/[escript]/trunk/escript/py_src/pdetools.py
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revision 1956 by gross, Mon Nov 3 05:08:42 2008 UTC revision 2693 by artak, Tue Sep 29 05:47:45 2009 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 412  class Locator: Line 424  class Locator:
424    
425  class SolverSchemeException(Exception):  class SolverSchemeException(Exception):
426     """     """
427     exceptions thrown by solvers     This is a generic exception thrown by solvers.
428     """     """
429     pass     pass
430    
431  class IndefinitePreconditioner(SolverSchemeException):  class IndefinitePreconditioner(SolverSchemeException):
432     """     """
433     the preconditioner is not positive definite.     Exception thrown if the preconditioner is not positive definite.
434     """     """
435     pass     pass
436    
437  class MaxIterReached(SolverSchemeException):  class MaxIterReached(SolverSchemeException):
438     """     """
439     maxium number of iteration steps is reached.     Exception thrown if the maximum number of iteration steps is reached.
440     """     """
441     pass     pass
442  class IterationBreakDown(SolverSchemeException):  
443    class CorrectionFailed(SolverSchemeException):
444     """     """
445     iteration scheme econouters an incurable breakdown.     Exception thrown if no convergence has been achieved in the solution
446       correction scheme.
447     """     """
448     pass     pass
449  class NegativeNorm(SolverSchemeException):  
450    class IterationBreakDown(SolverSchemeException):
451     """     """
452     a norm calculation returns a negative norm.     Exception thrown if the iteration scheme encountered an incurable breakdown.
453     """     """
454     pass     pass
455    
456  class IterationHistory(object):  class NegativeNorm(SolverSchemeException):
457     """     """
458     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.  
459     """     """
460     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.  
461    
462          def PCG(r, Aprod, x, Msolve, bilinearform, atol=0, rtol=1.e-8, iter_max=100, initial_guess=True, verbose=False):
463         @param norm_r: current residual norm     """
464         @type norm_r: non-negative C{float}     Solver for
        @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}  
465    
466         """     *Ax=b*
        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]  
467    
468     def stoppingcriterium2(self,norm_r,norm_b,solver="GMRES",TOL=None):     with a symmetric and positive definite operator A (more details required!).
469         """     It uses the conjugate gradient method with preconditioner M providing an
470         returns True if the C{norm_r} is C{tolerance}*C{norm_b}     approximation of A.
471    
472             The iteration is terminated if
        @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}  
473    
474         """     *|r| <= atol+rtol*|r0|*
        if TOL==None:  
           TOL=self.tolerance  
        self.history.append(norm_r)  
        if self.verbose: print "iter: %s:  norm(r) = %e"#(len(self.history)-1, self.history[-1])  
        return self.history[-1]<=TOL * norm_b  
475    
476  def PCG(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100):     where *r0* is the initial residual and *|.|* is the energy norm. In fact
    """  
    Solver for  
477    
478     M{Ax=b}     *|r| = sqrt( bilinearform(Msolve(r),r))*
   
    with a symmetric and positive definite operator A (more details required!).  
    It uses the conjugate gradient method with preconditioner M providing an approximation of A.  
   
    The iteration is terminated if the C{stoppingcriterium} function return C{True}.  
479    
480     For details on the preconditioned conjugate gradient method see the book:     For details on the preconditioned conjugate gradient method see the book:
481    
482     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,
483     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,
484     C. Romine, and H. van der Vorst.     C. Romine, and H. van der Vorst}.
485    
486     @param b: the right hand side of the liner system. C{b} is altered.     :param r: initial residual *r=b-Ax*. ``r`` is altered.
487     @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)
488     @param Aprod: returns the value Ax     :param x: an initial guess for the solution
489     @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)
490     @param Msolve: solves Mx=r     :param Aprod: returns the value Ax
491     @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
492  type like argument C{x}.                  argument ``x``. The returned object needs to be of the same type
493     @param bilinearform: inner product C{<x,r>}                  like argument ``r``.
494     @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
495     @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
496     @type stoppingcriterium: function that returns C{True} or C{False}                   argument ``r``. The returned object needs to be of the same
497     @param x: an initial guess for the solution. If no C{x} is given 0*b is used.                   type like argument ``x``.
498     @type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)     :param bilinearform: inner product ``<x,r>``
499     @param iter_max: maximum number of iteration steps.     :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
500     @type iter_max: C{int}                         type like argument ``x`` and ``r`` is. The returned value
501     @return: the solution approximation and the corresponding residual                         is a ``float``.
502     @rtype: C{tuple}     :param atol: absolute tolerance
503     @warning: C{b} and C{x} are altered.     :type atol: non-negative ``float``
504       :param rtol: relative tolerance
505       :type rtol: non-negative ``float``
506       :param iter_max: maximum number of iteration steps
507       :type iter_max: ``int``
508       :return: the solution approximation and the corresponding residual
509       :rtype: ``tuple``
510       :warning: ``r`` and ``x`` are altered.
511     """     """
512     iter=0     iter=0
    if x==None:  
       x=0*b  
    else:  
       b += (-1)*Aprod(x)  
    r=b  
513     rhat=Msolve(r)     rhat=Msolve(r)
514     d = rhat     d = rhat
515     rhat_dot_r = bilinearform(rhat, r)     rhat_dot_r = bilinearform(rhat, r)
516     if rhat_dot_r<0: raise NegativeNorm,"negative norm."     if rhat_dot_r<0: raise NegativeNorm,"negative norm."
517       norm_r0=math.sqrt(rhat_dot_r)
518       atol2=atol+rtol*norm_r0
519       if atol2<=0:
520          raise ValueError,"Non-positive tolarance."
521       atol2=max(atol2, 100. * util.EPSILON * norm_r0)
522    
523       if verbose: print "PCG: initial residual norm = %e (absolute tolerance = %e)"%(norm_r0, atol2)
524    
525     while not stoppingcriterium(math.sqrt(rhat_dot_r),r,x):  
526       while not math.sqrt(rhat_dot_r) <= atol2:
527         iter+=1         iter+=1
528         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
529    
530         q=Aprod(d)         q=Aprod(d)
531         alpha = rhat_dot_r / bilinearform(d, q)         alpha = rhat_dot_r / bilinearform(d, q)
532         x += alpha * d         x += alpha * d
533         r += (-alpha) * q         if isinstance(q,ArithmeticTuple):
534           r += q * (-alpha)      # Doing it the other way calls the float64.__mul__ not AT.__rmul__
535           else:
536               r += (-alpha) * q
537         rhat=Msolve(r)         rhat=Msolve(r)
538         rhat_dot_r_new = bilinearform(rhat, r)         rhat_dot_r_new = bilinearform(rhat, r)
539         beta = rhat_dot_r_new / rhat_dot_r         beta = rhat_dot_r_new / rhat_dot_r
# Line 557  type like argument C{x}. Line 542  type like argument C{x}.
542    
543         rhat_dot_r = rhat_dot_r_new         rhat_dot_r = rhat_dot_r_new
544         if rhat_dot_r<0: raise NegativeNorm,"negative norm."         if rhat_dot_r<0: raise NegativeNorm,"negative norm."
545           if verbose: print "PCG: iteration step %s: residual norm = %e"%(iter, math.sqrt(rhat_dot_r))
546     return x,r     if verbose: print "PCG: tolerance reached after %s steps."%iter
547       return x,r,math.sqrt(rhat_dot_r)
548    
549  class Defect(object):  class Defect(object):
550      """      """
551      defines a non-linear defect F(x) of a variable x      Defines a non-linear defect F(x) of a variable x.
552      """      """
553      def __init__(self):      def __init__(self):
554          """          """
555          initialize defect          Initializes defect.
556          """          """
557          self.setDerivativeIncrementLength()          self.setDerivativeIncrementLength()
558    
559      def bilinearform(self, x0, x1):      def bilinearform(self, x0, x1):
560          """          """
561          returns the inner product of x0 and x1          Returns the inner product of x0 and x1
562          @param x0: a value for x  
563          @param x1: a value for x          :param x0: value for x0
564          @return: the inner product of x0 and x1          :param x1: value for x1
565          @rtype: C{float}          :return: the inner product of x0 and x1
566            :rtype: ``float``
567          """          """
568          return 0          return 0
569          
570      def norm(self,x):      def norm(self,x):
571          """          """
572          the norm of argument C{x}          Returns the norm of argument ``x``.
573    
574          @param x: a value for x          :param x: a value
575          @return: norm of argument x          :return: norm of argument x
576          @rtype: C{float}          :rtype: ``float``
577          @note: by default C{sqrt(self.bilinearform(x,x)} is retrurned.          :note: by default ``sqrt(self.bilinearform(x,x)`` is returned.
578          """          """
579          s=self.bilinearform(x,x)          s=self.bilinearform(x,x)
580          if s<0: raise NegativeNorm,"negative norm."          if s<0: raise NegativeNorm,"negative norm."
581          return math.sqrt(s)          return math.sqrt(s)
582    
   
