/[escript]/trunk/escript/py_src/pdetools.py
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revision 1956 by gross, Mon Nov 3 05:08:42 2008 UTC revision 2862 by gross, Thu Jan 21 04:45:39 2010 UTC
# Line 1  Line 1 
1    
2  ########################################################  ########################################################
3  #  #
4  # Copyright (c) 2003-2008 by University of Queensland  # Copyright (c) 2003-2009 by University of Queensland
5  # Earth Systems Science Computational Center (ESSCC)  # Earth Systems Science Computational Center (ESSCC)
6  # http://www.uq.edu.au/esscc  # http://www.uq.edu.au/esscc
7  #  #
# Line 11  Line 11 
11  #  #
12  ########################################################  ########################################################
13    
14  __copyright__="""Copyright (c) 2003-2008 by University of Queensland  __copyright__="""Copyright (c) 2003-2009 by University of Queensland
15  Earth Systems Science Computational Center (ESSCC)  Earth Systems Science Computational Center (ESSCC)
16  http://www.uq.edu.au/esscc  http://www.uq.edu.au/esscc
17  Primary Business: Queensland, Australia"""  Primary Business: Queensland, Australia"""
18  __license__="""Licensed under the Open Software License version 3.0  __license__="""Licensed under the Open Software License version 3.0
19  http://www.opensource.org/licenses/osl-3.0.php"""  http://www.opensource.org/licenses/osl-3.0.php"""
20  __url__="http://www.uq.edu.au/esscc/escript-finley"  __url__="https://launchpad.net/escript-finley"
21    
22  """  """
23  Provides some tools related to PDEs.  Provides some tools related to PDEs.
24    
25  Currently includes:  Currently includes:
26      - Projector - to project a discontinuous      - Projector - to project a discontinuous function onto a continuous function
27      - Locator - to trace values in data objects at a certain location      - Locator - to trace values in data objects at a certain location
28      - TimeIntegrationManager - to handel extraplotion in time      - TimeIntegrationManager - to handle extrapolation in time
29      - SaddlePointProblem - solver for Saddle point problems using the inexact uszawa scheme      - SaddlePointProblem - solver for Saddle point problems using the inexact uszawa scheme
30    
31  @var __author__: name of author  :var __author__: name of author
32  @var __copyright__: copyrights  :var __copyright__: copyrights
33  @var __license__: licence agreement  :var __license__: licence agreement
34  @var __url__: url entry point on documentation  :var __url__: url entry point on documentation
35  @var __version__: version  :var __version__: version
36  @var __date__: date of the version  :var __date__: date of the version
37  """  """
38    
39  __author__="Lutz Gross, l.gross@uq.edu.au"  __author__="Lutz Gross, l.gross@uq.edu.au"
# Line 41  __author__="Lutz Gross, l.gross@uq.edu.a Line 41  __author__="Lutz Gross, l.gross@uq.edu.a
41    
42  import escript  import escript
43  import linearPDEs  import linearPDEs
44  import numarray  import numpy
45  import util  import util
46  import math  import math
47    
 ##### Added by Artak  
 # from Numeric import zeros,Int,Float64  
 ###################################  
   
   
48  class TimeIntegrationManager:  class TimeIntegrationManager:
49    """    """
50    a simple mechanism to manage time dependend values.    A simple mechanism to manage time dependend values.
51    
52    typical usage is::    Typical usage is::
53    
54       dt=0.1 # time increment       dt=0.1 # time increment
55       tm=TimeIntegrationManager(inital_value,p=1)       tm=TimeIntegrationManager(inital_value,p=1)
# Line 64  class TimeIntegrationManager: Line 59  class TimeIntegrationManager:
59           tm.checkin(dt,v)           tm.checkin(dt,v)
60           t+=dt           t+=dt
61    
62    @note: currently only p=1 is supported.    :note: currently only p=1 is supported.
63    """    """
64    def __init__(self,*inital_values,**kwargs):    def __init__(self,*inital_values,**kwargs):
65       """       """
66       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
67         and p is the order used for extrapolation.
68       """       """
69       if kwargs.has_key("p"):       if kwargs.has_key("p"):
70              self.__p=kwargs["p"]              self.__p=kwargs["p"]
# Line 85  class TimeIntegrationManager: Line 81  class TimeIntegrationManager:
81    
82    def getTime(self):    def getTime(self):
83        return self.__t        return self.__t
84    
85    def getValue(self):    def getValue(self):
86        out=self.__v_mem[0]        out=self.__v_mem[0]
87        if len(out)==1:        if len(out)==1:
# Line 94  class TimeIntegrationManager: Line 91  class TimeIntegrationManager:
91    
92    def checkin(self,dt,*values):    def checkin(self,dt,*values):
93        """        """
94        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.
95        """        """
96        o=min(self.__order+1,self.__p)        o=min(self.__order+1,self.__p)
97        self.__order=min(self.__order+1,self.__p)        self.__order=min(self.__order+1,self.__p)
# Line 111  class TimeIntegrationManager: Line 108  class TimeIntegrationManager:
108    
109    def extrapolate(self,dt):    def extrapolate(self,dt):
110        """        """
111        extrapolates to dt forward in time.        Extrapolates to ``dt`` forward in time.
112        """        """
113        if self.__order==0:        if self.__order==0:
114           out=self.__v_mem[0]           out=self.__v_mem[0]
# Line 126  class TimeIntegrationManager: Line 123  class TimeIntegrationManager:
123           return out[0]           return out[0]
124        else:        else:
125           return out           return out
126    
127    
128  class Projector:  class Projector:
129    """    """
130    The Projector is a factory which projects a discontiuous function onto a    The Projector is a factory which projects a discontinuous function onto a
131    continuous function on the a given domain.    continuous function on a given domain.
132    """    """
133    def __init__(self, domain, reduce = True, fast=True):    def __init__(self, domain, reduce=True, fast=True):
134      """      """
135      Create a continuous function space projector for a domain.      Creates a continuous function space projector for a domain.
136    
137      @param domain: Domain of the projection.      :param domain: Domain of the projection.
138      @param reduce: Flag to reduce projection order (default is True)      :param reduce: Flag to reduce projection order
139      @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
140      """      """
141      self.__pde = linearPDEs.LinearPDE(domain)      self.__pde = linearPDEs.LinearPDE(domain)
142      if fast:      if fast:
143        self.__pde.setSolverMethod(linearPDEs.LinearPDE.LUMPING)          self.__pde.getSolverOptions().setSolverMethod(linearPDEs.SolverOptions.LUMPING)
144      self.__pde.setSymmetryOn()      self.__pde.setSymmetryOn()
145      self.__pde.setReducedOrderTo(reduce)      self.__pde.setReducedOrderTo(reduce)
146      self.__pde.setValue(D = 1.)      self.__pde.setValue(D = 1.)
147      return      return
148      def getSolverOptions(self):
149        """
150        Returns the solver options of the PDE solver.
151        
152        :rtype: `linearPDEs.SolverOptions`
153        """
154        return self.__pde.getSolverOptions()
155    
156    def __call__(self, input_data):    def __call__(self, input_data):
157      """      """
158      Projects input_data onto a continuous function      Projects ``input_data`` onto a continuous function.
159    
160      @param input_data: The input_data to be projected.      :param input_data: the data to be projected
161      """      """
162      out=escript.Data(0.,input_data.getShape(),self.__pde.getFunctionSpaceForSolution())      out=escript.Data(0.,input_data.getShape(),self.__pde.getFunctionSpaceForSolution())
163      self.__pde.setValue(Y = escript.Data(), Y_reduced = escript.Data())      self.__pde.setValue(Y = escript.Data(), Y_reduced = escript.Data())
# Line 186  class Projector: Line 190  class Projector:
190    
191  class NoPDE:  class NoPDE:
192       """       """
193       solves the following problem for u:       Solves the following problem for u:
194    
195       M{kronecker[i,j]*D[j]*u[j]=Y[i]}       *kronecker[i,j]*D[j]*u[j]=Y[i]*
196    
197       with constraint       with constraint
198    
199       M{u[j]=r[j]}  where M{q[j]>0}       *u[j]=r[j]*  where *q[j]>0*
200    
201       where D, Y, r and q are given functions of rank 1.       where *D*, *Y*, *r* and *q* are given functions of rank 1.
202    
203       In the case of scalars this takes the form       In the case of scalars this takes the form
204    
205       M{D*u=Y}       *D*u=Y*
206    
207       with constraint       with constraint
208    
209       M{u=r}  where M{q>0}       *u=r* where *q>0*
210    
211       where D, Y, r and q are given scalar functions.       where *D*, *Y*, *r* and *q* are given scalar functions.
212    
213       The constraint is overwriting any other condition.       The constraint overwrites any other condition.
214    
215       @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
216              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
217              thing is a bit strange and I blame Robert.Woodcock@csiro.au for this.              in `Solution` or `ReducedSolution`.
218       """       """
219         # The whole thing is a bit strange and I blame Rob Woodcock (CSIRO) for
220         # this.
221       def __init__(self,domain,D=None,Y=None,q=None,r=None):       def __init__(self,domain,D=None,Y=None,q=None,r=None):
222           """           """
223           initialize the problem           Initializes the problem.
224    
225           @param domain: domain of the PDE.           :param domain: domain of the PDE
226           @type domain: L{Domain}           :type domain: `Domain`
227           @param D: coefficient of the solution.           :param D: coefficient of the solution
228           @type D: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type D: ``float``, ``int``, ``numpy.ndarray``, `Data`
229           @param Y: right hand side           :param Y: right hand side
230           @type Y: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type Y: ``float``, ``int``, ``numpy.ndarray``, `Data`
231           @param q: location of constraints           :param q: location of constraints
232           @type q: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type q: ``float``, ``int``, ``numpy.ndarray``, `Data`
233           @param r: value of solution at locations of constraints           :param r: value of solution at locations of constraints
234           @type r: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type r: ``float``, ``int``, ``numpy.ndarray``, `Data`
235           """           """
236           self.__domain=domain           self.__domain=domain
237           self.__D=D           self.__D=D
# Line 234  class NoPDE: Line 240  class NoPDE:
240           self.__r=r           self.__r=r
241           self.__u=None           self.__u=None
242           self.__function_space=escript.Solution(self.__domain)           self.__function_space=escript.Solution(self.__domain)
243    
244       def setReducedOn(self):       def setReducedOn(self):
245           """           """
246           sets the L{FunctionSpace} of the solution to L{ReducedSolution}           Sets the `FunctionSpace` of the solution to `ReducedSolution`.
247           """           """
248           self.__function_space=escript.ReducedSolution(self.__domain)           self.__function_space=escript.ReducedSolution(self.__domain)
249           self.__u=None           self.__u=None
250    
251       def setReducedOff(self):       def setReducedOff(self):
252           """           """
253           sets the L{FunctionSpace} of the solution to L{Solution}           Sets the `FunctionSpace` of the solution to `Solution`.
254           """           """
255           self.__function_space=escript.Solution(self.__domain)           self.__function_space=escript.Solution(self.__domain)
256           self.__u=None           self.__u=None
257            
258       def setValue(self,D=None,Y=None,q=None,r=None):       def setValue(self,D=None,Y=None,q=None,r=None):
259           """           """
260           assigns values to the parameters.           Assigns values to the parameters.
