/[escript]/trunk/escript/py_src/pdetools.py
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revision 1639 by gross, Mon Jul 14 08:55:25 2008 UTC revision 2793 by gross, Tue Dec 1 06:10:10 2009 UTC
# Line 1  Line 1 
1    
2    ########################################################
3  #  #
4  # $Id$  # Copyright (c) 2003-2009 by University of Queensland
5  #  # Earth Systems Science Computational Center (ESSCC)
6  #######################################################  # http://www.uq.edu.au/esscc
7  #  #
8  #           Copyright 2003-2007 by ACceSS MNRF  # Primary Business: Queensland, Australia
9  #       Copyright 2007 by University of Queensland  # Licensed under the Open Software License version 3.0
10  #  # http://www.opensource.org/licenses/osl-3.0.php
11  #                http://esscc.uq.edu.au  #
12  #        Primary Business: Queensland, Australia  ########################################################
13  #  Licensed under the Open Software License version 3.0  
14  #     http://www.opensource.org/licenses/osl-3.0.php  __copyright__="""Copyright (c) 2003-2009 by University of Queensland
15  #  Earth Systems Science Computational Center (ESSCC)
16  #######################################################  http://www.uq.edu.au/esscc
17  #  Primary Business: Queensland, Australia"""
18    __license__="""Licensed under the Open Software License version 3.0
19    http://www.opensource.org/licenses/osl-3.0.php"""
20    __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"
 __copyright__="""  Copyright (c) 2006 by ACcESS MNRF  
                     http://www.access.edu.au  
                 Primary Business: Queensland, Australia"""  
 __license__="""Licensed under the Open Software License version 3.0  
              http://www.opensource.org/licenses/osl-3.0.php"""  
 __url__="http://www.iservo.edu.au/esys"  
 __version__="$Revision$"  
 __date__="$Date$"  
40    
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 67  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 88  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 97  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 114  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 129  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 189  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 237  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 279  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 299  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 337  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 353  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 378  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 413  class Locator: Line 422  class Locator:
422          else:          else:
423             return data             return data
424    
425    
426    def getInfLocator(arg):
427        """
428        Return a Locator for a point with the inf value over all arg.
429        """
430        if not isinstance(arg, escript.Data):
431        raise TypeError,"getInfLocator: Unknown argument type."
432        a_inf=util.inf(arg)
433        loc=util.length(arg-a_inf).minGlobalDataPoint() # This gives us the location but not coords
434        x=arg.getFunctionSpace().getX()
435        x_min=x.getTupleForGlobalDataPoint(*loc)
436        return Locator(arg.getFunctionSpace(),x_min)
437    
438    def getSupLocator(arg):
439        """
440        Return a Locator for a point with the sup value over all arg.
441        """
442        if not isinstance(arg, escript.Data):
443        raise TypeError,"getInfLocator: Unknown argument type."
444        a_inf=util.sup(arg)
445        loc=util.length(arg-a_inf).minGlobalDataPoint() # This gives us the location but not coords
446        x=arg.getFunctionSpace().getX()
447        x_min=x.getTupleForGlobalDataPoint(*loc)
448        return Locator(arg.getFunctionSpace(),x_min)
449        
450    
451  class SolverSchemeException(Exception):  class SolverSchemeException(Exception):
452     """     """
453     exceptions thrown by solvers     This is a generic exception thrown by solvers.
454     """     """
455     pass     pass
456    
457  class IndefinitePreconditioner(SolverSchemeException):  class IndefinitePreconditioner(SolverSchemeException):
458     """     """
459     the preconditioner is not positive definite.     Exception thrown if the preconditioner is not positive definite.
460     """     """
461     pass     pass
462    
463  class MaxIterReached(SolverSchemeException):  class MaxIterReached(SolverSchemeException):
464     """     """
465     maxium number of iteration steps is reached.     Exception thrown if the maximum number of iteration steps is reached.
466     """     """
467     pass     pass
468  class IterationBreakDown(SolverSchemeException):  
469    class CorrectionFailed(SolverSchemeException):
470     """     """
471     iteration scheme econouters an incurable breakdown.     Exception thrown if no convergence has been achieved in the solution
472       correction scheme.
473     """     """
474     pass     pass
475  class NegativeNorm(SolverSchemeException):  
476    class IterationBreakDown(SolverSchemeException):
477     """     """
478     a norm calculation returns a negative norm.     Exception thrown if the iteration scheme encountered an incurable breakdown.