583      def eval(self,x):      def eval(self,x):
584          """          """
585          returns the value F of a given x          Returns the value F of a given ``x``.
586    
587          @param x: value for which the defect C{F} is evalulated.          :param x: value for which the defect ``F`` is evaluated
588          @return: value of the defect at C{x}          :return: value of the defect at ``x``
589          """          """
590          return 0          return 0
591    
592      def __call__(self,x):      def __call__(self,x):
593          return self.eval(x)          return self.eval(x)
594    
595      def setDerivativeIncrementLength(self,inc=math.sqrt(util.EPSILON)):      def setDerivativeIncrementLength(self,inc=1000.*math.sqrt(util.EPSILON)):
596          """          """
597          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
598          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
599            direction of v with x as a starting point.
600    
601          @param inc: relative increment length          :param inc: relative increment length
602          @type inc: positive C{float}          :type inc: positive ``float``
603          """          """
604          if inc<=0: raise ValueError,"positive increment required."          if inc<=0: raise ValueError,"positive increment required."
605          self.__inc=inc          self.__inc=inc
606    
607      def getDerivativeIncrementLength(self):      def getDerivativeIncrementLength(self):
608          """          """
609          returns the relative increment length used to approximate the derivative of the defect          Returns the relative increment length used to approximate the
610          @return: value of the defect at C{x}          derivative of the defect.
611          @rtype: positive C{float}          :return: value of the defect at ``x``
612            :rtype: positive ``float``
613          """          """
614          return self.__inc          return self.__inc
615    
616      def derivative(self, F0, x0, v, v_is_normalised=True):      def derivative(self, F0, x0, v, v_is_normalised=True):
617          """          """
618          returns the directional derivative at x0 in the direction of v          Returns the directional derivative at ``x0`` in the direction of ``v``.
619    
620          @param F0: value of this defect at x0          :param F0: value of this defect at x0
621          @param x0: value at which derivative is calculated.          :param x0: value at which derivative is calculated
622          @param v: direction          :param v: direction
623          @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
624          @return: derivative of this defect at x0 in the direction of C{v}                                  (self.norm(v)=0)
625          @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``
626          maybe oepsnew verwritten to use exact evalution.          :note: by default numerical evaluation (self.eval(x0+eps*v)-F0)/eps is
627                   used but this method maybe overwritten to use exact evaluation.
628          """          """
629          normx=self.norm(x0)          normx=self.norm(x0)
630          if normx>0:          if normx>0:
# Line 651  class Defect(object): Line 640  class Defect(object):
640          F1=self.eval(x0 + epsnew * v)          F1=self.eval(x0 + epsnew * v)
641          return (F1-F0)/epsnew          return (F1-F0)/epsnew
642    
643  ######################################      ######################################
644  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, sub_tol_max=0.5, gamma=0.9, verbose=False):
645     """     """
646     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
647       criterion:
648    
649       *norm(F(x) <= atol + rtol * norm(F(x0)*
650    
651     M{norm(F(x) <= atol + rtol * norm(F(x0)}     where *x0* is the initial guess.
652      
653     where M{x0} is the initial guess.     :param defect: object defining the function *F*. ``defect.norm`` defines the
654                      *norm* used in the stopping criterion.
655     @param defect: object defining the the function M{F}, C{defect.norm} defines the M{norm} used in the stopping criterion.     :type defect: `Defect`
656     @type defect: L{Defect}     :param x: initial guess for the solution, ``x`` is altered.
657     @param x: initial guess for the solution, C{x} is altered.     :type x: any object type allowing basic operations such as
658     @type x: any object type allowing basic operations such as  L{numarray.NumArray}, L{Data}              ``numpy.ndarray``, `Data`
659     @param iter_max: maximum number of iteration steps     :param iter_max: maximum number of iteration steps
660     @type iter_max: positive C{int}     :type iter_max: positive ``int``
661     @param sub_iter_max:     :param sub_iter_max: maximum number of inner iteration steps
662     @type sub_iter_max:     :type sub_iter_max: positive ``int``
663     @param atol: absolute tolerance for the solution     :param atol: absolute tolerance for the solution
664     @type atol: positive C{float}     :type atol: positive ``float``
665     @param rtol: relative tolerance for the solution     :param rtol: relative tolerance for the solution
666     @type rtol: positive C{float}     :type rtol: positive ``float``
667     @param gamma: tolerance safety factor for inner iteration     :param gamma: tolerance safety factor for inner iteration
668     @type gamma: positive C{float}, less than 1     :type gamma: positive ``float``, less than 1
669     @param sub_tol_max: upper bound for inner tolerance.     :param sub_tol_max: upper bound for inner tolerance
670     @type sub_tol_max: positive C{float}, less than 1     :type sub_tol_max: positive ``float``, less than 1
671     @return: an approximation of the solution with the desired accuracy     :return: an approximation of the solution with the desired accuracy
672     @rtype: same type as the initial guess.     :rtype: same type as the initial guess
673     """     """
674     lmaxit=iter_max     lmaxit=iter_max
675     if atol<0: raise ValueError,"atol needs to be non-negative."     if atol<0: raise ValueError,"atol needs to be non-negative."
# Line 697  def NewtonGMRES(defect, x, iter_max=100, Line 689  def NewtonGMRES(defect, x, iter_max=100,
689     # main iteration loop     # main iteration loop
690     #     #
691     while not fnrm<=stop_tol:     while not fnrm<=stop_tol:
692              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
693              #              #
694          #   adjust sub_tol_          #   adjust sub_tol_
695          #          #
696              if iter > 1:              if iter > 1:
697             rat=fnrm/fnrmo             rat=fnrm/fnrmo
# Line 712  def NewtonGMRES(defect, x, iter_max=100, Line 704  def NewtonGMRES(defect, x, iter_max=100,
704              #     if iter_max in __FDGMRES is reached MaxIterReached is thrown              #     if iter_max in __FDGMRES is reached MaxIterReached is thrown
705              #     if iter_restart -1 is returned as sub_iter              #     if iter_restart -1 is returned as sub_iter
706              #     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
707              #              #
708              #                #
709              if verbose: print "             subiteration (GMRES) is called with relative tolerance %e."%sub_tol              if verbose: print "             subiteration (GMRES) is called with relative tolerance %e."%sub_tol
710              try:              try:
711                 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, sub_tol*fnrm, iter_max=iter_max-iter, iter_restart=sub_iter_max)
# Line 734  def NewtonGMRES(defect, x, iter_max=100, Line 726  def NewtonGMRES(defect, x, iter_max=100,
726    
727  def __givapp(c,s,vin):  def __givapp(c,s,vin):
728      """      """
729      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
730      @warning: C{vin} is altered.      ``vin``
731    
732        :warning: ``vin`` is altered.
733      """      """
734      vrot=vin      vrot=vin
735      if isinstance(c,float):      if isinstance(c,float):
736          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]]
737      else:      else:
738          for i in range(len(c)):          for i in range(len(c)):
739              w1=c[i]*vrot[i]-s[i]*vrot[i+1]              w1=c[i]*vrot[i]-s[i]*vrot[i+1]
740          w2=s[i]*vrot[i]+c[i]*vrot[i+1]          w2=s[i]*vrot[i]+c[i]*vrot[i+1]
741              vrot[i:i+2]=w1,w2              vrot[i]=w1
742                vrot[i+1]=w2
743      return vrot      return vrot
744    
745  def __FDGMRES(F0, defect, x0, atol, iter_max=100, iter_restart=20):  def __FDGMRES(F0, defect, x0, atol, iter_max=100, iter_restart=20):
746     h=numarray.zeros((iter_restart,iter_restart),numarray.Float64)     h=numpy.zeros((iter_restart,iter_restart),numpy.float64)
747     c=numarray.zeros(iter_restart,numarray.Float64)     c=numpy.zeros(iter_restart,numpy.float64)
748     s=numarray.zeros(iter_restart,numarray.Float64)     s=numpy.zeros(iter_restart,numpy.float64)
749     g=numarray.zeros(iter_restart,numarray.Float64)     g=numpy.zeros(iter_restart,numpy.float64)
750     v=[]     v=[]
751    
752     rho=defect.norm(F0)     rho=defect.norm(F0)
753     if rho<=0.: return x0*0     if rho<=0.: return x0*0
754      
755     v.append(-F0/rho)     v.append(-F0/rho)
756     g[0]=rho     g[0]=rho
757     iter=0     iter=0
758     while rho > atol and iter<iter_restart-1:     while rho > atol and iter<iter_restart-1:
759            if iter  >= iter_max:
760      if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max              raise MaxIterReached,"maximum number of %s steps reached."%iter_max
761    
762          p=defect.derivative(F0,x0,v[iter], v_is_normalised=True)          p=defect.derivative(F0,x0,v[iter], v_is_normalised=True)
763      v.append(p)          v.append(p)
764    
765      v_norm1=defect.norm(v[iter+1])          v_norm1=defect.norm(v[iter+1])
766    
767          # Modified Gram-Schmidt          # Modified Gram-Schmidt
768      for j in range(iter+1):          for j in range(iter+1):
769           h[j][iter]=defect.bilinearform(v[j],v[iter+1])                h[j,iter]=defect.bilinearform(v[j],v[iter+1])
770           v[iter+1]-=h[j][iter]*v[j]              v[iter+1]-=h[j,iter]*v[j]
771          
772      h[iter+1][iter]=defect.norm(v[iter+1])          h[iter+1,iter]=defect.norm(v[iter+1])
773      v_norm2=h[iter+1][iter]          v_norm2=h[iter+1,iter]
774    
775          # Reorthogonalize if needed          # Reorthogonalize if needed
776      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)
777          for j in range(iter+1):                for j in range(iter+1):
778             hr=defect.bilinearform(v[j],v[iter+1])                  hr=defect.bilinearform(v[j],v[iter+1])
779                 h[j][iter]=h[j][iter]+hr                  h[j,iter]=h[j,iter]+hr
780                 v[iter+1] -= hr*v[j]                  v[iter+1] -= hr*v[j]
781    
782          v_norm2=defect.norm(v[iter+1])              v_norm2=defect.norm(v[iter+1])
783          h[iter+1][iter]=v_norm2              h[iter+1,iter]=v_norm2
784          #   watch out for happy breakdown          #   watch out for happy breakdown
785          if not v_norm2 == 0:          if not v_norm2 == 0:
786                  v[iter+1]=v[iter+1]/h[iter+1][iter]              v[iter+1]=v[iter+1]/h[iter+1,iter]
787    
788          #   Form and store the information for the new Givens rotation          #   Form and store the information for the new Givens rotation
789      if iter > 0 :          if iter > 0 :
790          hhat=numarray.zeros(iter+1,numarray.Float64)              hhat=numpy.zeros(iter+1,numpy.float64)
791          for i in range(iter+1) : hhat[i]=h[i][iter]              for i in range(iter+1) : hhat[i]=h[i,iter]
792          hhat=__givapp(c[0:iter],s[0:iter],hhat);              hhat=__givapp(c[0:iter],s[0:iter],hhat);
793              for i in range(iter+1) : h[i][iter]=hhat[i]              for i in range(iter+1) : h[i,iter]=hhat[i]
794    
795      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])
796    
797      if mu!=0 :          if mu!=0 :
798          c[iter]=h[iter][iter]/mu              c[iter]=h[iter,iter]/mu
799          s[iter]=-h[iter+1][iter]/mu              s[iter]=-h[iter+1,iter]/mu
800          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]
801          h[iter+1][iter]=0.0              h[iter+1,iter]=0.0
802          g[iter:iter+2]=__givapp(c[iter],s[iter],g[iter:iter+2])              gg=__givapp(c[iter],s[iter],[g[iter],g[iter+1]])
803                g[iter]=gg[0]
804                g[iter+1]=gg[1]
805    
806          # Update the residual norm          # Update the residual norm
807          rho=abs(g[iter+1])          rho=abs(g[iter+1])
808      iter+=1          iter+=1
809    
810     # At this point either iter > iter_max or rho < tol.     # At this point either iter > iter_max or rho < tol.
811     # It's time to compute x and leave.             # It's time to compute x and leave.
812     if iter > 0 :     if iter > 0 :
813       y=numarray.zeros(iter,numarray.Float64)           y=numpy.zeros(iter,numpy.float64)
814       y[iter-1] = g[iter-1] / h[iter-1][iter-1]       y[iter-1] = g[iter-1] / h[iter-1,iter-1]
815       if iter > 1 :         if iter > 1 :
816          i=iter-2            i=iter-2
817          while i>=0 :          while i>=0 :
818            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]
819            i=i-1            i=i-1
820       xhat=v[iter-1]*y[iter-1]       xhat=v[iter-1]*y[iter-1]
821       for i in range(iter-1):       for i in range(iter-1):
822      xhat += v[i]*y[i]      xhat += v[i]*y[i]
823     else :     else :
824        xhat=v[0] * 0        xhat=v[0] * 0
825    
826     if iter<iter_restart-1:     if iter<iter_restart-1:
827        stopped=iter        stopped=iter
828     else:     else:
829        stopped=-1        stopped=-1
830    
831     return xhat,stopped     return xhat,stopped
832    
833  ##############################################  def GMRES(r, Aprod, x, bilinearform, atol=0, rtol=1.e-8, iter_max=100, iter_restart=20, verbose=False,P_R=None):
834  def GMRES(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100, iter_restart=20):     """
835  ################################################     Solver for
836    
837       *Ax=b*
838    
839       with a general operator A (more details required!).
840       It uses the generalized minimum residual method (GMRES).
841    
842       The iteration is terminated if
843    
844       *|r| <= atol+rtol*|r0|*
845    
846       where *r0* is the initial residual and *|.|* is the energy norm. In fact
847    
848       *|r| = sqrt( bilinearform(r,r))*
849    
850       :param r: initial residual *r=b-Ax*. ``r`` is altered.
851       :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
852       :param x: an initial guess for the solution
853       :type x: same like ``r``
854       :param Aprod: returns the value Ax
855       :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
856                    argument ``x``. The returned object needs to be of the same
857                    type like argument ``r``.
858       :param bilinearform: inner product ``<x,r>``
859       :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
860                           type like argument ``x`` and ``r``. The returned value is
861                           a ``float``.
862       :param atol: absolute tolerance
863       :type atol: non-negative ``float``
864       :param rtol: relative tolerance
865       :type rtol: non-negative ``float``
866       :param iter_max: maximum number of iteration steps
867       :type iter_max: ``int``
868       :param iter_restart: in order to save memory the orthogonalization process
869                            is terminated after ``iter_restart`` steps and the
870                            iteration is restarted.
871       :type iter_restart: ``int``
872       :return: the solution approximation and the corresponding residual
873       :rtype: ``tuple``
874       :warning: ``r`` and ``x`` are altered.
875       """
876     m=iter_restart     m=iter_restart
877       restarted=False
878     iter=0     iter=0
879     xc=x     if rtol>0:
880          r_dot_r = bilinearform(r, r)
881          if r_dot_r<0: raise NegativeNorm,"negative norm."
882          atol2=atol+rtol*math.sqrt(r_dot_r)
883          if verbose: print "GMRES: norm of right hand side = %e (absolute tolerance = %e)"%(math.sqrt(r_dot_r), atol2)
884       else:
885          atol2=atol
886          if verbose: print "GMRES: absolute tolerance = %e"%atol2
887       if atol2<=0:
888          raise ValueError,"Non-positive tolarance."
889    
890     while True:     while True:
891        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
892        xc,stopped=__GMRESm(b*1, Aprod, Msolve, bilinearform, stoppingcriterium, x=xc*1, iter_max=iter_max-iter, iter_restart=m)        if restarted:
893             r2 = r-Aprod(x-x2)
894          else:
895             r2=1*r
896          x2=x*1.
897          x,stopped=_GMRESm(r2, Aprod, x, bilinearform, atol2, iter_max=iter_max-iter, iter_restart=m, verbose=verbose,P_R=P_R)
898          iter+=iter_restart
899        if stopped: break        if stopped: break
900        iter+=iter_restart            if verbose: print "GMRES: restart."
901     return xc        restarted=True
902       if verbose: print "GMRES: tolerance has been reached."
903       return x
904    
905  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):
906     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)  
907    
908     if x==None:     h=numpy.zeros((iter_restart+1,iter_restart),numpy.float64)
909        x=0*b     c=numpy.zeros(iter_restart,numpy.float64)
910     else:     s=numpy.zeros(iter_restart,numpy.float64)
911        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)  
912     v=[]     v=[]
913    
914       r_dot_r = bilinearform(r, r)
915       if r_dot_r<0: raise NegativeNorm,"negative norm."
916     rho=math.sqrt(r_dot_r)     rho=math.sqrt(r_dot_r)
917      
918     v.append(r/rho)     v.append(r/rho)
919     g[0]=rho     g[0]=rho
920    
921     while not (stoppingcriterium(rho,norm_b) or iter==iter_restart-1):     if verbose: print "GMRES: initial residual %e (absolute tolerance = %e)"%(rho,atol)
922       while not (rho<=atol or iter==iter_restart):
923    
924      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
925    
926      p=Msolve(Aprod(v[iter]))          if P_R!=None:
927                p=Aprod(P_R(v[iter]))
928            else:
929            p=Aprod(v[iter])
930      v.append(p)      v.append(p)
931    
932      v_norm1=math.sqrt(bilinearform(v[iter+1], v[iter+1]))        v_norm1=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
933    
934    # Modified Gram-Schmidt
935        for j in range(iter+1):
936          h[j,iter]=bilinearform(v[j],v[iter+1])
937          v[iter+1]-=h[j,iter]*v[j]
938    
939  # Modified Gram-Schmidt      h[iter+1,iter]=math.sqrt(bilinearform(v[iter+1],v[iter+1]))
940      for j in range(iter+1):      v_norm2=h[iter+1,iter]
       h[j][iter]=bilinearform(v[j],v[iter+1])    
       v[iter+1]-=h[j][iter]*v[j]  
         