261    
262           @param D: coefficient of the solution.           :param D: coefficient of the solution
263           @type D: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type D: ``float``, ``int``, ``numpy.ndarray``, `Data`
264           @param Y: right hand side           :param Y: right hand side
265           @type Y: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type Y: ``float``, ``int``, ``numpy.ndarray``, `Data`
266           @param q: location of constraints           :param q: location of constraints
267           @type q: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type q: ``float``, ``int``, ``numpy.ndarray``, `Data`
268           @param r: value of solution at locations of constraints           :param r: value of solution at locations of constraints
269           @type r: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type r: ``float``, ``int``, ``numpy.ndarray``, `Data`
270           """           """
271           if not D==None:           if not D==None:
272              self.__D=D              self.__D=D
# Line 276  class NoPDE: Line 283  class NoPDE:
283    
284       def getSolution(self):       def getSolution(self):
285           """           """
286           returns the solution           Returns the solution.
287            
288           @return: the solution of the problem           :return: the solution of the problem
289           @rtype: L{Data} object in the L{FunctionSpace} L{Solution} or L{ReducedSolution}.           :rtype: `Data` object in the `FunctionSpace` `Solution` or
290                     `ReducedSolution`
291           """           """
292           if self.__u==None:           if self.__u==None:
293              if self.__D==None:              if self.__D==None:
# Line 296  class NoPDE: Line 304  class NoPDE:
304                  self.__u*=(1.-q)                  self.__u*=(1.-q)
305                  if not self.__r==None: self.__u+=q*self.__r                  if not self.__r==None: self.__u+=q*self.__r
306           return self.__u           return self.__u
307                
308  class Locator:  class Locator:
309       """       """
310       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
311       spatial coordinate x.         coordinate x.
312        
313       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
314       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.  
315       """       """
316    
317       def __init__(self,where,x=numarray.zeros((3,))):       def __init__(self,where,x=numpy.zeros((3,))):
318         """         """
319         Initializes a Locator to access values in Data objects on the Doamin         Initializes a Locator to access values in Data objects on the Doamin
320         or FunctionSpace where for the sample point which         or FunctionSpace for the sample point which is closest to the given
321         closest to the given point x.         point x.
322    
323         @param where: function space         :param where: function space
324         @type where: L{escript.FunctionSpace}         :type where: `escript.FunctionSpace`
325         @param x: coefficient of the solution.         :param x: location(s) of the Locator
326         @type x: L{numarray.NumArray} or C{list} of L{numarray.NumArray}         :type x: ``numpy.ndarray`` or ``list`` of ``numpy.ndarray``
327         """         """
328         if isinstance(where,escript.FunctionSpace):         if isinstance(where,escript.FunctionSpace):
329            self.__function_space=where            self.__function_space=where
330         else:         else:
331            self.__function_space=escript.ContinuousFunction(where)            self.__function_space=escript.ContinuousFunction(where)
332           iterative=False
333         if isinstance(x, list):         if isinstance(x, list):
334               if len(x)==0:
335                  raise "ValueError", "At least one point must be given."
336               try:
337                 iter(x[0])
338                 iterative=True
339               except TypeError:
340                 iterative=False
341           if iterative:
342             self.__id=[]             self.__id=[]
343             for p in x:             for p in x:
344                self.__id.append(util.length(self.__function_space.getX()-p[:self.__function_space.getDim()]).minGlobalDataPoint())                self.__id.append(util.length(self.__function_space.getX()-p[:self.__function_space.getDim()]).minGlobalDataPoint())
# Line 334  class Locator: Line 350  class Locator:
350         Returns the coordinates of the Locator as a string.         Returns the coordinates of the Locator as a string.
351         """         """
352         x=self.getX()         x=self.getX()
353         if instance(x,list):         if isinstance(x,list):
354            out="["            out="["
355            first=True            first=True
356            for xx in x:            for xx in x:
# Line 350  class Locator: Line 366  class Locator:
366    
367       def getX(self):       def getX(self):
368          """          """
369      Returns the exact coordinates of the Locator.          Returns the exact coordinates of the Locator.
370      """          """
371          return self(self.getFunctionSpace().getX())          return self(self.getFunctionSpace().getX())
372    
373       def getFunctionSpace(self):       def getFunctionSpace(self):
374          """          """
375      Returns the function space of the Locator.          Returns the function space of the Locator.
376      """          """
377          return self.__function_space          return self.__function_space
378    
379       def getId(self,item=None):       def getId(self,item=None):
380          """          """
381      Returns the identifier of the location.          Returns the identifier of the location.
382      """          """
383          if item == None:          if item == None:
384             return self.__id             return self.__id
385          else:          else:
# Line 375  class Locator: Line 391  class Locator:
391    
392       def __call__(self,data):       def __call__(self,data):
393          """          """
394      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.
395      the object is returned.          """
     """  
396          return self.getValue(data)          return self.getValue(data)
397    
398       def getValue(self,data):       def getValue(self,data):
399          """          """
400      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`
401      otherwise the object is returned.          object otherwise the object is returned.
402      """          """
403          if isinstance(data,escript.Data):          if isinstance(data,escript.Data):
404             if data.getFunctionSpace()==self.getFunctionSpace():             dat=util.interpolate(data,self.getFunctionSpace())
              dat=data  
            else:  
              dat=data.interpolate(self.getFunctionSpace())  
405             id=self.getId()             id=self.getId()
406             r=data.getRank()             r=data.getRank()
407             if isinstance(id,list):             if isinstance(id,list):
408                 out=[]                 out=[]
409                 for i in id:                 for i in id:
410                    o=data.getValueOfGlobalDataPoint(*i)                    o=numpy.array(dat.getTupleForGlobalDataPoint(*i))
411                    if data.getRank()==0:                    if data.getRank()==0:
412                       out.append(o[0])                       out.append(o[0])
413                    else:                    else:
414                       out.append(o)                       out.append(o)
415                 return out                 return out
416             else:             else:
417               out=data.getValueOfGlobalDataPoint(*id)               out=numpy.array(dat.getTupleForGlobalDataPoint(*id))
418               if data.getRank()==0:               if data.getRank()==0:
419                  return out[0]                  return out[0]
420               else:               else:
# Line 410  class Locator: Line 422  class Locator:
422          else:          else:
423             return data             return data
424    
425    
426    def getInfLocator(arg):
427        """
428        Return a Locator for a point with the inf value over all arg.
429        """
430        if not isinstance(arg, escript.Data):
431        raise TypeError,"getInfLocator: Unknown argument type."
432        a_inf=util.inf(arg)
433        loc=util.length(arg-a_inf).minGlobalDataPoint() # This gives us the location but not coords
434        x=arg.getFunctionSpace().getX()
435        x_min=x.getTupleForGlobalDataPoint(*loc)
436        return Locator(arg.getFunctionSpace(),x_min)
437    
438    def getSupLocator(arg):
439        """
440        Return a Locator for a point with the sup value over all arg.
441        """
442        if not isinstance(arg, escript.Data):
443        raise TypeError,"getInfLocator: Unknown argument type."
444        a_inf=util.sup(arg)
445        loc=util.length(arg-a_inf).minGlobalDataPoint() # This gives us the location but not coords
446        x=arg.getFunctionSpace().getX()
447        x_min=x.getTupleForGlobalDataPoint(*loc)
448        return Locator(arg.getFunctionSpace(),x_min)
449        
450    
451  class SolverSchemeException(Exception):  class SolverSchemeException(Exception):
452     """     """
453     exceptions thrown by solvers     This is a generic exception thrown by solvers.
454     """     """
455     pass     pass
456    
457  class IndefinitePreconditioner(SolverSchemeException):  class IndefinitePreconditioner(SolverSchemeException):
458     """     """
459     the preconditioner is not positive definite.     Exception thrown if the preconditioner is not positive definite.
460     """     """
461     pass     pass
462    
463  class MaxIterReached(SolverSchemeException):  class MaxIterReached(SolverSchemeException):
464     """     """
465     maxium number of iteration steps is reached.     Exception thrown if the maximum number of iteration steps is reached.
466     """     """
467     pass     pass
468  class IterationBreakDown(SolverSchemeException):  
469    class CorrectionFailed(SolverSchemeException):
470     """     """
471     iteration scheme econouters an incurable breakdown.     Exception thrown if no convergence has been achieved in the solution
472       correction scheme.
473     """     """
474     pass     pass
475  class NegativeNorm(SolverSchemeException):  
476    class IterationBreakDown(SolverSchemeException):
477     """     """
478     a norm calculation returns a negative norm.     Exception thrown if the iteration scheme encountered an incurable breakdown.
479     """     """
480     pass     pass
481    
482  class IterationHistory(object):  class NegativeNorm(SolverSchemeException):
483     """     """
484     The IterationHistory class is used to define a stopping criterium. It keeps track of the     Exception thrown if a norm calculation returns a negative norm.
    residual norms. The stoppingcriterium indicates termination if the residual norm has been reduced by  
    a given tolerance.  
485     """     """
486     def __init__(self,tolerance=math.sqrt(util.EPSILON),verbose=False):     pass
       """  
       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]  
487    
488     def stoppingcriterium2(self,norm_r,norm_b,solver="GMRES",TOL=None):  def PCG(r, Aprod, x, Msolve, bilinearform, atol=0, rtol=1.e-8, iter_max=100, initial_guess=True, verbose=False):
489         """     """
490         returns True if the C{norm_r} is C{tolerance}*C{norm_b}     Solver for
491    
492             *Ax=b*
        @param norm_r: current residual norm  
        @type norm_r: non-negative C{float}  
        @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}  
493    
494         """     with a symmetric and positive definite operator A (more details required!).
495         if TOL==None:     It uses the conjugate gradient method with preconditioner M providing an
496            TOL=self.tolerance     approximation of A.
        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  
497    
498  def PCG(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100):     The iteration is terminated if
    """  
    Solver for  
499    
500     M{Ax=b}     *|r| <= atol+rtol*|r0|*
501    
502     with a symmetric and positive definite operator A (more details required!).     where *r0* is the initial residual and *|.|* is the energy norm. In fact
    It uses the conjugate gradient method with preconditioner M providing an approximation of A.  
503    
504     The iteration is terminated if the C{stoppingcriterium} function return C{True}.     *|r| = sqrt( bilinearform(Msolve(r),r))*
505    
506     For details on the preconditioned conjugate gradient method see the book:     For details on the preconditioned conjugate gradient method see the book:
507    
508     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,
509     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,
510     C. Romine, and H. van der Vorst.     C. Romine, and H. van der Vorst}.
511    
512     @param b: the right hand side of the liner system. C{b} is altered.     :param r: initial residual *r=b-Ax*. ``r`` is altered.
513     @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)
514     @param Aprod: returns the value Ax     :param x: an initial guess for the solution
515     @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)
516     @param Msolve: solves Mx=r     :param Aprod: returns the value Ax
517     @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
518  type like argument C{x}.                  argument ``x``. The returned object needs to be of the same type
519     @param bilinearform: inner product C{<x,r>}                  like argument ``r``.
520     @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
521     @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
522     @type stoppingcriterium: function that returns C{True} or C{False}                   argument ``r``. The returned object needs to be of the same
523     @param x: an initial guess for the solution. If no C{x} is given 0*b is used.                   type like argument ``x``.
524     @type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)     :param bilinearform: inner product ``<x,r>``
525     @param iter_max: maximum number of iteration steps.     :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
526     @type iter_max: C{int}                         type like argument ``x`` and ``r`` is. The returned value
527     @return: the solution approximation and the corresponding residual                         is a ``float``.