479     """     """
480     pass     pass
481    
482  class IterationHistory(object):  class NegativeNorm(SolverSchemeException):
483     """     """
484     The IterationHistory class is used to define a stopping criterium. It keeps track of the     Exception thrown if a norm calculation returns a negative norm.
    residual norms. The stoppingcriterium indicates termination if the residual norm has been reduced by  
    a given tolerance.  
485     """     """
486     def __init__(self,tolerance=math.sqrt(util.EPSILON),verbose=False):     pass
       """  
       Initialization  
   
       @param tolerance: tolerance  
       @type tolerance: positive C{float}  
       @param verbose: switches on the printing out some information  
       @type verbose: C{bool}  
       """  
       if not tolerance>0.:  
           raise ValueError,"tolerance needs to be positive."  
       self.tolerance=tolerance  
       self.verbose=verbose  
       self.history=[]  
    def stoppingcriterium(self,norm_r,r,x):  
        """  
        returns True if the C{norm_r} is C{tolerance}*C{norm_r[0]} where C{norm_r[0]}  is the residual norm at the first call.  
   
         
        @param norm_r: current residual norm  
        @type norm_r: non-negative C{float}  
        @param r: current residual (not used)  
        @param x: current solution approximation (not used)  
        @return: C{True} is the stopping criterium is fullfilled. Otherwise C{False} is returned.  
        @rtype: C{bool}  
   
        """  
        self.history.append(norm_r)  
        if self.verbose: print "iter: %s:  inner(rhat,r) = %e"#(len(self.history)-1, self.history[-1])  
        return self.history[-1]<=self.tolerance * self.history[0]  
487    
488     def stoppingcriterium2(self,norm_r,norm_b,solver="GMRES",TOL=None):  def PCG(r, Aprod, x, Msolve, bilinearform, atol=0, rtol=1.e-8, iter_max=100, initial_guess=True, verbose=False):
489         """     """
490         returns True if the C{norm_r} is C{tolerance}*C{norm_b}     Solver for
491    
492             *Ax=b*
        @param norm_r: current residual norm  
        @type norm_r: non-negative C{float}  
        @param norm_b: norm of right hand side  
        @type norm_b: non-negative C{float}  
        @return: C{True} is the stopping criterium is fullfilled. Otherwise C{False} is returned.  
        @rtype: C{bool}  
493    
494         """     with a symmetric and positive definite operator A (more details required!).
495         if TOL==None:     It uses the conjugate gradient method with preconditioner M providing an
496            TOL=self.tolerance     approximation of A.
        self.history.append(norm_r)  
        if self.verbose: print "iter: %s:  norm(r) = %e"#(len(self.history)-1, self.history[-1])  
        return self.history[-1]<=TOL * norm_b  
497    
498  def PCG(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100):     The iteration is terminated if
    """  
    Solver for  
499    
500     M{Ax=b}     *|r| <= atol+rtol*|r0|*
501    
502     with a symmetric and positive definite operator A (more details required!).     where *r0* is the initial residual and *|.|* is the energy norm. In fact
    It uses the conjugate gradient method with preconditioner M providing an approximation of A.  
503    
504     The iteration is terminated if the C{stoppingcriterium} function return C{True}.     *|r| = sqrt( bilinearform(Msolve(r),r))*
505    
506     For details on the preconditioned conjugate gradient method see the book:     For details on the preconditioned conjugate gradient method see the book:
507    
508     Templates for the Solution of Linear Systems by R. Barrett, M. Berry,     I{Templates for the Solution of Linear Systems by R. Barrett, M. Berry,
509     T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo,     T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo,
510     C. Romine, and H. van der Vorst.     C. Romine, and H. van der Vorst}.
511    
512     @param b: the right hand side of the liner system. C{b} is altered.     :param r: initial residual *r=b-Ax*. ``r`` is altered.
513     @type b: any object supporting inplace add (x+=y) and scaling (x=scalar*y)     :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
514     @param Aprod: returns the value Ax     :param x: an initial guess for the solution
515     @type Aprod: function C{Aprod(x)} where C{x} is of the same object like argument C{x}. The returned object needs to be of the same type like argument C{b}.     :type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
516     @param Msolve: solves Mx=r     :param Aprod: returns the value Ax
517     @type Msolve: function C{Msolve(r)} where C{r} is of the same type like argument C{b}. The returned object needs to be of the same     :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
518  type like argument C{x}.                  argument ``x``. The returned object needs to be of the same type
519     @param bilinearform: inner product C{<x,r>}                  like argument ``r``.