     h[iter+1][iter]=math.sqrt(bilinearform(v[iter+1],v[iter+1]))  
     v_norm2=h[iter+1][iter]  
941    
942  # Reorthogonalize if needed  # Reorthogonalize if needed
943      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)
944       for j in range(iter+1):         for j in range(iter+1):
945          hr=bilinearform(v[j],v[iter+1])          hr=bilinearform(v[j],v[iter+1])
946              h[j][iter]=h[j][iter]+hr              h[j,iter]=h[j,iter]+hr
947              v[iter+1] -= hr*v[j]              v[iter+1] -= hr*v[j]
948    
949       v_norm2=math.sqrt(bilinearform(v[iter+1], v[iter+1]))         v_norm2=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
950       h[iter+1][iter]=v_norm2       h[iter+1,iter]=v_norm2
951    
952  #   watch out for happy breakdown  #   watch out for happy breakdown
953          if not v_norm2 == 0:          if not v_norm2 == 0:
954           v[iter+1]=v[iter+1]/h[iter+1][iter]           v[iter+1]=v[iter+1]/h[iter+1,iter]
955    
956  #   Form and store the information for the new Givens rotation  #   Form and store the information for the new Givens rotation
957      if iter > 0 :      if iter > 0: h[:iter+1,iter]=__givapp(c[:iter],s[:iter],h[:iter+1,iter])
958          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])  
959    
960      if mu!=0 :      if mu!=0 :
961          c[iter]=h[iter][iter]/mu          c[iter]=h[iter,iter]/mu
962          s[iter]=-h[iter+1][iter]/mu          s[iter]=-h[iter+1,iter]/mu
963          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]
964          h[iter+1][iter]=0.0          h[iter+1,iter]=0.0
965          g[iter:iter+2]=__givapp(c[iter],s[iter],g[iter:iter+2])                  gg=__givapp(c[iter],s[iter],[g[iter],g[iter+1]])
966                    g[iter]=gg[0]
967                    g[iter+1]=gg[1]
968  # Update the residual norm  # Update the residual norm
969                  
970          rho=abs(g[iter+1])          rho=abs(g[iter+1])
971            if verbose: print "GMRES: iteration step %s: residual %e"%(iter,rho)
972      iter+=1      iter+=1
973    
974  # At this point either iter > iter_max or rho < tol.  # At this point either iter > iter_max or rho < tol.
975  # It's time to compute x and leave.          # It's time to compute x and leave.
976    
977     if iter > 0 :     if verbose: print "GMRES: iteration stopped after %s step."%iter
978       y=numarray.zeros(iter,numarray.Float64)         if iter > 0 :
979       y[iter-1] = g[iter-1] / h[iter-1][iter-1]       y=numpy.zeros(iter,numpy.float64)
980       if iter > 1 :         y[iter-1] = g[iter-1] / h[iter-1,iter-1]
981          i=iter-2         if iter > 1 :
982            i=iter-2
983          while i>=0 :          while i>=0 :
984            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]
985            i=i-1            i=i-1
986       xhat=v[iter-1]*y[iter-1]       xhat=v[iter-1]*y[iter-1]
987       for i in range(iter-1):       for i in range(iter-1):
988      xhat += v[i]*y[i]      xhat += v[i]*y[i]
989     else : xhat=v[0]     else:
990         xhat=v[0] * 0
991     x += xhat     if P_R!=None:
992     if iter<iter_restart-1:        x += P_R(xhat)
993        stopped=True     else:
994     else:        x += xhat
995       if iter<iter_restart-1:
996          stopped=True
997       else:
998        stopped=False        stopped=False
999    
1000     return x,stopped     return x,stopped
1001    
1002  #################################################  def MINRES(r, Aprod, x, Msolve, bilinearform, atol=0, rtol=1.e-8, iter_max=100):
1003  def MINRES(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100):      """
1004  #################################################      Solver for
1005      #  
1006      #  minres solves the system of linear equations Ax = b      *Ax=b*
1007      #  where A is a symmetric matrix (possibly indefinite or singular)  
1008      #  and b is a given vector.      with a symmetric and positive definite operator A (more details required!).
1009      #        It uses the minimum residual method (MINRES) with preconditioner M
1010      #  "A" may be a dense or sparse matrix (preferably sparse!)      providing an approximation of A.
1011      #  or the name of a function such that  
1012      #               y = A(x)      The iteration is terminated if
1013      #  returns the product y = Ax for any given vector x.  
1014      #      *|r| <= atol+rtol*|r0|*
1015      #  "M" defines a positive-definite preconditioner M = C C'.  
1016      #  "M" may be a dense or sparse matrix (preferably sparse!)      where *r0* is the initial residual and *|.|* is the energy norm. In fact
1017      #  or the name of a function such that  
1018      #  solves the system My = x for any given vector x.      *|r| = sqrt( bilinearform(Msolve(r),r))*
1019      #  
1020      #      For details on the preconditioned conjugate gradient method see the book:
1021        
1022        I{Templates for the Solution of Linear Systems by R. Barrett, M. Berry,
1023        T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo,
1024        C. Romine, and H. van der Vorst}.
1025    
1026        :param r: initial residual *r=b-Ax*. ``r`` is altered.
1027        :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1028        :param x: an initial guess for the solution
1029        :type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1030        :param Aprod: returns the value Ax
1031        :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
1032                     argument ``x``. The returned object needs to be of the same
1033                     type like argument ``r``.
1034        :param Msolve: solves Mx=r
1035        :type Msolve: function ``Msolve(r)`` where ``r`` is of the same type like
1036                      argument ``r``. The returned object needs to be of the same
1037                      type like argument ``x``.
1038        :param bilinearform: inner product ``<x,r>``
1039        :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
1040                            type like argument ``x`` and ``r`` is. The returned value
1041                            is a ``float``.
1042        :param atol: absolute tolerance
1043        :type atol: non-negative ``float``
1044        :param rtol: relative tolerance
1045        :type rtol: non-negative ``float``
1046        :param iter_max: maximum number of iteration steps
1047        :type iter_max: ``int``
1048        :return: the solution approximation and the corresponding residual
1049        :rtype: ``tuple``
1050        :warning: ``r`` and ``x`` are altered.
1051        """
1052      #------------------------------------------------------------------      #------------------------------------------------------------------
1053      # Set up y and v for the first Lanczos vector v1.      # Set up y and v for the first Lanczos vector v1.
1054      # y  =  beta1 P' v1,  where  P = C**(-1).      # y  =  beta1 P' v1,  where  P = C**(-1).
1055      # v is really P' v1.      # v is really P' v1.
1056      #------------------------------------------------------------------      #------------------------------------------------------------------
1057      if x==None:      r1    = r
1058        x=0*b      y = Msolve(r)
1059      else:      beta1 = bilinearform(y,r)
       b += (-1)*Aprod(x)  
1060    
     r1    = b  
     y = Msolve(b)  
     beta1 = bilinearform(y,b)  
   