528     @rtype: C{tuple}     :param atol: absolute tolerance
529     @warning: C{b} and C{x} are altered.     :type atol: non-negative ``float``
530       :param rtol: relative tolerance
531       :type rtol: non-negative ``float``
532       :param iter_max: maximum number of iteration steps
533       :type iter_max: ``int``
534       :return: the solution approximation and the corresponding residual
535       :rtype: ``tuple``
536       :warning: ``r`` and ``x`` are altered.
537     """     """
538     iter=0     iter=0
    if x==None:  
       x=0*b  
    else:  
       b += (-1)*Aprod(x)  
    r=b  
539     rhat=Msolve(r)     rhat=Msolve(r)
540     d = rhat     d = rhat
541     rhat_dot_r = bilinearform(rhat, r)     rhat_dot_r = bilinearform(rhat, r)
542     if rhat_dot_r<0: raise NegativeNorm,"negative norm."     if rhat_dot_r<0: raise NegativeNorm,"negative norm."
543       norm_r0=math.sqrt(rhat_dot_r)
544       atol2=atol+rtol*norm_r0
545       if atol2<=0:
546          raise ValueError,"Non-positive tolarance."
547       atol2=max(atol2, 100. * util.EPSILON * norm_r0)
548    
549       if verbose: print "PCG: initial residual norm = %e (absolute tolerance = %e)"%(norm_r0, atol2)
550    
551    
552     while not stoppingcriterium(math.sqrt(rhat_dot_r),r,x):     while not math.sqrt(rhat_dot_r) <= atol2:
553         iter+=1         iter+=1
554         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
555    
556         q=Aprod(d)         q=Aprod(d)
557         alpha = rhat_dot_r / bilinearform(d, q)         alpha = rhat_dot_r / bilinearform(d, q)
558         x += alpha * d         x += alpha * d
559         r += (-alpha) * q         if isinstance(q,ArithmeticTuple):
560           r += q * (-alpha)      # Doing it the other way calls the float64.__mul__ not AT.__rmul__
561           else:
562               r += (-alpha) * q
563         rhat=Msolve(r)         rhat=Msolve(r)
564         rhat_dot_r_new = bilinearform(rhat, r)         rhat_dot_r_new = bilinearform(rhat, r)
565         beta = rhat_dot_r_new / rhat_dot_r         beta = rhat_dot_r_new / rhat_dot_r
# Line 557  type like argument C{x}. Line 568  type like argument C{x}.
568    
569         rhat_dot_r = rhat_dot_r_new         rhat_dot_r = rhat_dot_r_new
570         if rhat_dot_r<0: raise NegativeNorm,"negative norm."         if rhat_dot_r<0: raise NegativeNorm,"negative norm."
571           if verbose: print "PCG: iteration step %s: residual norm = %e"%(iter, math.sqrt(rhat_dot_r))
572     return x,r     if verbose: print "PCG: tolerance reached after %s steps."%iter
573       return x,r,math.sqrt(rhat_dot_r)
574    
575  class Defect(object):  class Defect(object):
576      """      """
577      defines a non-linear defect F(x) of a variable x      Defines a non-linear defect F(x) of a variable x.
578      """      """
579      def __init__(self):      def __init__(self):
580          """          """
581          initialize defect          Initializes defect.
582          """          """
583          self.setDerivativeIncrementLength()          self.setDerivativeIncrementLength()
584    
585      def bilinearform(self, x0, x1):      def bilinearform(self, x0, x1):
586          """          """
587          returns the inner product of x0 and x1          Returns the inner product of x0 and x1
588          @param x0: a value for x  
589          @param x1: a value for x          :param x0: value for x0
590          @return: the inner product of x0 and x1          :param x1: value for x1
591          @rtype: C{float}          :return: the inner product of x0 and x1
592            :rtype: ``float``
593          """          """
594          return 0          return 0
595          
596      def norm(self,x):      def norm(self,x):
597          """          """
598          the norm of argument C{x}          Returns the norm of argument ``x``.
599    
600          @param x: a value for x          :param x: a value
601          @return: norm of argument x          :return: norm of argument x
602          @rtype: C{float}          :rtype: ``float``
603          @note: by default C{sqrt(self.bilinearform(x,x)} is retrurned.          :note: by default ``sqrt(self.bilinearform(x,x)`` is returned.
604          """          """
605          s=self.bilinearform(x,x)          s=self.bilinearform(x,x)
606          if s<0: raise NegativeNorm,"negative norm."          if s<0: raise NegativeNorm,"negative norm."
607          return math.sqrt(s)          return math.sqrt(s)
608    
   
609      def eval(self,x):      def eval(self,x):
610          """          """
611          returns the value F of a given x          Returns the value F of a given ``x``.
612    
613          @param x: value for which the defect C{F} is evalulated.          :param x: value for which the defect ``F`` is evaluated
614          @return: value of the defect at C{x}          :return: value of the defect at ``x``
615          """          """
616          return 0          return 0
617    
618      def __call__(self,x):      def __call__(self,x):
619          return self.eval(x)          return self.eval(x)
620    
621      def setDerivativeIncrementLength(self,inc=math.sqrt(util.EPSILON)):      def setDerivativeIncrementLength(self,inc=1000.*math.sqrt(util.EPSILON)):
622          """          """
623          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
624          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
625            direction of v with x as a starting point.
626    
627          @param inc: relative increment length          :param inc: relative increment length
628          @type inc: positive C{float}          :type inc: positive ``float``
629          """          """
630          if inc<=0: raise ValueError,"positive increment required."          if inc<=0: raise ValueError,"positive increment required."
631          self.__inc=inc          self.__inc=inc
632    
633      def getDerivativeIncrementLength(self):      def getDerivativeIncrementLength(self):
634          """          """
635          returns the relative increment length used to approximate the derivative of the defect          Returns the relative increment length used to approximate the
636          @return: value of the defect at C{x}          derivative of the defect.
637          @rtype: positive C{float}          :return: value of the defect at ``x``
638            :rtype: positive ``float``
639          """          """
640          return self.__inc          return self.__inc
641    
642      def derivative(self, F0, x0, v, v_is_normalised=True):      def derivative(self, F0, x0, v, v_is_normalised=True):
643          """          """
644          returns the directional derivative at x0 in the direction of v          Returns the directional derivative at ``x0`` in the direction of ``v``.
645    
646          @param F0: value of this defect at x0          :param F0: value of this defect at x0
647          @param x0: value at which derivative is calculated.          :param x0: value at which derivative is calculated
648          @param v: direction          :param v: direction
649          @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
650          @return: derivative of this defect at x0 in the direction of C{v}                                  (self.norm(v)=0)
651          @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``
652          maybe oepsnew verwritten to use exact evalution.          :note: by default numerical evaluation (self.eval(x0+eps*v)-F0)/eps is
653                   used but this method maybe overwritten to use exact evaluation.
654          """          """
655          normx=self.norm(x0)          normx=self.norm(x0)
656          if normx>0:          if normx>0:
# Line 651  class Defect(object): Line 666  class Defect(object):
666          F1=self.eval(x0 + epsnew * v)          F1=self.eval(x0 + epsnew * v)
667          return (F1-F0)/epsnew          return (F1-F0)/epsnew
668    
669  ######################################      ######################################
670  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):
671     """     """
672     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
673       criterion:
674    
675     M{norm(F(x) <= atol + rtol * norm(F(x0)}     *norm(F(x) <= atol + rtol * norm(F(x0)*
676      
677     where M{x0} is the initial guess.     where *x0* is the initial guess.
678    
679     @param defect: object defining the the function M{F}, C{defect.norm} defines the M{norm} used in the stopping criterion.     :param defect: object defining the function *F*. ``defect.norm`` defines the
680     @type defect: L{Defect}                    *norm* used in the stopping criterion.
681     @param x: initial guess for the solution, C{x} is altered.     :type defect: `Defect`
682     @type x: any object type allowing basic operations such as  L{numarray.NumArray}, L{Data}     :param x: initial guess for the solution, ``x`` is altered.
683     @param iter_max: maximum number of iteration steps     :type x: any object type allowing basic operations such as
684     @type iter_max: positive C{int}              ``numpy.ndarray``, `Data`
685     @param sub_iter_max:     :param iter_max: maximum number of iteration steps
686     @type sub_iter_max:     :type iter_max: positive ``int``
687     @param atol: absolute tolerance for the solution     :param sub_iter_max: maximum number of inner iteration steps
688     @type atol: positive C{float}     :type sub_iter_max: positive ``int``
689     @param rtol: relative tolerance for the solution     :param atol: absolute tolerance for the solution
690     @type rtol: positive C{float}     :type atol: positive ``float``
691     @param gamma: tolerance safety factor for inner iteration     :param rtol: relative tolerance for the solution
692     @type gamma: positive C{float}, less than 1     :type rtol: positive ``float``
693     @param sub_tol_max: upper bound for inner tolerance.     :param gamma: tolerance safety factor for inner iteration
694     @type sub_tol_max: positive C{float}, less than 1     :type gamma: positive ``float``, less than 1
695     @return: an approximation of the solution with the desired accuracy     :param subtol_max: upper bound for inner tolerance
696     @rtype: same type as the initial guess.     :type subtol_max: positive ``float``, less than 1
697       :return: an approximation of the solution with the desired accuracy
698       :rtype: same type as the initial guess
699     """     """
700     lmaxit=iter_max     lmaxit=iter_max
701     if atol<0: raise ValueError,"atol needs to be non-negative."     if atol<0: raise ValueError,"atol needs to be non-negative."
702     if rtol<0: raise ValueError,"rtol needs to be non-negative."     if rtol<0: raise ValueError,"rtol needs to be non-negative."
703     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."
704     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
705     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
706    
707     F=defect(x)     F=defect(x)
708     fnrm=defect.norm(F)     fnrm=defect.norm(F)
709     stop_tol=atol + rtol*fnrm     stop_tol=atol + rtol*fnrm
710     sub_tol=sub_tol_max     subtol=subtol_max
711     if verbose: print "NewtonGMRES: initial residual = %e."%fnrm     if verbose: print "NewtonGMRES: initial residual = %e."%fnrm
712     if verbose: print "             tolerance = %e."%sub_tol     if verbose: print "             tolerance = %e."%subtol
713     iter=1     iter=1
714     #     #
715     # main iteration loop     # main iteration loop
716     #     #
717     while not fnrm<=stop_tol:     while not fnrm<=stop_tol:
718              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
719              #              #
720          #   adjust sub_tol_          #   adjust subtol_
721          #          #
722              if iter > 1:              if iter > 1:
723             rat=fnrm/fnrmo             rat=fnrm/fnrmo
724                 sub_tol_old=sub_tol                 subtol_old=subtol
725             sub_tol=gamma*rat**2             subtol=gamma*rat**2
726             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)
727             sub_tol=max(min(sub_tol,sub_tol_max), .5*stop_tol/fnrm)             subtol=max(min(subtol,subtol_max), .5*stop_tol/fnrm)
728          #          #
729          # calculate newton increment xc          # calculate newton increment xc
730              #     if iter_max in __FDGMRES is reached MaxIterReached is thrown              #     if iter_max in __FDGMRES is reached MaxIterReached is thrown
731              #     if iter_restart -1 is returned as sub_iter              #     if iter_restart -1 is returned as sub_iter
732              #     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
733              #              #
734              #                #
735              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
736              try:              try:
737                 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)
738              except MaxIterReached:              except MaxIterReached:
739                 raise MaxIterReached,"maximum number of %s steps reached."%iter_max                 raise MaxIterReached,"maximum number of %s steps reached."%iter_max
740              if sub_iter<0:              if sub_iter<0:
# Line 734  def NewtonGMRES(defect, x, iter_max=100, Line 752  def NewtonGMRES(defect, x, iter_max=100,
752    
753  def __givapp(c,s,vin):  def __givapp(c,s,vin):
754      """      """
755      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
756      @warning: C{vin} is altered.      ``vin``
757    
758        :warning: ``vin`` is altered.
