter>1:  
                dg2=g_new-g  
                df2=f_new-f  
                norm_dg2=util.sqrt(self.inner(dg2,dg2))  
                norm_df2=util.Lsup(df2)  
                # print norm_g_new, norm_g, norm_dg, norm_p, tolerance  
                tol_eq_g=tolerance*norm_dg2/(norm_g*abs(self.relaxation))*norm_p_new  
                tol_eq_f=tolerance_u*norm_df2/norm_f*norm_u_new  
                if norm_g_new <= tol_eq_g and norm_f_new <= tol_eq_f:  
                    converged=True  
             f, norm_f, u, norm_u, g, norm_g, p, norm_p = f_new, norm_f_new, u_new, norm_u_new, g_new, norm_g_new, p_new, norm_p_new  
         self.trace("convergence after %s steps."%self.iter)  
         return u,p  

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