1061      if beta1< 0: raise NegativeNorm,"negative norm."      if beta1< 0: raise NegativeNorm,"negative norm."
1062    
1063      #  If b = 0 exactly, stop with x = 0.      #  If r = 0 exactly, stop with x
1064      if beta1==0: return x*0.      if beta1==0: return x
1065    
1066      if beta1> 0:      if beta1> 0: beta1  = math.sqrt(beta1)
       beta1  = math.sqrt(beta1)        
1067    
1068      #------------------------------------------------------------------      #------------------------------------------------------------------
1069      # Initialize quantities.      # Initialize quantities.
# Line 1008  def MINRES(b, Aprod, Msolve, bilinearfor Line 1083  def MINRES(b, Aprod, Msolve, bilinearfor
1083      ynorm2 = 0      ynorm2 = 0
1084      cs     = -1      cs     = -1
1085      sn     = 0      sn     = 0
1086      w      = b*0.      w      = r*0.
1087      w2     = b*0.      w2     = r*0.
1088      r2     = r1      r2     = r1
1089      eps    = 0.0001      eps    = 0.0001
1090    
1091      #---------------------------------------------------------------------      #---------------------------------------------------------------------
1092      # Main iteration loop.      # Main iteration loop.
1093      # --------------------------------------------------------------------      # --------------------------------------------------------------------
1094      while not stoppingcriterium(rnorm,Anorm*ynorm,'MINRES'):    #  checks ||r|| < (||A|| ||x||) * TOL      while not rnorm<=atol+rtol*Anorm*ynorm:    #  checks ||r|| < (||A|| ||x||) * TOL
1095    
1096      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
1097          iter    = iter  +  1          iter    = iter  +  1
# Line 1035  def MINRES(b, Aprod, Msolve, bilinearfor Line 1110  def MINRES(b, Aprod, Msolve, bilinearfor
1110          #-----------------------------------------------------------------          #-----------------------------------------------------------------
1111          s = 1/beta                 # Normalize previous vector (in y).          s = 1/beta                 # Normalize previous vector (in y).
1112          v = s*y                    # v = vk if P = I          v = s*y                    # v = vk if P = I
1113        
1114          y      = Aprod(v)          y      = Aprod(v)
1115        
1116          if iter >= 2:          if iter >= 2:
1117            y = y - (beta/oldb)*r1            y = y - (beta/oldb)*r1
1118    
1119          alfa   = bilinearform(v,y)              # alphak          alfa   = bilinearform(v,y)              # alphak
1120          y      += (- alfa/beta)*r2          y      += (- alfa/beta)*r2
1121          r1     = r2          r1     = r2
1122          r2     = y          r2     = y
1123          y = Msolve(r2)          y = Msolve(r2)
# Line 1052  def MINRES(b, Aprod, Msolve, bilinearfor Line 1127  def MINRES(b, Aprod, Msolve, bilinearfor
1127    
1128          beta   = math.sqrt( beta )          beta   = math.sqrt( beta )
1129          tnorm2 = tnorm2 + alfa*alfa + oldb*oldb + beta*beta          tnorm2 = tnorm2 + alfa*alfa + oldb*oldb + beta*beta
1130            
1131          if iter==1:                 # Initialize a few things.          if iter==1:                 # Initialize a few things.
1132            gmax   = abs( alfa )      # alpha1            gmax   = abs( alfa )      # alpha1
1133            gmin   = gmax             # alpha1            gmin   = gmax             # alpha1
# Line 1060  def MINRES(b, Aprod, Msolve, bilinearfor Line 1135  def MINRES(b, Aprod, Msolve, bilinearfor
1135          # Apply previous rotation Qk-1 to get          # Apply previous rotation Qk-1 to get
1136          #   [deltak epslnk+1] = [cs  sn][dbark    0   ]          #   [deltak epslnk+1] = [cs  sn][dbark    0   ]
1137          #   [gbar k dbar k+1]   [sn -cs][alfak betak+1].          #   [gbar k dbar k+1]   [sn -cs][alfak betak+1].
1138        
1139          oldeps = epsln          oldeps = epsln
1140          delta  = cs * dbar  +  sn * alfa  # delta1 = 0         deltak          delta  = cs * dbar  +  sn * alfa  # delta1 = 0         deltak
1141          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 1145  def MINRES(b, Aprod, Msolve, bilinearfor
1145          # Compute the next plane rotation Qk          # Compute the next plane rotation Qk
1146    
1147          gamma  = math.sqrt(gbar*gbar+beta*beta)  # gammak          gamma  = math.sqrt(gbar*gbar+beta*beta)  # gammak
1148          gamma  = max(gamma,eps)          gamma  = max(gamma,eps)
1149          cs     = gbar / gamma             # ck          cs     = gbar / gamma             # ck
1150          sn     = beta / gamma             # sk          sn     = beta / gamma             # sk
1151          phi    = cs * phibar              # phik          phi    = cs * phibar              # phik
# Line 1078  def MINRES(b, Aprod, Msolve, bilinearfor Line 1153  def MINRES(b, Aprod, Msolve, bilinearfor
1153    
1154          # Update  x.          # Update  x.
1155    
1156          denom = 1/gamma          denom = 1/gamma
1157          w1    = w2          w1    = w2
1158          w2    = w          w2    = w
1159          w     = (v - oldeps*w1 - delta*w2) * denom          w     = (v - oldeps*w1 - delta*w2) * denom
1160          x     +=  phi*w          x     +=  phi*w
1161    
# Line 1095  def MINRES(b, Aprod, Msolve, bilinearfor Line 1170  def MINRES(b, Aprod, Msolve, bilinearfor
1170    
1171          # Estimate various norms and test for convergence.          # Estimate various norms and test for convergence.
1172    
1173          Anorm  = math.sqrt( tnorm2 )          Anorm  = math.sqrt( tnorm2 )
1174          ynorm  = math.sqrt( ynorm2 )          ynorm  = math.sqrt( ynorm2 )
1175    
1176          rnorm  = phibar          rnorm  = phibar
1177    
1178      return x      return x
1179    
1180  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):
1181      """
1182      Solver for
1183    
1184      *Ax=b*
1185    
1186  # TFQMR solver for linear systems    with a general operator A (more details required!).
1187  #    It uses the Transpose-Free Quasi-Minimal Residual method (TFQMR).
 #  
 # initialization  
 #  
   errtol = math.sqrt(bilinearform(b,b))  
   norm_b=errtol  
   kmax  = iter_max  
   error = []  
   