759      """      """
760      vrot=vin      vrot=vin
761      if isinstance(c,float):      if isinstance(c,float):
762          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]]
763      else:      else:
764          for i in range(len(c)):          for i in range(len(c)):
765              w1=c[i]*vrot[i]-s[i]*vrot[i+1]              w1=c[i]*vrot[i]-s[i]*vrot[i+1]
766          w2=s[i]*vrot[i]+c[i]*vrot[i+1]          w2=s[i]*vrot[i]+c[i]*vrot[i+1]
767              vrot[i:i+2]=w1,w2              vrot[i]=w1
768                vrot[i+1]=w2
769      return vrot      return vrot
770    
771  def __FDGMRES(F0, defect, x0, atol, iter_max=100, iter_restart=20):  def __FDGMRES(F0, defect, x0, atol, iter_max=100, iter_restart=20):
772     h=numarray.zeros((iter_restart,iter_restart),numarray.Float64)     h=numpy.zeros((iter_restart,iter_restart),numpy.float64)
773     c=numarray.zeros(iter_restart,numarray.Float64)     c=numpy.zeros(iter_restart,numpy.float64)
774     s=numarray.zeros(iter_restart,numarray.Float64)     s=numpy.zeros(iter_restart,numpy.float64)
775     g=numarray.zeros(iter_restart,numarray.Float64)     g=numpy.zeros(iter_restart,numpy.float64)
776     v=[]     v=[]
777    
778     rho=defect.norm(F0)     rho=defect.norm(F0)
779     if rho<=0.: return x0*0     if rho<=0.: return x0*0
780      
781     v.append(-F0/rho)     v.append(-F0/rho)
782     g[0]=rho     g[0]=rho
783     iter=0     iter=0
784     while rho > atol and iter<iter_restart-1:     while rho > atol and iter<iter_restart-1:
785            if iter  >= iter_max:
786      if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max              raise MaxIterReached,"maximum number of %s steps reached."%iter_max
787    
788          p=defect.derivative(F0,x0,v[iter], v_is_normalised=True)          p=defect.derivative(F0,x0,v[iter], v_is_normalised=True)
789      v.append(p)          v.append(p)
790    
791            v_norm1=defect.norm(v[iter+1])
792    
793      v_norm1=defect.norm(v[iter+1])          # Modified Gram-Schmidt
794            for j in range(iter+1):
795                h[j,iter]=defect.bilinearform(v[j],v[iter+1])
796                v[iter+1]-=h[j,iter]*v[j]
797    
798          # Modified Gram-Schmidt          h[iter+1,iter]=defect.norm(v[iter+1])
799      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]  
800    
801          # Reorthogonalize if needed          # Reorthogonalize if needed
802      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)
803          for j in range(iter+1):                for j in range(iter+1):
804             hr=defect.bilinearform(v[j],v[iter+1])                  hr=defect.bilinearform(v[j],v[iter+1])
805                 h[j][iter]=h[j][iter]+hr                  h[j,iter]=h[j,iter]+hr
806                 v[iter+1] -= hr*v[j]                  v[iter+1] -= hr*v[j]
807    
808          v_norm2=defect.norm(v[iter+1])              v_norm2=defect.norm(v[iter+1])
809          h[iter+1][iter]=v_norm2              h[iter+1,iter]=v_norm2
810          #   watch out for happy breakdown          #   watch out for happy breakdown
811          if not v_norm2 == 0:          if not v_norm2 == 0:
812                  v[iter+1]=v[iter+1]/h[iter+1][iter]              v[iter+1]=v[iter+1]/h[iter+1,iter]
813    
814          #   Form and store the information for the new Givens rotation          #   Form and store the information for the new Givens rotation
815      if iter > 0 :          if iter > 0 :
816          hhat=numarray.zeros(iter+1,numarray.Float64)              hhat=numpy.zeros(iter+1,numpy.float64)
817          for i in range(iter+1) : hhat[i]=h[i][iter]              for i in range(iter+1) : hhat[i]=h[i,iter]
818          hhat=__givapp(c[0:iter],s[0:iter],hhat);              hhat=__givapp(c[0:iter],s[0:iter],hhat);
819              for i in range(iter+1) : h[i][iter]=hhat[i]              for i in range(iter+1) : h[i,iter]=hhat[i]
820    
821      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])
822    
823      if mu!=0 :          if mu!=0 :
824          c[iter]=h[iter][iter]/mu              c[iter]=h[iter,iter]/mu
825          s[iter]=-h[iter+1][iter]/mu              s[iter]=-h[iter+1,iter]/mu
826          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]
827          h[iter+1][iter]=0.0              h[iter+1,iter]=0.0
828          g[iter:iter+2]=__givapp(c[iter],s[iter],g[iter:iter+2])              gg=__givapp(c[iter],s[iter],[g[iter],g[iter+1]])
829                g[iter]=gg[0]
830                g[iter+1]=gg[1]
831    
832          # Update the residual norm          # Update the residual norm
833          rho=abs(g[iter+1])          rho=abs(g[iter+1])
834      iter+=1          iter+=1
835    
836     # At this point either iter > iter_max or rho < tol.     # At this point either iter > iter_max or rho < tol.
837     # It's time to compute x and leave.             # It's time to compute x and leave.
838     if iter > 0 :     if iter > 0 :
839       y=numarray.zeros(iter,numarray.Float64)           y=numpy.zeros(iter,numpy.float64)
840       y[iter-1] = g[iter-1] / h[iter-1][iter-1]       y[iter-1] = g[iter-1] / h[iter-1,iter-1]
841       if iter > 1 :         if iter > 1 :
842          i=iter-2            i=iter-2
843          while i>=0 :          while i>=0 :
844            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]
845            i=i-1            i=i-1
846       xhat=v[iter-1]*y[iter-1]       xhat=v[iter-1]*y[iter-1]
847       for i in range(iter-1):       for i in range(iter-1):
848      xhat += v[i]*y[i]      xhat += v[i]*y[i]
849     else :     else :
850        xhat=v[0] * 0        xhat=v[0] * 0
851    
852     if iter<iter_restart-1:     if iter<iter_restart-1:
853        stopped=iter        stopped=iter
854     else:     else:
855        stopped=-1        stopped=-1
856    
857     return xhat,stopped     return xhat,stopped
858    
859  ##############################################  def GMRES(r, Aprod, x, bilinearform, atol=0, rtol=1.e-8, iter_max=100, iter_restart=20, verbose=False,P_R=None):
860  def GMRES(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100, iter_restart=20):     """
861  ################################################     Solver for
862    
863       *Ax=b*
864    
865       with a general operator A (more details required!).
866       It uses the generalized minimum residual method (GMRES).
867    
868       The iteration is terminated if
869    
870       *|r| <= atol+rtol*|r0|*
871    
872       where *r0* is the initial residual and *|.|* is the energy norm. In fact
873    
874       *|r| = sqrt( bilinearform(r,r))*
875    
876       :param r: initial residual *r=b-Ax*. ``r`` is altered.
877       :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
878       :param x: an initial guess for the solution
879       :type x: same like ``r``
880       :param Aprod: returns the value Ax
881       :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
882                    argument ``x``. The returned object needs to be of the same
883                    type like argument ``r``.
884       :param bilinearform: inner product ``<x,r>``
885       :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
886                           type like argument ``x`` and ``r``. The returned value is
887                           a ``float``.
888       :param atol: absolute tolerance
889       :type atol: non-negative ``float``
890       :param rtol: relative tolerance
891       :type rtol: non-negative ``float``
892       :param iter_max: maximum number of iteration steps
893       :type iter_max: ``int``
894       :param iter_restart: in order to save memory the orthogonalization process
895                            is terminated after ``iter_restart`` steps and the
896                            iteration is restarted.
897       :type iter_restart: ``int``
898       :return: the solution approximation and the corresponding residual
899       :rtype: ``tuple``
900       :warning: ``r`` and ``x`` are altered.
901       """
902     m=iter_restart     m=iter_restart
903       restarted=False
904     iter=0     iter=0
905     xc=x     if rtol>0:
906          r_dot_r = bilinearform(r, r)
907          if r_dot_r<0: raise NegativeNorm,"negative norm."
908          atol2=atol+rtol*math.sqrt(r_dot_r)
909          if verbose: print "GMRES: norm of right hand side = %e (absolute tolerance = %e)"%(math.sqrt(r_dot_r), atol2)
910       else:
911          atol2=atol
912          if verbose: print "GMRES: absolute tolerance = %e"%atol2
913       if atol2<=0:
914          raise ValueError,"Non-positive tolarance."
915    
916     while True:     while True:
917        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
918        xc,stopped=__GMRESm(b*1, Aprod, Msolve, bilinearform, stoppingcriterium, x=xc*1, iter_max=iter_max-iter, iter_restart=m)        if restarted:
919             r2 = r-Aprod(x-x2)
920          else:
921             r2=1*r
922          x2=x*1.
923          x,stopped=_GMRESm(r2, Aprod, x, bilinearform, atol2, iter_max=iter_max-iter, iter_restart=m, verbose=verbose,P_R=P_R)
924          iter+=iter_restart
925        if stopped: break        if stopped: break
926        iter+=iter_restart            if verbose: print "GMRES: restart."
927     return xc        restarted=True
928       if verbose: print "GMRES: tolerance has been reached."
929       return x
930    
931  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):
932     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)  
933    
934     if x==None:     h=numpy.zeros((iter_restart+1,iter_restart),numpy.float64)
935        x=0*b     c=numpy.zeros(iter_restart,numpy.float64)
936     else:     s=numpy.zeros(iter_restart,numpy.float64)
937        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)  
938     v=[]     v=[]
939    
940       r_dot_r = bilinearform(r, r)
941       if r_dot_r<0: raise NegativeNorm,"negative norm."