520     @type bilinearform: function C{bilinearform(x,r)} where C{x} is of the same type like argument C{x} and C{r} is . The returned value is a C{float}.     :param Msolve: solves Mx=r
521     @param stoppingcriterium: function which returns True if a stopping criterium is meet. C{stoppingcriterium} has the arguments C{norm_r}, C{r} and C{x} giving the current norm of the residual (=C{sqrt(bilinearform(Msolve(r),r)}), the current residual and the current solution approximation. C{stoppingcriterium} is called in each iteration step.     :type Msolve: function ``Msolve(r)`` where ``r`` is of the same type like
522     @type stoppingcriterium: function that returns C{True} or C{False}                   argument ``r``. The returned object needs to be of the same
523     @param x: an initial guess for the solution. If no C{x} is given 0*b is used.                   type like argument ``x``.
524     @type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)     :param bilinearform: inner product ``<x,r>``
525     @param iter_max: maximum number of iteration steps.     :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
526     @type iter_max: C{int}                         type like argument ``x`` and ``r`` is. The returned value
527     @return: the solution approximation and the corresponding residual                         is a ``float``.
528     @rtype: C{tuple}     :param atol: absolute tolerance
529     @warning: C{b} and C{x} are altered.     :type atol: non-negative ``float``
530       :param rtol: relative tolerance
531       :type rtol: non-negative ``float``
532       :param iter_max: maximum number of iteration steps
533       :type iter_max: ``int``
534       :return: the solution approximation and the corresponding residual
535       :rtype: ``tuple``
536       :warning: ``r`` and ``x`` are altered.
537     """     """
538     iter=0     iter=0
    if x==None:  
       x=0*b  
    else:  
       b += (-1)*Aprod(x)  
    r=b  
539     rhat=Msolve(r)     rhat=Msolve(r)
540     d = rhat     d = rhat
541     rhat_dot_r = bilinearform(rhat, r)     rhat_dot_r = bilinearform(rhat, r)
542     if rhat_dot_r<0: raise NegativeNorm,"negative norm."     if rhat_dot_r<0: raise NegativeNorm,"negative norm."
543       norm_r0=math.sqrt(rhat_dot_r)
544       atol2=atol+rtol*norm_r0
545       if atol2<=0:
546          raise ValueError,"Non-positive tolarance."
547       atol2=max(atol2, 100. * util.EPSILON * norm_r0)
548    
549     while not stoppingcriterium(math.sqrt(rhat_dot_r),r,x):     if verbose: print "PCG: initial residual norm = %e (absolute tolerance = %e)"%(norm_r0, atol2)
550    
551    
552       while not math.sqrt(rhat_dot_r) <= atol2:
553         iter+=1         iter+=1
554         if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max         if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
555    
556         q=Aprod(d)         q=Aprod(d)
557         alpha = rhat_dot_r / bilinearform(d, q)         alpha = rhat_dot_r / bilinearform(d, q)
558         x += alpha * d         x += alpha * d
559         r += (-alpha) * q         if isinstance(q,ArithmeticTuple):
560           r += q * (-alpha)      # Doing it the other way calls the float64.__mul__ not AT.__rmul__
561           else:
562               r += (-alpha) * q
563         rhat=Msolve(r)         rhat=Msolve(r)
564         rhat_dot_r_new = bilinearform(rhat, r)         rhat_dot_r_new = bilinearform(rhat, r)
565         beta = rhat_dot_r_new / rhat_dot_r         beta = rhat_dot_r_new / rhat_dot_r
# Line 560  type like argument C{x}. Line 568  type like argument C{x}.
568    
569         rhat_dot_r = rhat_dot_r_new         rhat_dot_r = rhat_dot_r_new
570         if rhat_dot_r<0: raise NegativeNorm,"negative norm."         if rhat_dot_r<0: raise NegativeNorm,"negative norm."
571           if verbose: print "PCG: iteration step %s: residual norm = %e"%(iter, math.sqrt(rhat_dot_r))
572       if verbose: print "PCG: tolerance reached after %s steps."%iter
573       return x,r,math.sqrt(rhat_dot_r)
574    
575     return x,r  class Defect(object):
576        """
577        Defines a non-linear defect F(x) of a variable x.
578        """
579        def __init__(self):
580            """
581            Initializes defect.
582            """
583            self.setDerivativeIncrementLength()
584    
585        def bilinearform(self, x0, x1):
586            """
587            Returns the inner product of x0 and x1
588    
589  ############################          :param x0: value for x0
590  # Added by Artak          :param x1: value for x1
591  #################################3          :return: the inner product of x0 and x1
592            :rtype: ``float``
593            """
594            return 0
595    
596  #Apply a sequence of k Givens rotations, used within gmres codes      def norm(self,x):
597  # vrot=givapp(c, s, vin, k)          """
598  def givapp(c,s,vin):          Returns the norm of argument ``x``.