   if math.sqrt(bilinearform(x,x)) != 0.0:  
     r = b - Aprod(x)  
   else:  
     r = b  
1188    
1189    r=Msolve(r)    The iteration is terminated if
1190    
1191      *|r| <= atol+rtol*|r0|*
1192    
1193      where *r0* is the initial residual and *|.|* is the energy norm. In fact
1194    
1195      *|r| = sqrt( bilinearform(r,r))*
1196    
1197      :param r: initial residual *r=b-Ax*. ``r`` is altered.
1198      :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1199      :param x: an initial guess for the solution
1200      :type x: same like ``r``
1201      :param Aprod: returns the value Ax
1202      :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
1203                   argument ``x``. The returned object needs to be of the same type
1204                   like argument ``r``.
1205      :param bilinearform: inner product ``<x,r>``
1206      :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
1207                          type like argument ``x`` and ``r``. The returned value is
1208                          a ``float``.
1209      :param atol: absolute tolerance
1210      :type atol: non-negative ``float``
1211      :param rtol: relative tolerance
1212      :type rtol: non-negative ``float``
1213      :param iter_max: maximum number of iteration steps
1214      :type iter_max: ``int``
1215      :rtype: ``tuple``
1216      :warning: ``r`` and ``x`` are altered.
1217      """
1218    u1=0    u1=0
1219    u2=0    u2=0
1220    y1=0    y1=0
1221    y2=0    y2=0
1222    
1223    w = r    w = r
1224    y1 = r    y1 = r
1225    iter = 0    iter = 0
1226    d = 0    d = 0
1227        v = Aprod(y1)
   v = Msolve(Aprod(y1))  
1228    u1 = v    u1 = v
1229      
1230    theta = 0.0;    theta = 0.0;
1231    eta = 0.0;    eta = 0.0;
1232    tau = math.sqrt(bilinearform(r,r))    rho=bilinearform(r,r)
1233    error = [ error, tau ]    if rho < 0: raise NegativeNorm,"negative norm."
1234    rho = tau * tau    tau = math.sqrt(rho)
1235    m=1    norm_r0=tau
1236  #    while tau>atol+rtol*norm_r0:
 #  TFQMR iteration  
 #  
 #  while ( iter < kmax-1 ):  
     
   while not stoppingcriterium(tau*math.sqrt ( m + 1 ),norm_b,'TFQMR'):  
1237      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
1238    
1239      sigma = bilinearform(r,v)      sigma = bilinearform(r,v)
1240        if sigma == 0.0: raise IterationBreakDown,'TFQMR breakdown, sigma=0'
     if ( sigma == 0.0 ):  
       raise 'TFQMR breakdown, sigma=0'  
       
1241    
1242      alpha = rho / sigma      alpha = rho / sigma
1243    
# Line 1162  def TFQMR(b, Aprod, Msolve, bilinearform Line 1247  def TFQMR(b, Aprod, Msolve, bilinearform
1247  #  #
1248        if ( j == 1 ):        if ( j == 1 ):
1249          y2 = y1 - alpha * v          y2 = y1 - alpha * v
1250          u2 = Msolve(Aprod(y2))          u2 = Aprod(y2)
1251    
1252        m = 2 * (iter+1) - 2 + (j+1)        m = 2 * (iter+1) - 2 + (j+1)
1253        if j==0:        if j==0:
1254           w = w - alpha * u1           w = w - alpha * u1
1255           d = y1 + ( theta * theta * eta / alpha ) * d           d = y1 + ( theta * theta * eta / alpha ) * d
1256        if j==1:        if j==1:
# Line 1180  def TFQMR(b, Aprod, Msolve, bilinearform Line 1265  def TFQMR(b, Aprod, Msolve, bilinearform
1265  #  #
1266  #  Try to terminate the iteration at each pass through the loop  #  Try to terminate the iteration at each pass through the loop
1267  #  #
1268       # 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'  
       