942     rho=math.sqrt(r_dot_r)     rho=math.sqrt(r_dot_r)
943      
944     v.append(r/rho)     v.append(r/rho)
945     g[0]=rho     g[0]=rho
946    
947     while not (stoppingcriterium(rho,norm_b) or iter==iter_restart-1):     if verbose: print "GMRES: initial residual %e (absolute tolerance = %e)"%(rho,atol)
948       while not (rho<=atol or iter==iter_restart):
949    
950      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
951    
952      p=Msolve(Aprod(v[iter]))          if P_R!=None:
953                p=Aprod(P_R(v[iter]))
954            else:
955            p=Aprod(v[iter])
956      v.append(p)      v.append(p)
957    
958      v_norm1=math.sqrt(bilinearform(v[iter+1], v[iter+1]))        v_norm1=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
959    
960  # Modified Gram-Schmidt  # Modified Gram-Schmidt
961      for j in range(iter+1):      for j in range(iter+1):
962        h[j][iter]=bilinearform(v[j],v[iter+1])          h[j,iter]=bilinearform(v[j],v[iter+1])
963        v[iter+1]-=h[j][iter]*v[j]        v[iter+1]-=h[j,iter]*v[j]
964          
965      h[iter+1][iter]=math.sqrt(bilinearform(v[iter+1],v[iter+1]))      h[iter+1,iter]=math.sqrt(bilinearform(v[iter+1],v[iter+1]))
966      v_norm2=h[iter+1][iter]      v_norm2=h[iter+1,iter]
967    
968  # Reorthogonalize if needed  # Reorthogonalize if needed
969      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)
970       for j in range(iter+1):         for j in range(iter+1):
971          hr=bilinearform(v[j],v[iter+1])          hr=bilinearform(v[j],v[iter+1])
972              h[j][iter]=h[j][iter]+hr              h[j,iter]=h[j,iter]+hr
973              v[iter+1] -= hr*v[j]              v[iter+1] -= hr*v[j]
974    
975       v_norm2=math.sqrt(bilinearform(v[iter+1], v[iter+1]))         v_norm2=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
976       h[iter+1][iter]=v_norm2       h[iter+1,iter]=v_norm2
977    
978  #   watch out for happy breakdown  #   watch out for happy breakdown
979          if not v_norm2 == 0:          if not v_norm2 == 0:
980           v[iter+1]=v[iter+1]/h[iter+1][iter]           v[iter+1]=v[iter+1]/h[iter+1,iter]
981    
982  #   Form and store the information for the new Givens rotation  #   Form and store the information for the new Givens rotation
983      if iter > 0 :      if iter > 0: h[:iter+1,iter]=__givapp(c[:iter],s[:iter],h[:iter+1,iter])
984          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])  
985    
986      if mu!=0 :      if mu!=0 :
987          c[iter]=h[iter][iter]/mu          c[iter]=h[iter,iter]/mu
988          s[iter]=-h[iter+1][iter]/mu          s[iter]=-h[iter+1,iter]/mu
989          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]
990          h[iter+1][iter]=0.0          h[iter+1,iter]=0.0
991          g[iter:iter+2]=__givapp(c[iter],s[iter],g[iter:iter+2])                  gg=__givapp(c[iter],s[iter],[g[iter],g[iter+1]])
992                    g[iter]=gg[0]
993                    g[iter+1]=gg[1]
994  # Update the residual norm  # Update the residual norm
995                  
996          rho=abs(g[iter+1])          rho=abs(g[iter+1])
997            if verbose: print "GMRES: iteration step %s: residual %e"%(iter,rho)
998      iter+=1      iter+=1
999    
1000  # At this point either iter > iter_max or rho < tol.  # At this point either iter > iter_max or rho < tol.
1001  # It's time to compute x and leave.          # It's time to compute x and leave.
1002    
1003     if iter > 0 :     if verbose: print "GMRES: iteration stopped after %s step."%iter
1004       y=numarray.zeros(iter,numarray.Float64)         if iter > 0 :
1005       y[iter-1] = g[iter-1] / h[iter-1][iter-1]       y=numpy.zeros(iter,numpy.float64)
1006       if iter > 1 :         y[iter-1] = g[iter-1] / h[iter-1,iter-1]
1007          i=iter-2         if iter > 1 :
1008            i=iter-2
1009          while i>=0 :          while i>=0 :
1010            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]
1011            i=i-1            i=i-1
1012       xhat=v[iter-1]*y[iter-1]       xhat=v[iter-1]*y[iter-1]
1013       for i in range(iter-1):       for i in range(iter-1):
1014      xhat += v[i]*y[i]      xhat += v[i]*y[i]
1015     else : xhat=v[0]     else:
1016         xhat=v[0] * 0
1017     x += xhat     if P_R!=None:
1018     if iter<iter_restart-1:        x += P_R(xhat)
1019        stopped=True     else:
1020     else:        x += xhat
1021       if iter<iter_restart-1:
1022          stopped=True
1023       else:
1024        stopped=False        stopped=False
1025    
1026     return x,stopped     return x,stopped
1027    
1028  #################################################  def MINRES(r, Aprod, x, Msolve, bilinearform, atol=0, rtol=1.e-8, iter_max=100):
1029  def MINRES(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100):      """
1030  #################################################      Solver for
1031      #  
1032      #  minres solves the system of linear equations Ax = b      *Ax=b*
1033      #  where A is a symmetric matrix (possibly indefinite or singular)  
1034      #  and b is a given vector.      with a symmetric and positive definite operator A (more details required!).
1035      #        It uses the minimum residual method (MINRES) with preconditioner M
1036      #  "A" may be a dense or sparse matrix (preferably sparse!)      providing an approximation of A.
1037      #  or the name of a function such that  
1038      #               y = A(x)      The iteration is terminated if
1039      #  returns the product y = Ax for any given vector x.  
1040      #      *|r| <= atol+rtol*|r0|*
1041      #  "M" defines a positive-definite preconditioner M = C C'.  
1042      #  "M" may be a dense or sparse matrix (preferably sparse!)      where *r0* is the initial residual and *|.|* is the energy norm. In fact
1043      #  or the name of a function such that  
1044      #  solves the system My = x for any given vector x.      *|r| = sqrt( bilinearform(Msolve(r),r))*
1045      #  
1046      #      For details on the preconditioned conjugate gradient method see the book:
1047        
1048        I{Templates for the Solution of Linear Systems by R. Barrett, M. Berry,
1049        T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo,
1050        C. Romine, and H. van der Vorst}.
1051    
1052        :param r: initial residual *r=b-Ax*. ``r`` is altered.
1053        :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1054        :param x: an initial guess for the solution
1055        :type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1056        :param Aprod: returns the value Ax
1057        :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
1058                     argument ``x``. The returned object needs to be of the same
1059                     type like argument ``r``.
1060        :param Msolve: solves Mx=r
1061        :type Msolve: function ``Msolve(r)`` where ``r`` is of the same type like
1062                      argument ``r``. The returned object needs to be of the same
1063                      type like argument ``x``.
1064        :param bilinearform: inner product ``<x,r>``
1065        :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
1066                            type like argument ``x`` and ``r`` is. The returned value
1067                            is a ``float``.
1068        :param atol: absolute tolerance
1069        :type atol: non-negative ``float``
1070        :param rtol: relative tolerance
1071        :type rtol: non-negative ``float``
1072        :param iter_max: maximum number of iteration steps
1073        :type iter_max: ``int``
1074        :return: the solution approximation and the corresponding residual
1075        :rtype: ``tuple``
1076        :warning: ``r`` and ``x`` are altered.
1077        """
1078      #------------------------------------------------------------------      #------------------------------------------------------------------
1079      # Set up y and v for the first Lanczos vector v1.      # Set up y and v for the first Lanczos vector v1.
1080      # y  =  beta1 P' v1,  where  P = C**(-1).      # y  =  beta1 P' v1,  where  P = C**(-1).
1081      # v is really P' v1.      # v is really P' v1.
1082      #------------------------------------------------------------------      #------------------------------------------------------------------
1083      if x==None:      r1    = r
1084        x=0*b      y = Msolve(r)
1085      else:      beta1 = bilinearform(y,r)
       b += (-1)*Aprod(x)  
1086    
     r1    = b  
     y = Msolve(b)  
     beta1 = bilinearform(y,b)  
   
1087      if beta1< 0: raise NegativeNorm,"negative norm."      if beta1< 0: raise NegativeNorm,"negative norm."
1088    
1089      #  If b = 0 exactly, stop with x = 0.      #  If r = 0 exactly, stop with x
1090      if beta1==0: return x*0.      if beta1==0: return x
1091    
1092      if beta1> 0:      if beta1> 0: beta1  = math.sqrt(beta1)
       beta1  = math.sqrt(beta1)        
1093    
1094      #------------------------------------------------------------------      #------------------------------------------------------------------
1095      # Initialize quantities.      # Initialize quantities.
# Line 1008  def MINRES(b, Aprod, Msolve, bilinearfor Line 1109  def MINRES(b, Aprod, Msolve, bilinearfor
1109      ynorm2 = 0      ynorm2 = 0
1110      cs     = -1      cs     = -1
1111      sn     = 0      sn     = 0
1112      w      = b*0.      w      = r*0.
1113      w2     = b*0.      w2     = r*0.
1114      r2     = r1      r2     = r1
1115      eps    = 0.0001      eps    = 0.0001
1116    
1117      #---------------------------------------------------------------------      #---------------------------------------------------------------------
1118      # Main iteration loop.      # Main iteration loop.
1119      # --------------------------------------------------------------------      # --------------------------------------------------------------------
1120      while not stoppingcriterium(rnorm,Anorm*ynorm,'MINRES'):    #  checks ||r|| < (||A|| ||x||) * TOL      while not rnorm<=atol+rtol*Anorm*ynorm:    #  checks ||r|| < (||A|| ||x||) * TOL
1121    
1122      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
1123          iter    = iter  +  1          iter    = iter  +  1
# Line 1035  def MINRES(b, Aprod, Msolve, bilinearfor Line 1136  def MINRES(b, Aprod, Msolve, bilinearfor
1136          #-----------------------------------------------------------------          #-----------------------------------------------------------------
1137          s = 1/beta                 # Normalize previous vector (in y).          s = 1/beta                 # Normalize previous vector (in y).
1138          v = s*y                    # v = vk if P = I          v = s*y                    # v = vk if P = I
1139        
1140          y      = Aprod(v)          y      = Aprod(v)
1141        
1142          if iter >= 2:          if iter >= 2:
1143            y = y - (beta/oldb)*r1            y = y - (beta/oldb)*r1
1144    
1145          alfa   = bilinearform(v,y)              # alphak          alfa   = bilinearform(v,y)              # alphak
1146          y      += (- alfa/beta)*r2          y      += (- alfa/beta)*r2
1147          r1     = r2          r1     = r2
1148          r2     = y          r2     = y
1149          y = Msolve(r2)          y = Msolve(r2)
# Line 1052  def MINRES(b, Aprod, Msolve, bilinearfor Line 1153  def MINRES(b, Aprod, Msolve, bilinearfor
1153    
1154          beta   = math.sqrt( beta )          beta   = math.sqrt( beta )
1155          tnorm2 = tnorm2 + alfa*alfa + oldb*oldb + beta*beta          tnorm2 = tnorm2 + alfa*alfa + oldb*oldb + beta*beta
1156            
1157          if iter==1:                 # Initialize a few things.          if iter==1:                 # Initialize a few things.
1158            gmax   = abs( alfa )      # alpha1            gmax   = abs( alfa )      # alpha1
1159            gmin   = gmax             # alpha1            gmin   = gmax             # alpha1
# Line 1060  def MINRES(b, Aprod, Msolve, bilinearfor Line 1161  def MINRES(b, Aprod, Msolve, bilinearfor
1161          # Apply previous rotation Qk-1 to get          # Apply previous rotation Qk-1 to get
1162          #   [deltak epslnk+1] = [cs  sn][dbark    0   ]          #   [deltak epslnk+1] = [cs  sn][dbark    0   ]
1163          #   [gbar k dbar k+1]   [sn -cs][alfak betak+1].          #   [gbar k dbar k+1]   [sn -cs][alfak betak+1].
1164        
1165          oldeps = epsln          oldeps = epsln
1166          delta  = cs * dbar  +  sn * alfa  # delta1 = 0         deltak          delta  = cs * dbar  +  sn * alfa  # delta1 = 0         deltak
1167          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 1171  def MINRES(b, Aprod, Msolve, bilinearfor
1171          # Compute the next plane rotation Qk          # Compute the next plane rotation Qk
1172    
1173          gamma  = math.sqrt(gbar*gbar+beta*beta)  # gammak          gamma  = math.sqrt(gbar*gbar+beta*beta)  # gammak
1174          gamma  = max(gamma,eps)          gamma  = max(gamma,eps)
1175          cs     = gbar / gamma             # ck          cs     = gbar / gamma             # ck
1176          sn     = beta / gamma             # sk          sn     = beta / gamma             # sk
1177          phi    = cs * phibar              # phik          phi    = cs * phibar              # phik
# Line 1078  def MINRES(b, Aprod, Msolve, bilinearfor Line 1179  def MINRES(b, Aprod, Msolve, bilinearfor
1179    
1180          # Update  x.          # Update  x.