1269    
1270      rhon = bilinearform(r,w)      rhon = bilinearform(r,w)
1271      beta = rhon / rho;      beta = rhon / rho;
1272      rho = rhon;      rho = rhon;
1273      y1 = w + beta * y2;      y1 = w + beta * y2;
1274      u1 = Msolve(Aprod(y1))      u1 = Aprod(y1)
1275      v = u1 + beta * ( u2 + beta * v )      v = u1 + beta * ( u2 + beta * v )
1276      error = [ error, tau ]  
1277      total_iters = iter      iter += 1
       
     iter = iter + 1  
1278    
1279    return x    return x
1280    
# Line 1208  def TFQMR(b, Aprod, Msolve, bilinearform Line 1283  def TFQMR(b, Aprod, Msolve, bilinearform
1283    
1284  class ArithmeticTuple(object):  class ArithmeticTuple(object):
1285     """     """
1286     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
1287       ArithmeticTuple and ``a`` is a float.
1288    
1289     example of usage:     Example of usage::
1290    
1291     from esys.escript import Data         from esys.escript import Data
1292     from numarray import array         from numpy import array
1293     a=Data(...)         a=Data(...)
1294     b=array([1.,4.])         b=array([1.,4.])
1295     x=ArithmeticTuple(a,b)         x=ArithmeticTuple(a,b)
1296     y=5.*x         y=5.*x
1297    
1298     """     """
1299     def __init__(self,*args):     def __init__(self,*args):
1300         """         """
1301         initialize object with elements args.         Initializes object with elements ``args``.
1302    
1303         @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
1304                        scaling (x=a*y)
1305         """         """
1306         self.__items=list(args)         self.__items=list(args)
1307    
1308     def __len__(self):     def __len__(self):
1309         """         """
1310         number of items         Returns the number of items.
1311    
1312         @return: number of items         :return: number of items
1313         @rtype: C{int}         :rtype: ``int``
1314         """         """
1315         return len(self.__items)         return len(self.__items)
1316    
1317     def __getitem__(self,index):     def __getitem__(self,index):
1318         """         """
1319         get an item         Returns item at specified position.
1320    
1321         @param index: item to be returned         :param index: index of item to be returned
1322         @type index: C{int}         :type index: ``int``
1323         @return: item with index C{index}         :return: item with index ``index``
1324         """         """
1325         return self.__items.__getitem__(index)         return self.__items.__getitem__(index)
1326    
1327     def __mul__(self,other):     def __mul__(self,other):
1328         """         """
1329         scaling from the right         Scales by ``other`` from the right.
1330    
1331         @param other: scaling factor         :param other: scaling factor
1332         @type other: C{float}         :type other: ``float``
1333         @return: itemwise self*other         :return: itemwise self*other
1334         @rtype: L{ArithmeticTuple}         :rtype: `ArithmeticTuple`
1335         """         """
1336         out=[]         out=[]
1337         try:           try:
1338             l=len(other)             l=len(other)
1339             if l!=len(self):             if l!=len(self):
1340                 raise ValueError,"length of of arguments don't match."                 raise ValueError,"length of arguments don't match."
1341             for i in range(l): out.append(self[i]*other[i])             for i in range(l): out.append(self[i]*other[i])
1342         except TypeError:         except TypeError:
1343         for i in range(len(self)): out.append(self[i]*other)             for i in range(len(self)): out.append(self[i]*other)
1344         return ArithmeticTuple(*tuple(out))         return ArithmeticTuple(*tuple(out))
1345    
1346     def __rmul__(self,other):     def __rmul__(self,other):
1347         """         """
1348         scaling from the left         Scales by ``other`` from the left.
1349    
1350         @param other: scaling factor         :param other: scaling factor
1351         @type other: C{float}         :type other: ``float``
1352         @return: itemwise other*self         :return: itemwise other*self
1353         @rtype: L{ArithmeticTuple}         :rtype: `ArithmeticTuple`
1354         """         """
1355         out=[]         out=[]
1356         try:           try:
1357             l=len(other)             l=len(other)
1358             if l!=len(self):             if l!=len(self):
1359                 raise ValueError,"length of of arguments don't match."                 raise ValueError,"length of arguments don't match."
1360             for i in range(l): out.append(other[i]*self[i])             for i in range(l): out.append(other[i]*self[i])
1361         except TypeError:         except TypeError:
1362         for i in range(len(self)): out.append(other*self[i])             for i in range(len(self)): out.append(other*self[i])
1363         return ArithmeticTuple(*tuple(out))         return ArithmeticTuple(*tuple(out))
1364    
1365     def __div__(self,other):     def __div__(self,other):
1366         """         """
1367         dividing from the right         Scales by (1/``other``) from the right.
1368    
1369         @param other: scaling factor         :param other: scaling factor
1370         @type other: C{float}         :type other: ``float``
1371         @return: itemwise self/other         :return: itemwise self/other
1372         @rtype: L{ArithmeticTuple}         :rtype: `ArithmeticTuple`
1373         """         """
1374         return self*(1/other)         return self*(1/other)
1375    
1376     def __rdiv__(self,other):     def __rdiv__(self,other):
1377         """         """
1378         dividing from the left         Scales by (1/``other``) from the left.
1379    
1380         @param other: scaling factor         :param other: scaling factor
1381         @type other: C{float}         :type other: ``float``
1382         @return: itemwise other/self         :return: itemwise other/self
1383         @rtype: L{ArithmeticTuple}         :rtype: `ArithmeticTuple`
1384         """         """
1385         out=[]         out=[]
1386         try:           try:
1387             l=len(other)             l=len(other)
1388             if l!=len(self):             if l!=len(self):
1389                 raise ValueError,"length of of arguments don't match."                 raise ValueError,"length of arguments don't match."
1390             for i in range(l): out.append(other[i]/self[i])             for i in range(l): out.append(other[i]/self[i])
1391         except TypeError:         except TypeError:
1392         for i in range(len(self)): out.append(other/self[i])             for i in range(len(self)): out.append(other/self[i])
1393         return ArithmeticTuple(*tuple(out))         return ArithmeticTuple(*tuple(out))
1394      
1395     def __iadd__(self,other):     def __iadd__(self,other):
1396         """         """
1397         in-place add of other to self         Inplace addition of ``other`` to self.
1398    
1399         @param other: increment         :param other: increment
1400         @type other: C{ArithmeticTuple}         :type other: ``ArithmeticTuple``
1401         """         """
1402         if len(self) != len(other):         if len(self) != len(other):
1403             raise ValueError,"tuple length must match."             raise ValueError,"tuple lengths must match."
1404         for i in range(len(self)):         for i in range(len(self)):
1405             self.__items[i]+=other[i]             self.__items[i]+=other[i]
1406         return self         return self
1407    
1408     def __add__(self,other):     def __add__(self,other):
1409         """         """
1410         add other to self         Adds ``other`` to self.
1411    
1412         @param other: increment         :param other: increment
1413         @type other: C{ArithmeticTuple}         :type other: ``ArithmeticTuple``
1414         """         """
1415         out=[]         out=[]
1416         try:           try:
1417             l=len(other)             l=len(other)
1418             if l!=len(self):             if l!=len(self):
1419                 raise ValueError,"length of of arguments don't match."                 raise ValueError,"length of arguments don't match."
1420             for i in range(l): out.append(self[i]+other[i])             for i in range(l): out.append(self[i]+other[i])
1421         except TypeError:         except TypeError:
1422         for i in range(len(self)): out.append(self[i]+other)             for i in range(len(self)): out.append(self[i]+other)
1423         return ArithmeticTuple(*tuple(out))         return ArithmeticTuple(*tuple(out))
1424    
1425     def __sub__(self,other):     def __sub__(self,other):
1426         """         """
1427         subtract other from self         Subtracts ``other`` from self.
1428    
1429         @param other: increment         :param other: decrement
1430         @type other: C{ArithmeticTuple}         :type other: ``ArithmeticTuple``
1431         """         """
1432         out=[]         out=[]
1433         try:           try:
1434             l=len(other)             l=len(other)
1435             if l!=len(self):             if l!=len(self):
1436                 raise ValueError,"length of of arguments don't match."                 raise ValueError,"length of arguments don't match."
1437             for i in range(l): out.append(self[i]-other[i])             for i in range(l): out.append(self[i]-other[i])
1438         except TypeError:         except TypeError:
1439         for i in range(len(self)): out.append(self[i]-other)             for i in range(len(self)): out.append(self[i]-other)
1440         return ArithmeticTuple(*tuple(out))         return ArithmeticTuple(*tuple(out))
1441      
1442     def __isub__(self,other):     def __isub__(self,other):
1443         """         """
1444         subtract other from self         Inplace subtraction of ``other`` from self.
1445    
1446         @param other: increment         :param other: decrement
1447         @type other: C{ArithmeticTuple}         :type other: ``ArithmeticTuple``
1448         """         """
1449         if len(self) != len(other):         if len(self) != len(other):
1450             raise ValueError,"tuple length must match."             raise ValueError,"tuple length must match."
# Line 1377  class ArithmeticTuple(object): Line 1454  class ArithmeticTuple(object):
1454    
1455     def __neg__(self):     def __neg__(self):
1456         """         """
1457         negate         Negates values.
   