1181    
1182          denom = 1/gamma          denom = 1/gamma
1183          w1    = w2          w1    = w2
1184          w2    = w          w2    = w
1185          w     = (v - oldeps*w1 - delta*w2) * denom          w     = (v - oldeps*w1 - delta*w2) * denom
1186          x     +=  phi*w          x     +=  phi*w
1187    
# Line 1095  def MINRES(b, Aprod, Msolve, bilinearfor Line 1196  def MINRES(b, Aprod, Msolve, bilinearfor
1196    
1197          # Estimate various norms and test for convergence.          # Estimate various norms and test for convergence.
1198    
1199          Anorm  = math.sqrt( tnorm2 )          Anorm  = math.sqrt( tnorm2 )
1200          ynorm  = math.sqrt( ynorm2 )          ynorm  = math.sqrt( ynorm2 )
1201    
1202          rnorm  = phibar          rnorm  = phibar
1203    
1204      return x      return x
1205    
1206  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):
1207      """
1208      Solver for
1209    
1210  # TFQMR solver for linear systems    *Ax=b*
1211  #  
1212  #    with a general operator A (more details required!).
1213  # initialization    It uses the Transpose-Free Quasi-Minimal Residual method (TFQMR).
 #  
   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  
1214    
1215    r=Msolve(r)    The iteration is terminated if
1216    
1217      *|r| <= atol+rtol*|r0|*
1218    
1219      where *r0* is the initial residual and *|.|* is the energy norm. In fact
1220    
1221      *|r| = sqrt( bilinearform(r,r))*
1222    
1223      :param r: initial residual *r=b-Ax*. ``r`` is altered.
1224      :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1225      :param x: an initial guess for the solution
1226      :type x: same like ``r``
1227      :param Aprod: returns the value Ax
1228      :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
1229                   argument ``x``. The returned object needs to be of the same type
1230                   like argument ``r``.
1231      :param bilinearform: inner product ``<x,r>``
1232      :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
1233                          type like argument ``x`` and ``r``. The returned value is
1234                          a ``float``.
1235      :param atol: absolute tolerance
1236      :type atol: non-negative ``float``
1237      :param rtol: relative tolerance
1238      :type rtol: non-negative ``float``
1239      :param iter_max: maximum number of iteration steps
1240      :type iter_max: ``int``
1241      :rtype: ``tuple``
1242      :warning: ``r`` and ``x`` are altered.
1243      """
1244    u1=0    u1=0
1245    u2=0    u2=0
1246    y1=0    y1=0
1247    y2=0    y2=0
1248    
1249    w = r    w = r
1250    y1 = r    y1 = r
1251    iter = 0    iter = 0
1252    d = 0    d = 0
1253        v = Aprod(y1)
   v = Msolve(Aprod(y1))  
1254    u1 = v    u1 = v
1255      
1256    theta = 0.0;    theta = 0.0;
1257    eta = 0.0;    eta = 0.0;
1258    tau = math.sqrt(bilinearform(r,r))    rho=bilinearform(r,r)
1259    error = [ error, tau ]    if rho < 0: raise NegativeNorm,"negative norm."
1260    rho = tau * tau    tau = math.sqrt(rho)
1261    m=1    norm_r0=tau
1262  #    while tau>atol+rtol*norm_r0:
 #  TFQMR iteration  
 #  
 #  while ( iter < kmax-1 ):  
     
   while not stoppingcriterium(tau*math.sqrt ( m + 1 ),norm_b,'TFQMR'):  
1263      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
1264    
1265      sigma = bilinearform(r,v)      sigma = bilinearform(r,v)
1266        if sigma == 0.0: raise IterationBreakDown,'TFQMR breakdown, sigma=0'
     if ( sigma == 0.0 ):  
       raise 'TFQMR breakdown, sigma=0'  
       
1267    
1268      alpha = rho / sigma      alpha = rho / sigma
1269    
# Line 1162  def TFQMR(b, Aprod, Msolve, bilinearform Line 1273  def TFQMR(b, Aprod, Msolve, bilinearform
1273  #  #
1274        if ( j == 1 ):        if ( j == 1 ):
1275          y2 = y1 - alpha * v          y2 = y1 - alpha * v
1276          u2 = Msolve(Aprod(y2))          u2 = Aprod(y2)
1277    
1278        m = 2 * (iter+1) - 2 + (j+1)        m = 2 * (iter+1) - 2 + (j+1)
1279        if j==0:        if j==0:
1280           w = w - alpha * u1           w = w - alpha * u1
1281           d = y1 + ( theta * theta * eta / alpha ) * d           d = y1 + ( theta * theta * eta / alpha ) * d
1282        if j==1:        if j==1:
# Line 1180  def TFQMR(b, Aprod, Msolve, bilinearform Line 1291  def TFQMR(b, Aprod, Msolve, bilinearform
1291  #  #
1292  #  Try to terminate the iteration at each pass through the loop  #  Try to terminate the iteration at each pass through the loop
1293  #  #
1294       # 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'  
       
1295    
1296      rhon = bilinearform(r,w)      rhon = bilinearform(r,w)
1297      beta = rhon / rho;      beta = rhon / rho;
1298      rho = rhon;      rho = rhon;
1299      y1 = w + beta * y2;      y1 = w + beta * y2;
1300      u1 = Msolve(Aprod(y1))      u1 = Aprod(y1)
1301      v = u1 + beta * ( u2 + beta * v )      v = u1 + beta * ( u2 + beta * v )
1302      error = [ error, tau ]  
1303      total_iters = iter      iter += 1
       
     iter = iter + 1  
1304    
1305    return x    return x
1306    
# Line 1208  def TFQMR(b, Aprod, Msolve, bilinearform Line 1309  def TFQMR(b, Aprod, Msolve, bilinearform
1309    
1310  class ArithmeticTuple(object):  class ArithmeticTuple(object):
1311     """     """
1312     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
1313       ArithmeticTuple and ``a`` is a float.
1314    
1315     example of usage:     Example of usage::
1316    
1317     from esys.escript import Data         from esys.escript import Data
1318     from numarray import array         from numpy import array
1319     a=Data(...)         a=Data(...)
1320     b=array([1.,4.])         b=array([1.,4.])
1321     x=ArithmeticTuple(a,b)         x=ArithmeticTuple(a,b)
1322     y=5.*x         y=5.*x
1323    
1324     """     """
1325     def __init__(self,*args):     def __init__(self,*args):
1326         """         """
1327         initialize object with elements args.         Initializes object with elements ``args``.
1328    
1329         @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
1330                        scaling (x=a*y)
1331         """         """
1332         self.__items=list(args)         self.__items=list(args)
1333    
1334     def __len__(self):     def __len__(self):
1335         """         """
1336         number of items         Returns the number of items.
1337    
1338         @return: number of items         :return: number of items
1339         @rtype: C{int}         :rtype: ``int``
1340         """         """
1341         return len(self.__items)         return len(self.__items)
1342    
1343     def __getitem__(self,index):     def __getitem__(self,index):
1344         """         """
1345         get an item         Returns item at specified position.
1346    
1347         @param index: item to be returned         :param index: index of item to be returned
1348         @type index: C{int}         :type index: ``int``
1349         @return: item with index C{index}         :return: item with index ``index``
1350         """         """
1351         return self.__items.__getitem__(index)         return self.__items.__getitem__(index)
1352    
1353     def __mul__(self,other):     def __mul__(self,other):
1354         """         """
1355         scaling from the right         Scales by ``other`` from the right.
1356    
1357         @param other: scaling factor         :param other: scaling factor
1358         @type other: C{float}         :type other: ``float``
1359         @return: itemwise self*other         :return: itemwise self*other
1360         @rtype: L{ArithmeticTuple}         :rtype: `ArithmeticTuple`
1361         """         """
1362         out=[]         out=[]
1363         try:           try:
1364             l=len(other)             l=len(other)
1365             if l!=len(self):             if l!=len(self):
1366                 raise ValueError,"length of of arguments don't match."                 raise ValueError,"length of arguments don't match."
1367             for i in range(l): out.append(self[i]*other[i])             for i in range(l): out.append(self[i]*other[i])
1368         except TypeError:         except TypeError:
1369         for i in range(len(self)): out.append(self[i]*other)             for i in range(len(self)): out.append(self[i]*other)
1370         return ArithmeticTuple(*tuple(out))         return ArithmeticTuple(*tuple(out))
1371    
1372     def __rmul__(self,other):     def __rmul__(self,other):
1373         """         """
1374         scaling from the left         Scales by ``other`` from the left.
1375    
1376         @param other: scaling factor         :param other: scaling factor
1377         @type other: C{float}         :type other: ``float``
1378         @return: itemwise other*self         :return: itemwise other*self
1379         @rtype: L{ArithmeticTuple}         :rtype: `ArithmeticTuple`
1380         """         """
1381         out=[]         out=[]
1382         try:           try:
1383             l=len(other)             l=len(other)
1384             if l!=len(self):             if l!=len(self):
1385                 raise ValueError,"length of of arguments don't match."                 raise ValueError,"length of arguments don't match."
1386             for i in range(l): out.append(other[i]*self[i])             for i in range(l): out.append(other[i]*self[i])
1387         except TypeError:         except TypeError:
1388         for i in range(len(self)): out.append(other*self[i])             for i in range(len(self)): out.append(other*self[i])
1389         return ArithmeticTuple(*tuple(out))         return ArithmeticTuple(*tuple(out))
1390    
1391     def __div__(self,other):     def __div__(self,other):
1392         """         """
1393         dividing from the right         Scales by (1/``other``) from the right.
1394    
1395         @param other: scaling factor         :param other: scaling factor
1396         @type other: C{float}         :type other: ``float``
1397         @return: itemwise self/other         :return: itemwise self/other
1398         @rtype: L{ArithmeticTuple}         :rtype: `ArithmeticTuple`
1399         """         """
1400         return self*(1/other)         return self*(1/other)
1401    
1402     def __rdiv__(self,other):     def __rdiv__(self,other):
1403         """         """
1404         dividing from the left         Scales by (1/``other``) from the left.
1405    
1406         @param other: scaling factor         :param other: scaling factor
1407         @type other: C{float}         :type other: ``float``
1408         @return: itemwise other/self         :return: itemwise other/self
1409         @rtype: L{ArithmeticTuple}         :rtype: `ArithmeticTuple`
1410         """         """
1411         out=[]         out=[]
1412         try:           try:
1413             l=len(other)             l=len(other)
1414             if l!=len(self):             if l!=len(self):
1415                 raise ValueError,"length of of arguments don't match."                 raise ValueError,"length of arguments don't match."
1416             for i in range(l): out.append(other[i]/self[i])             for i in range(l): out.append(other[i]/self[i])
1417         except TypeError:         except TypeError:
1418         for i in range(len(self)): out.append(other/self[i])             for i in range(len(self)): out.append(other/self[i])
1419         return ArithmeticTuple(*tuple(out))         return ArithmeticTuple(*tuple(out))
1420      
1421     def __iadd__(self,other):     def __iadd__(self,other):
1422         """         """
1423         in-place add of other to self         Inplace addition of ``other`` to self.