1458         """         """
1459         out=[]         out=[]
1460         for i in range(len(self)):         for i in range(len(self)):
# Line 1388  class ArithmeticTuple(object): Line 1464  class ArithmeticTuple(object):
1464    
1465  class HomogeneousSaddlePointProblem(object):  class HomogeneousSaddlePointProblem(object):
1466        """        """
1467        This provides a framwork for solving linear homogeneous saddle point problem of the form        This class provides a framework for solving linear homogeneous saddle
1468          point problems of the form::
              Av+B^*p=f  
              Bv    =0  
1469    
1470        for the unknowns v and p and given operators A and B and given right hand side f.            *Av+B^*p=f*
1471        B^* is the adjoint operator of B is the given inner product.            *Bv     =0*
1472    
1473          for the unknowns *v* and *p* and given operators *A* and *B* and
1474          given right hand side *f*. *B^** is the adjoint operator of *B*.
1475        """        """
1476        def __init__(self,**kwargs):        def __init__(self, adaptSubTolerance=True, **kwargs):
1477        """
1478        initializes the saddle point problem
1479        
1480        :param adaptSubTolerance: If True the tolerance for subproblem is set automatically.
1481        :type adaptSubTolerance: ``bool``
1482        """
1483          self.setTolerance()          self.setTolerance()
1484          self.setToleranceReductionFactor()          self.setAbsoluteTolerance()
1485        self.__adaptSubTolerance=adaptSubTolerance
1486          #=============================================================
1487        def initialize(self):        def initialize(self):
1488          """          """
1489          initialize the problem (overwrite)          Initializes the problem (overwrite).
1490          """          """
1491          pass          pass
1492        def B(self,v):  
1493          def inner_pBv(self,p,Bv):
1494           """           """
1495           returns Bv (overwrite)           Returns inner product of element p and Bv (overwrite).
          @rtype: equal to the type of p  
1496    
1497           @note: boundary conditions on p should be zero!           :param p: a pressure increment
1498             :param Bv: a residual
1499             :return: inner product of element p and Bv
1500             :rtype: ``float``
1501             :note: used if PCG is applied.
1502           """           """
1503           pass           raise NotImplementedError,"no inner product for p and Bv implemented."
1504    
1505        def inner(self,p0,p1):        def inner_p(self,p0,p1):
1506           """           """
1507           returns inner product of two element p0 and p1  (overwrite)           Returns inner product of p0 and p1 (overwrite).
           
          @type p0: equal to the type of p  
          @type p1: equal to the type of p  
          @rtype: C{float}  
1508    
1509           @rtype: equal to the type of p           :param p0: a pressure
1510             :param p1: a pressure
1511             :return: inner product of p0 and p1
1512             :rtype: ``float``
1513           """           """
1514           pass           raise NotImplementedError,"no inner product for p implemented."
1515      
1516          def norm_v(self,v):
1517             """
1518             Returns the norm of v (overwrite).
1519    
1520        def solve_A(self,u,p):           :param v: a velovity
1521             :return: norm of v
1522             :rtype: non-negative ``float``
1523             """
1524             raise NotImplementedError,"no norm of v implemented."
1525          def getV(self, p, v0):
1526           """           """
1527           solves Av=f-Au-B^*p with accuracy self.getReducedTolerance() (overwrite)           return the value for v for a given p (overwrite)
1528    
1529           @rtype: equal to the type of v           :param p: a pressure
1530           @note: boundary conditions on v should be zero!           :param v0: a initial guess for the value v to return.
1531             :return: v given as *v= A^{-1} (f-B^*p)*
1532           """           """
1533           pass           raise NotImplementedError,"no v calculation implemented."
1534    
1535        def solve_prec(self,p):          
1536          def Bv(self,v):
1537            """
1538            Returns Bv (overwrite).
1539    
1540            :rtype: equal to the type of p
1541            :note: boundary conditions on p should be zero!
1542            """
1543            raise NotImplementedError, "no operator B implemented."
1544    
1545          def norm_Bv(self,Bv):
1546            """
1547            Returns the norm of Bv (overwrite).
1548    
1549            :rtype: equal to the type of p
1550            :note: boundary conditions on p should be zero!
1551            """
1552            raise NotImplementedError, "no norm of Bv implemented."
1553    
1554          def solve_AinvBt(self,p):
1555           """           """
1556           provides a preconditioner for BA^{-1}B^* with accuracy self.getReducedTolerance() (overwrite)           Solves *Av=B^*p* with accuracy `self.getSubProblemTolerance()`
1557             (overwrite).
1558    
1559           @rtype: equal to the type of p           :param p: a pressure increment
1560             :return: the solution of *Av=B^*p*
1561             :note: boundary conditions on v should be zero!
1562           """           """
1563           pass           raise NotImplementedError,"no operator A implemented."
1564    
1565        def stoppingcriterium(self,Bv,v,p):        def solve_prec(self,Bv):
1566           """           """
1567           returns a True if iteration is terminated. (overwrite)           Provides a preconditioner for *(BA^{-1}B^ * )* applied to Bv with accuracy
1568             `self.getSubProblemTolerance()` (overwrite).
1569    
1570           @rtype: C{bool}           :rtype: equal to the type of p
1571             :note: boundary conditions on p should be zero!
1572             """
1573             raise NotImplementedError,"no preconditioner for Schur complement implemented."
1574          def setSubProblemTolerance(self):
1575             """
1576         Updates the tolerance for subproblems
1577         :note: method is typically the method is overwritten.
1578           """           """
1579           pass           pass
1580                      #=============================================================
1581        def __inner(self,p,r):        def __Aprod_PCG(self,p):
1582           return self.inner(p,r[1])            dv=self.solve_AinvBt(p)
1583              return ArithmeticTuple(dv,self.Bv(dv))
       def __inner_p(self,p1,p2):  
          return self.inner(p1,p2)  
         
       def __inner_a(self,a1,a2):  
          return self.inner_a(a1,a2)  
   