1424    
1425         @param other: increment         :param other: increment
1426         @type other: C{ArithmeticTuple}         :type other: ``ArithmeticTuple``
1427         """         """
1428         if len(self) != len(other):         if len(self) != len(other):
1429             raise ValueError,"tuple length must match."             raise ValueError,"tuple lengths must match."
1430         for i in range(len(self)):         for i in range(len(self)):
1431             self.__items[i]+=other[i]             self.__items[i]+=other[i]
1432         return self         return self
1433    
1434     def __add__(self,other):     def __add__(self,other):
1435         """         """
1436         add other to self         Adds ``other`` to self.
1437    
1438         @param other: increment         :param other: increment
1439         @type other: C{ArithmeticTuple}         :type other: ``ArithmeticTuple``
1440         """         """
1441         out=[]         out=[]
1442         try:           try:
1443             l=len(other)             l=len(other)
1444             if l!=len(self):             if l!=len(self):
1445                 raise ValueError,"length of of arguments don't match."                 raise ValueError,"length of arguments don't match."
1446             for i in range(l): out.append(self[i]+other[i])             for i in range(l): out.append(self[i]+other[i])
1447         except TypeError:         except TypeError:
1448         for i in range(len(self)): out.append(self[i]+other)             for i in range(len(self)): out.append(self[i]+other)
1449         return ArithmeticTuple(*tuple(out))         return ArithmeticTuple(*tuple(out))
1450    
1451     def __sub__(self,other):     def __sub__(self,other):
1452         """         """
1453         subtract other from self         Subtracts ``other`` from self.
1454    
1455         @param other: increment         :param other: decrement
1456         @type other: C{ArithmeticTuple}         :type other: ``ArithmeticTuple``
1457         """         """
1458         out=[]         out=[]
1459         try:           try:
1460             l=len(other)             l=len(other)
1461             if l!=len(self):             if l!=len(self):
1462                 raise ValueError,"length of of arguments don't match."                 raise ValueError,"length of arguments don't match."
1463             for i in range(l): out.append(self[i]-other[i])             for i in range(l): out.append(self[i]-other[i])
1464         except TypeError:         except TypeError:
1465         for i in range(len(self)): out.append(self[i]-other)             for i in range(len(self)): out.append(self[i]-other)
1466         return ArithmeticTuple(*tuple(out))         return ArithmeticTuple(*tuple(out))
1467      
1468     def __isub__(self,other):     def __isub__(self,other):
1469         """         """
1470         subtract other from self         Inplace subtraction of ``other`` from self.
1471    
1472         @param other: increment         :param other: decrement
1473         @type other: C{ArithmeticTuple}         :type other: ``ArithmeticTuple``
1474         """         """
1475         if len(self) != len(other):         if len(self) != len(other):
1476             raise ValueError,"tuple length must match."             raise ValueError,"tuple length must match."
# Line 1377  class ArithmeticTuple(object): Line 1480  class ArithmeticTuple(object):
1480    
1481     def __neg__(self):     def __neg__(self):
1482         """         """
1483         negate         Negates values.
   
1484         """         """
1485         out=[]         out=[]
1486         for i in range(len(self)):         for i in range(len(self)):
# Line 1388  class ArithmeticTuple(object): Line 1490  class ArithmeticTuple(object):
1490    
1491  class HomogeneousSaddlePointProblem(object):  class HomogeneousSaddlePointProblem(object):
1492        """        """
1493        This provides a framwork for solving linear homogeneous saddle point problem of the form        This class provides a framework for solving linear homogeneous saddle
1494          point problems of the form::
1495    
1496               Av+B^*p=f            *Av+B^*p=f*
1497               Bv    =0            *Bv     =0*
   
       for the unknowns v and p and given operators A and B and given right hand side f.  
       B^* is the adjoint operator of B is the given inner product.  
1498    
1499          for the unknowns *v* and *p* and given operators *A* and *B* and
1500          given right hand side *f*. *B^** is the adjoint operator of *B*.
1501          *A* may depend weakly on *v* and *p*.
1502        """        """
1503        def __init__(self,**kwargs):        def __init__(self, **kwargs):
1504        """
1505        initializes the saddle point problem
1506        """
1507            self.resetControlParameters()
1508          self.setTolerance()          self.setTolerance()
1509          self.setToleranceReductionFactor()          self.setAbsoluteTolerance()
1510          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):
       def initialize(self):  
         """  
         initialize the problem (overwrite)  
         """  
         pass  
       def B(self,v):  
1511           """           """
1512           returns Bv (overwrite)           sets a control parameter
          @rtype: equal to the type of p  
1513    
1514           @note: boundary conditions on p should be zero!           :param K_p: initial value for constant to adjust pressure tolerance
1515             :type K_p: ``float``
1516             :param K_v: initial value for constant to adjust velocity tolerance
1517             :type K_v: ``float``
1518             :param rtol_max: maximuim relative tolerance used to calculate presssure and velocity increment.
1519             :type rtol_max: ``float``
1520             :param chi_max: maximum tolerable converegence rate.
1521             :type chi_max: ``float``
1522             :param reduction_factor: reduction factor for adjustment factors.
1523             :type reduction_factor: ``float``
1524           """           """
1525           pass           self.setControlParameter(K_p, K_v, rtol_max, rtol_min, chi_max, reduction_factor, theta)
1526    
1527        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):
1528           """           """
1529           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}  
1530    
          @rtype: equal to the type of p  
          """  
          pass  
1531    
1532        def solve_A(self,u,p):           :param K_p: initial value for constant to adjust pressure tolerance
1533             :type K_p: ``float``
1534             :param K_v: initial value for constant to adjust velocity tolerance
1535             :type K_v: ``float``
1536             :param rtol_max: maximuim relative tolerance used to calculate presssure and velocity increment.
1537             :type rtol_max: ``float``
1538             :param chi_max: maximum tolerable converegence rate.
1539             :type chi_max: ``float``
1540             :type reduction_factor: ``float``
1541           """           """
1542           solves Av=f-Au-B^*p with accuracy self.getReducedTolerance() (overwrite)           if not K_p == None:
1543                if K_p<1:
1544                   raise ValueError,"K_p need to be greater or equal to 1."
1545             else:
1546                K_p=self.__K_p
1547    
1548             if not K_v == None:
1549                if K_v<1:
1550                   raise ValueError,"K_v need to be greater or equal to 1."
1551             else:
1552                K_v=self.__K_v
1553    
1554             if not rtol_max == None:
1555                if rtol_max<=0 or rtol_max>=1:
1556                   raise ValueError,"rtol_max needs to be positive and less than 1."
1557             else:
1558                rtol_max=self.__rtol_max
1559    
1560             if not rtol_min == None:
1561                if rtol_min<=0 or rtol_min>=1:
1562                   raise ValueError,"rtol_min needs to be positive and less than 1."
1563             else:
1564                rtol_min=self.__rtol_min
1565    
1566             if not chi_max == None:
1567                if chi_max<=0 or chi_max>=1:
1568                   raise ValueError,"chi_max needs to be positive and less than 1."
1569             else:
1570                chi_max = self.__chi_max
1571    
1572             if not reduction_factor == None:
1573                if reduction_factor<=0 or reduction_factor>1:
1574                   raise ValueError,"reduction_factor need to be between zero and one."
1575             else:
1576                reduction_factor=self.__reduction_factor
1577    
1578             if not theta == None:
1579                if theta<=0 or theta>1:
1580                   raise ValueError,"theta need to be between zero and one."
1581             else:
1582                theta=self.__theta
1583    
1584             if rtol_min>=rtol_max:
1585                 raise ValueError,"rtol_max = %e needs to be greater than rtol_min = %e"%(rtol_max,rtol_min)
1586             self.__chi_max = chi_max
1587             self.__rtol_max = rtol_max
1588             self.__K_p = K_p
1589             self.__K_v = K_v
1590             self.__reduction_factor = reduction_factor
1591             self.__theta = theta
1592             self.__rtol_min=rtol_min
1593    
1594           @rtype: equal to the type of v        #=============================================================
1595           @note: boundary conditions on v should be zero!        def inner_pBv(self,p,Bv):
1596           """           """
1597           pass           Returns inner product of element p and Bv (overwrite).
1598    
1599        def solve_prec(self,p):           :param p: a pressure increment
1600             :param Bv: a residual
1601             :return: inner product of element p and Bv
1602             :rtype: ``float``
1603             :note: used if PCG is applied.
1604           """           """
1605           provides a preconditioner for BA^{-1}B^* with accuracy self.getReducedTolerance() (overwrite)           raise NotImplementedError,"no inner product for p and Bv implemented."
1606    
1607           @rtype: equal to the type of p        def inner_p(self,p0,p1):
1608           """           """
1609           pass           Returns inner product of p0 and p1 (overwrite).
1610    
1611        def stoppingcriterium(self,Bv,v,p):           :param p0: a pressure
1612             :param p1: a pressure
1613             :return: inner product of p0 and p1
1614             :rtype: ``float``
1615             """
1616             raise NotImplementedError,"no inner product for p implemented."
1617      
1618          def norm_v(self,v):
1619           """           """
1620           returns a True if iteration is terminated. (overwrite)           Returns the norm of v (overwrite).
1621    
1622           @rtype: C{bool}           :param v: a velovity
1623             :return: norm of v
1624             :rtype: non-negative ``float``
1625           """           """
1626           pass           raise NotImplementedError,"no norm of v implemented."
1627                      def getDV(self, p, v, tol):
1628        def __inner(self,p,r):           """
1629           return self.inner(p,r[1])           return a correction to the value for a given v and a given p with accuracy `tol` (overwrite)
1630    
1631        def __inner_p(self,p1,p2):           :param p: pressure
1632           return self.inner(p1,p2)           :param v: pressure
1633                   :return: dv given as *dv= A^{-1} (f-A v-B^*p)*
1634        def __inner_a(self,a1,a2):           :note: Only *A* may depend on *v* and *p*
1635           return self.inner_a(a1,a2)           """
1636             raise NotImplementedError,"no dv calculation implemented."
1637    
1638        def __inner_a1(self,a1,a2):          
1639           return self.inner(a1[1],a2[1])        def Bv(self,v, tol):
1640            """
1641            Returns Bv with accuracy `tol` (overwrite)
1642    
1643        def __stoppingcriterium(self,norm_r,r,p):          :rtype: equal to the type of p
1644            return self.stoppingcriterium(r[1],r[0],p)          :note: boundary conditions on p should be zero!
1645            """
1646            raise NotImplementedError, "no operator B implemented."
1647    
1648        def __stoppingcriterium2(self,norm_r,norm_b,solver='GMRES',TOL=None):        def norm_Bv(self,Bv):
1649            return self.stoppingcriterium2(norm_r,norm_b,solver,TOL)          """
1650            Returns the norm of Bv (overwrite).
1651    
1652        def setTolerance(self,tolerance=1.e-8):          :rtype: equal to the type of p
1653                self.__tol=tolerance          :note: boundary conditions on p should be zero!
1654        def getTolerance(self):          """
1655                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)  
1656    
1657          def solve_AinvBt(self,dp, tol):
1658             """
1659             Solves *A dv=B^*dp* with accuracy `tol`
1660    
1661  def MaskFromBoundaryTag(domain,*tags):           :param dp: a pressure increment
1662     """           :return: the solution of *A dv=B^*dp*
1663     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.