       def __inner_a1(self,a1,a2):  
          return self.inner(a1[1],a2[1])  
1584    
1585        def __stoppingcriterium(self,norm_r,r,p):        def __inner_PCG(self,p,r):
1586            return self.stoppingcriterium(r[1],r[0],p)           return self.inner_pBv(p,r[1])
1587    
1588        def __stoppingcriterium2(self,norm_r,norm_b,solver='GMRES',TOL=None):        def __Msolve_PCG(self,r):
1589            return self.stoppingcriterium2(norm_r,norm_b,solver,TOL)            return self.solve_prec(r[1])
1590          #=============================================================
1591          def __Aprod_GMRES(self,p):
1592              return self.solve_prec(self.Bv(self.solve_AinvBt(p)))
1593          def __inner_GMRES(self,p0,p1):
1594             return self.inner_p(p0,p1)
1595    
1596          #=============================================================
1597          def norm_p(self,p):
1598              """
1599              calculates the norm of ``p``
1600              
1601              :param p: a pressure
1602              :return: the norm of ``p`` using the inner product for pressure
1603              :rtype: ``float``
1604              """
1605              f=self.inner_p(p,p)
1606              if f<0: raise ValueError,"negative pressure norm."
1607              return math.sqrt(f)
1608          def adaptSubTolerance(self):
1609          """
1610          Returns True if tolerance adaption for subproblem is choosen.
1611          """
1612              return self.__adaptSubTolerance
1613          
1614          def solve(self,v,p,max_iter=20, verbose=False, usePCG=True, iter_restart=20, max_correction_steps=10):
1615             """
1616             Solves the saddle point problem using initial guesses v and p.
1617    
1618        def setTolerance(self,tolerance=1.e-8):           :param v: initial guess for velocity
1619                self.__tol=tolerance           :param p: initial guess for pressure
1620        def getTolerance(self):           :type v: `Data`
1621                return self.__tol           :type p: `Data`
1622        def setToleranceReductionFactor(self,reduction=0.01):           :param usePCG: indicates the usage of the PCG rather than GMRES scheme.
1623                self.__reduction=reduction           :param max_iter: maximum number of iteration steps per correction
1624        def getSubProblemTolerance(self):                            attempt
1625                return self.__reduction*self.getTolerance()           :param verbose: if True, shows information on the progress of the
1626                             saddlepoint problem solver.
1627             :param iter_restart: restart the iteration after ``iter_restart`` steps
1628                                  (only used if useUzaw=False)
1629             :type usePCG: ``bool``
1630             :type max_iter: ``int``
1631             :type verbose: ``bool``
1632             :type iter_restart: ``int``
1633             :rtype: ``tuple`` of `Data` objects
1634             """
1635             self.verbose=verbose
1636             rtol=self.getTolerance()
1637             atol=self.getAbsoluteTolerance()
1638         if self.adaptSubTolerance(): self.setSubProblemTolerance()
1639             correction_step=0
1640             converged=False
1641             while not converged:
1642                  # calculate velocity for current pressure:
1643                  v=self.getV(p,v)
1644                  Bv=self.Bv(v)
1645                  norm_v=self.norm_v(v)
1646                  norm_Bv=self.norm_Bv(Bv)
1647                  ATOL=norm_v*rtol+atol
1648                  if self.verbose: print "HomogeneousSaddlePointProblem: norm v= %e, norm_Bv= %e, tolerance = %e."%(norm_v, norm_Bv,ATOL)
1649                  if not ATOL>0: raise ValueError,"overall absolute tolerance needs to be positive."
1650                  if norm_Bv <= ATOL:
1651                     converged=True
1652                  else:
1653                     correction_step+=1
1654                     if correction_step>max_correction_steps:
1655                          raise CorrectionFailed,"Given up after %d correction steps."%correction_step
1656                     dp=self.solve_prec(Bv)
1657                     if usePCG:
1658                       norm2=self.inner_pBv(dp,Bv)
1659                       if norm2<0: raise ValueError,"negative PCG norm."
1660                       norm2=math.sqrt(norm2)
1661                     else:
1662                       norm2=self.norm_p(dp)
1663                     ATOL_ITER=ATOL/norm_Bv*norm2*0.5
1664                     if self.verbose: print "HomogeneousSaddlePointProblem: tolerance for solver: %e"%ATOL_ITER
1665                     if usePCG:
1666                           p,v0,a_norm=PCG(ArithmeticTuple(v,Bv),self.__Aprod_PCG,p,self.__Msolve_PCG,self.__inner_PCG,atol=ATOL_ITER, rtol=0.,iter_max=max_iter, verbose=self.verbose)
1667                     else:
1668                           p=GMRES(dp,self.__Aprod_GMRES, p, self.__inner_GMRES,atol=ATOL_ITER, rtol=0.,iter_max=max_iter, iter_restart=iter_restart, verbose=self.verbose)
1669             if self.verbose: print "HomogeneousSaddlePointProblem: tolerance reached."
1670         return v,p
1671    
1672        def solve(self,v,p,max_iter=20, verbose=False, show_details=False, solver='PCG',iter_restart=20):        #========================================================================
1673                """        def setTolerance(self,tolerance=1.e-4):
1674                solves the saddle point problem using initial guesses v and p.           """
1675             Sets the relative tolerance for (v,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)  
1676    
1677                # print "RESULT div(u)=",util.Lsup(self.B(u)),util.Lsup(u)           :param tolerance: tolerance to be used
1678             :type tolerance: non-negative ``float``
1679             """
1680             if tolerance<0:
1681                 raise ValueError,"tolerance must be positive."
1682             self.__rtol=tolerance
1683    
1684            return u,p        def getTolerance(self):
1685             """
1686             Returns the relative tolerance.
1687    
1688        def __Msolve(self,r):           :return: relative tolerance
1689            return self.solve_prec(r[1])           :rtype: ``float``
1690             """
1691             return self.__rtol
1692    
1693        def __Msolve2(self,r):        def setAbsoluteTolerance(self,tolerance=0.):
1694            return self.solve_prec(r*1)           """
1695             Sets the absolute tolerance.
1696    
1697        def __Mempty(self,r):           :param tolerance: tolerance to be used
1698            return r           :type tolerance: non-negative ``float``
1699             """
1700             if tolerance<0:
1701                 raise ValueError,"tolerance must be non-negative."
1702             self.__atol=tolerance
1703    
1704          def getAbsoluteTolerance(self):
1705             """
1706             Returns the absolute tolerance.
1707    
1708        def __Aprod(self,p):           :return: absolute tolerance
1709            # return BA^-1B*p           :rtype: ``float``
1710            #solve Av =B^*p as Av =f-Az-B^*(-p)           """
1711            v=self.solve_A(self.__z,-p)           return self.__atol
           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)  
1712    
1713          def getSubProblemTolerance(self):
1714             """
1715             Sets the relative tolerance to solve the subproblem(s).
1716             """
1717             return max(200.*util.EPSILON,self.getTolerance()**2)
1718    
1719  def MaskFromBoundaryTag(domain,*tags):  def MaskFromBoundaryTag(domain,*tags):
1720     """     """
1721     creates a mask on the Solution(domain) function space which one for samples     Creates a mask on the Solution(domain) function space where the value is
1722     that touch the boundary tagged by tags.     one for samples that touch the boundary tagged by tags.
1723    
1724     usage: m=MaskFromBoundaryTag(domain,"left", "right")     Usage: m=MaskFromBoundaryTag(domain, "left", "right")
1725    
1726     @param domain: a given domain     :param domain: domain to be used
1727     @type domain: L{escript.Domain}     :type domain: `escript.Domain`
1728     @param tags: boundray tags     :param tags: boundary tags
1729     @type tags: C{str}     :type tags: ``str``
1730     @return: a mask which marks samples that are touching the boundary tagged by any of the given tags.     :return: a mask which marks samples that are touching the boundary tagged
1731     @rtype: L{escript.Data} of rank 0              by any of the given tags
1732       :rtype: `escript.Data` of rank 0
1733     """     """
1734     pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)     pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)
1735     d=escript.Scalar(0.,escript.FunctionOnBoundary(domain))     d=escript.Scalar(0.,escript.FunctionOnBoundary(domain))
1736     for t in tags: d.setTaggedValue(t,1.)     for t in tags: d.setTaggedValue(t,1.)
1737     pde.setValue(y=d)     pde.setValue(y=d)
1738     return util.whereNonZero(pde.getRightHandSide())     return util.whereNonZero(pde.getRightHandSide())
 #============================================================================================================================================  
 class SaddlePointProblem(object):  
    """  
    This implements a solver for a saddlepoint problem  
   
    M{f(u,p)=0}  
    M{g(u)=0}  
   
    for u and p. The problem is solved with an inexact Uszawa scheme for p:  
1739    
1740     M{Q_f (u^{k+1}-u^{k}) = - f(u^{k},p^{k})}  def MaskFromTag(domain,*tags):
    M{Q_g (p^{k+1}-p^{k}) =   g(u^{k+1})}  
   
    where Q_f is an approximation of the Jacobiean A_f of f with respect to u  and Q_f is an approximation of  
    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'  
    Q_g can be difficult, non-linear conjugate gradient method is applied to solve for p, so Q_g plays  
    in fact the role of a preconditioner.  
1741     """     """
1742     def __init__(self,verbose=False,*args):     Creates a mask on the Solution(domain) function space where the value is
1743         """     one for samples that touch regions tagged by tags.
        initializes the problem  
1744    
1745         @param verbose: switches on the printing out some information     Usage: m=MaskFromTag(domain, "ham")
        @type verbose: C{bool}  
        @note: this method may be overwritten by a particular saddle point problem  
        """  
        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.")  
   
    def trace(self,text):  
        """  
        prints text if verbose has been set  
   
        @param text: a text message  
        @type text: C{str}  
        """  
        if self.__verbose: print "%s: %s"%(str(self),text)  
   
    def solve_f(self,u,p,tol=1.e-8):  
        """  
        solves  
1746    
1747         A_f du = f(u,p)     :param domain: domain to be used
1748       :type domain: `escript.Domain`
1749         with tolerance C{tol} and return du. A_f is Jacobiean of f with respect to u.     :param tags: boundary tags
1750       :type tags: ``str``
1751         @param u: current approximation of u     :return: a mask which marks samples that are touching the boundary tagged
1752         @type u: L{escript.Data}              by any of the given tags
1753         @param p: current approximation of p     :rtype: `escript.Data` of rank 0
1754         @type p: L{escript.Data}     """
1755         @param tol: tolerance expected for du     pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)
1756         @type tol: C{float}     d=escript.Scalar(0.,escript.Function(domain))
1757         @return: increment du     for t in tags: d.setTaggedValue(t,1.)
1758         @rtype: L{escript.Data}     pde.setValue(Y=d)
1759         @note: this method has to be overwritten by a particular saddle point problem     return util.whereNonZero(pde.getRightHandSide())
        """  
        pass  
   
    def solve_g(self,u,tol=1.e-8):  
        """  
        solves  
   
        Q_g dp = g(u)  
   
        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.  
   
        @param u: current approximation of u  
        @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  
   
    def inner(self,p0,p1):  
        """  
        inner product of p0 and p1 approximating p. Typically this returns integrate(p0*p1)  
        @return: inner product of p0 and p1  
        @rtype: C{float}  
        """  
        pass  
1760    
    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  
1761    
         @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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