1664     that touch the boundary tagged by tags.           """
1665             raise NotImplementedError,"no operator A implemented."
1666    
1667     usage: m=MaskFromBoundaryTag(domain,"left", "right")        def solve_prec(self,Bv, tol):
1668             """
1669             Provides a preconditioner for *(BA^{-1}B^ * )* applied to Bv with accuracy `tol`
1670    
1671     @param domain: a given domain           :rtype: equal to the type of p
1672     @type domain: L{escript.Domain}           :note: boundary conditions on p should be zero!
1673     @param tags: boundray tags           """
1674     @type tags: C{str}           raise NotImplementedError,"no preconditioner for Schur complement implemented."
1675     @return: a mask which marks samples that are touching the boundary tagged by any of the given tags.        #=============================================================
1676     @rtype: L{escript.Data} of rank 0        def __Aprod_PCG(self,dp):
1677     """            dv=self.solve_AinvBt(dp, self.__subtol)
1678     pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)            return ArithmeticTuple(dv,self.Bv(dv, self.__subtol))
1679     d=escript.Scalar(0.,escript.FunctionOnBoundary(domain))  
1680     for t in tags: d.setTaggedValue(t,1.)        def __inner_PCG(self,p,r):
1681     pde.setValue(y=d)           return self.inner_pBv(p,r[1])
1682     return util.whereNonZero(pde.getRightHandSide())  
1683  #============================================================================================================================================        def __Msolve_PCG(self,r):
1684  class SaddlePointProblem(object):            return self.solve_prec(r[1], self.__subtol)
1685     """        #=============================================================
1686     This implements a solver for a saddlepoint problem        def __Aprod_GMRES(self,p):
1687              return self.solve_prec(self.Bv(self.solve_AinvBt(p, self.__subtol), self.__subtol), self.__subtol)
1688          def __inner_GMRES(self,p0,p1):
1689             return self.inner_p(p0,p1)
1690    
1691          #=============================================================
1692          def norm_p(self,p):
1693              """
1694              calculates the norm of ``p``
1695              
1696              :param p: a pressure
1697              :return: the norm of ``p`` using the inner product for pressure
1698              :rtype: ``float``
1699              """
1700              f=self.inner_p(p,p)
1701              if f<0: raise ValueError,"negative pressure norm."
1702              return math.sqrt(f)
1703          
1704          def solve(self,v,p,max_iter=20, verbose=False, usePCG=True, iter_restart=20, max_correction_steps=10):
1705             """
1706             Solves the saddle point problem using initial guesses v and p.
1707    
1708     M{f(u,p)=0}           :param v: initial guess for velocity
1709     M{g(u)=0}           :param p: initial guess for pressure
1710             :type v: `Data`
1711             :type p: `Data`
1712             :param usePCG: indicates the usage of the PCG rather than GMRES scheme.
1713             :param max_iter: maximum number of iteration steps per correction
1714                              attempt
1715             :param verbose: if True, shows information on the progress of the
1716                             saddlepoint problem solver.
1717             :param iter_restart: restart the iteration after ``iter_restart`` steps
1718                                  (only used if useUzaw=False)
1719             :type usePCG: ``bool``
1720             :type max_iter: ``int``
1721             :type verbose: ``bool``
1722             :type iter_restart: ``int``
1723             :rtype: ``tuple`` of `Data` objects
1724             :note: typically this method is overwritten by a subclass. It provides a wrapper for the ``_solve`` method.
1725             """
1726             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)
1727    
1728     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):
1729             """
1730             see `_solve` method.
1731             """
1732             self.verbose=verbose
1733             rtol=self.getTolerance()
1734             atol=self.getAbsoluteTolerance()
1735    
1736             K_p=self.__K_p
1737             K_v=self.__K_v
1738             correction_step=0
1739             converged=False
1740             chi=None
1741             eps=None
1742    
1743             if self.verbose: print "HomogeneousSaddlePointProblem: start iteration: rtol= %e, atol=%e"%(rtol, atol)
1744             while not converged:
1745    
1746                 # get tolerance for velecity increment:
1747                 if chi == None:
1748                    rtol_v=self.__rtol_max
1749                 else:
1750                    rtol_v=min(chi/K_v,self.__rtol_max)
1751                 rtol_v=max(rtol_v, self.__rtol_min)
1752                 if self.verbose: print "HomogeneousSaddlePointProblem: step %s: rtol_v= %e"%(correction_step,rtol_v)
1753                 # get velocity increment:
1754                 dv1=self.getDV(p,v,rtol_v)
1755                 v1=v+dv1
1756                 Bv1=self.Bv(v1, rtol_v)
1757                 norm_Bv1=self.norm_Bv(Bv1)
1758                 norm_dv1=self.norm_v(dv1)
1759                 if self.verbose: print "HomogeneousSaddlePointProblem: step %s: norm_Bv1 = %e, norm_dv1 = %e"%(correction_step, norm_Bv1, norm_dv1)
1760                 if norm_dv1*self.__theta < norm_Bv1:
1761                    # get tolerance for pressure increment:
1762                    large_Bv1=True
1763                    if chi == None or eps == None:
1764                       rtol_p=self.__rtol_max
1765                    else:
1766                       rtol_p=min(chi**2*eps/K_p/norm_Bv1, self.__rtol_max)
1767                    self.__subtol=max(rtol_p**2, self.__rtol_min)
1768                    if self.verbose: print "HomogeneousSaddlePointProblem: step %s: rtol_p= %e"%(correction_step,rtol_p)
1769                    # now we solve for the pressure increment dp from B*A^{-1}B^* dp = Bv1
1770                    if usePCG:
1771                        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)
1772                        v2=r[0]
1773                        Bv2=r[1]
1774                    else:
1775                        # don't use!!!!
1776                        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)
1777                        dv2=self.solve_AinvBt(dp, self.__subtol)
1778                        v2=v1-dv2
1779                        Bv2=self.Bv(v2, self.__subtol)
1780                    p2=p+dp
1781                 else:
1782                    large_Bv1=False
1783                    v2=v1
1784                    p2=p
1785                 # update business:
1786                 norm_dv2=self.norm_v(v2-v)
1787                 norm_v2=self.norm_v(v2)
1788                 if self.verbose: print "HomogeneousSaddlePointProblem: step %s: v2 = %e, norm_dv2 = %e"%(correction_step, norm_v2, self.norm_v(v2-v))
1789                 eps, eps_old = max(norm_Bv1, norm_dv2), eps
1790                 if eps_old == None:
1791                      chi, chi_old = None, chi
1792                 else:
1793                      chi, chi_old = min(eps/ eps_old, self.__chi_max), chi
1794                 if eps != None:
1795                     if chi !=None:
1796                        if self.verbose: print "HomogeneousSaddlePointProblem: step %s: convergence rate = %e, correction = %e"%(correction_step,chi, eps)
1797                     else:
1798                        if self.verbose: print "HomogeneousSaddlePointProblem: step %s: correction = %e"%(correction_step, eps)
1799                 if eps <= rtol*norm_v2+atol :
1800                     converged = True
1801                 else:
1802                     if correction_step>=max_correction_steps:
1803                          raise CorrectionFailed,"Given up after %d correction steps."%correction_step
1804                     if chi_old!=None:
1805                        K_p=max(1,self.__reduction_factor*K_p,(chi-chi_old)/chi_old**2*K_p)
1806                        K_v=max(1,self.__reduction_factor*K_v,(chi-chi_old)/chi_old**2*K_p)
1807                        if self.verbose: print "HomogeneousSaddlePointProblem: step %s: new adjustment factor K = %e"%(correction_step,K_p)
1808                 correction_step+=1
1809                 v,p =v2, p2
1810             if self.verbose: print "HomogeneousSaddlePointProblem: tolerance reached after %s steps."%correction_step
1811         return v,p
1812          #========================================================================
1813          def setTolerance(self,tolerance=1.e-4):
1814             """
1815             Sets the relative tolerance for (v,p).
1816    
1817     M{Q_f (u^{k+1}-u^{k}) = - f(u^{k},p^{k})}           :param tolerance: tolerance to be used
1818     M{Q_g (p^{k+1}-p^{k}) =   g(u^{k+1})}           :type tolerance: non-negative ``float``
1819             """
1820             if tolerance<0:
1821                 raise ValueError,"tolerance must be positive."
1822             self.__rtol=tolerance
1823    
1824     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):
1825     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'           """
1826     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  
1827    
1828         @param verbose: switches on the printing out some information           :return: relative tolerance
1829         @type verbose: C{bool}           :rtype: ``float``
1830         @note: this method may be overwritten by a particular saddle point problem           """
1831         """           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.")  
1832    
1833     def trace(self,text):        def setAbsoluteTolerance(self,tolerance=0.):
1834         """           """
1835         prints text if verbose has been set           Sets the absolute tolerance.
1836    
1837         @param text: a text message           :param tolerance: tolerance to be used
1838         @type text: C{str}           :type tolerance: non-negative ``float``
1839         """           """
1840         if self.__verbose: print "%s: %s"%(str(self),text)           if tolerance<0:
1841                 raise ValueError,"tolerance must be non-negative."
1842             self.__atol=tolerance
1843    
1844     def solve_f(self,u,p,tol=1.e-8):        def getAbsoluteTolerance(self):
1845         """           """
1846         solves           Returns the absolute tolerance.
1847    
1848         A_f du = f(u,p)           :return: absolute tolerance
1849             :rtype: ``float``
1850             """
1851             return self.__atol
1852    
        with tolerance C{tol} and return du. A_f is Jacobiean of f with respect to u.  
1853    
1854         @param u: current approximation of u  def MaskFromBoundaryTag(domain,*tags):
1855         @type u: L{escript.Data}     """
1856         @param p: current approximation of p     Creates a mask on the Solution(domain) function space where the value is
1857         @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  
1858    
1859     def solve_g(self,u,tol=1.e-8):     Usage: m=MaskFromBoundaryTag(domain, "left", "right")
        """  
        solves  
1860    
1861         Q_g dp = g(u)     :param domain: domain to be used
1862       :type domain: `escript.Domain`
1863       :param tags: boundary tags
1864       :type tags: ``str``
1865       :return: a mask which marks samples that are touching the boundary tagged
1866                by any of the given tags
1867       :rtype: `escript.Data` of rank 0
1868       """
1869       pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)
1870       d=escript.Scalar(0.,escript.FunctionOnBoundary(domain))
1871       for t in tags: d.setTaggedValue(t,1.)
1872       pde.setValue(y=d)
1873       return util.whereNonZero(pde.getRightHandSide())
1874    
1875         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):
1876       """
1877       Creates a mask on the Solution(domain) function space where the value is
1878       one for samples that touch regions tagged by tags.
1879    
1880         @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  
1881    
1882     def inner(self,p0,p1):     :param domain: domain to be used
1883         """     :type domain: `escript.Domain`
1884         inner product of p0 and p1 approximating p. Typically this returns integrate(p0*p1)     :param tags: boundary tags
1885         @return: inner product of p0 and p1     :type tags: ``str``
1886         @rtype: C{float}     :return: a mask which marks samples that are touching the boundary tagged
1887         """              by any of the given tags
1888         pass     :rtype: `escript.Data` of rank 0
1889       """
1890       pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)
1891       d=escript.Scalar(0.,escript.Function(domain))
1892       for t in tags: d.setTaggedValue(t,1.)
1893       pde.setValue(Y=d)
1894       return util.whereNonZero(pde.getRightHandSide())
1895    
    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  
1896    
         @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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