     vrot=vin # warning: vin is altered!!!!  
     if isinstance(c,float):  
         vrot=[c*vrot[0]-s*vrot[1],s*vrot[0]+c*vrot[1]]  
     else:  
         for i in range(len(c)):  
             w1=c[i]*vrot[i]-s[i]*vrot[i+1]  
         w2=s[i]*vrot[i]+c[i]*vrot[i+1]  
             vrot[i:i+2]=w1,w2  
     return vrot  
599    
600  ##############################################          :param x: a value
601  def GMRES(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100, iter_restart=20):          :return: norm of argument x
602  ################################################          :rtype: ``float``
603     m=iter_restart          :note: by default ``sqrt(self.bilinearform(x,x)`` is returned.
604     iter=0          """
605     xc=x          s=self.bilinearform(x,x)
606     while True:          if s<0: raise NegativeNorm,"negative norm."
607        if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached"%iter_max          return math.sqrt(s)
       xc,stopped=GMRESm(b*1, Aprod, Msolve, bilinearform, stoppingcriterium, x=xc*1, iter_max=iter_max-iter, iter_restart=m)  
       if stopped: break  
       iter+=iter_restart      
    return xc  
608    
609  def GMRESm(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100, iter_restart=20):      def eval(self,x):
610     iter=0          """
611     r=Msolve(b)          Returns the value F of a given ``x``.
    r_dot_r = bilinearform(r, r)  
    if r_dot_r<0: raise NegativeNorm,"negative norm."  
    norm_b=math.sqrt(r_dot_r)  
612    
613     if x==None:          :param x: value for which the defect ``F`` is evaluated
614        x=0*b          :return: value of the defect at ``x``
615     else:          """
616        r=Msolve(b-Aprod(x))          return 0
       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)  
    v=[]  
617    
618     rho=math.sqrt(r_dot_r)      def __call__(self,x):
619              return self.eval(x)
    v.append(r/rho)  
    g[0]=rho  
620    
621     while not (stoppingcriterium(rho,norm_b) or iter==iter_restart-1):      def setDerivativeIncrementLength(self,inc=1000.*math.sqrt(util.EPSILON)):
622            """
623            Sets the relative length of the increment used to approximate the
624            derivative of the defect. The increment is inc*norm(x)/norm(v)*v in the
625            direction of v with x as a starting point.
626    
627      if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max          :param inc: relative increment length
628            :type inc: positive ``float``
629            """
630            if inc<=0: raise ValueError,"positive increment required."
631            self.__inc=inc
632    
633      p=Msolve(Aprod(v[iter]))      def getDerivativeIncrementLength(self):
634            """
635            Returns the relative increment length used to approximate the
636            derivative of the defect.
637            :return: value of the defect at ``x``
638            :rtype: positive ``float``
639            """
640            return self.__inc
641    
642      v.append(p)      def derivative(self, F0, x0, v, v_is_normalised=True):
643            """
644            Returns the directional derivative at ``x0`` in the direction of ``v``.
645    
646      v_norm1=math.sqrt(bilinearform(v[iter+1], v[iter+1]))            :param F0: value of this defect at x0
647            :param x0: value at which derivative is calculated
648            :param v: direction
649            :param v_is_normalised: True to indicate that ``v`` is nomalized
650                                    (self.norm(v)=0)
651            :return: derivative of this defect at x0 in the direction of ``v``
652            :note: by default numerical evaluation (self.eval(x0+eps*v)-F0)/eps is
653                   used but this method maybe overwritten to use exact evaluation.
654            """
655            normx=self.norm(x0)
656            if normx>0:
657                 epsnew = self.getDerivativeIncrementLength() * normx
658            else:
659                 epsnew = self.getDerivativeIncrementLength()
660            if not v_is_normalised:
661                normv=self.norm(v)
662                if normv<=0:
663                   return F0*0
664                else:
665                   epsnew /= normv
666            F1=self.eval(x0 + epsnew * v)
667            return (F1-F0)/epsnew
668    
669  # Modified Gram-Schmidt  ######################################
670      for j in range(iter+1):  def NewtonGMRES(defect, x, iter_max=100, sub_iter_max=20, atol=0,rtol=1.e-4, subtol_max=0.5, gamma=0.9, verbose=False):
671        h[j][iter]=bilinearform(v[j],v[iter+1])       """
672        v[iter+1]-=h[j][iter]*v[j]     Solves a non-linear problem *F(x)=0* for unknown *x* using the stopping
673             criterion:
     h[iter+1][iter]=math.sqrt(bilinearform(v[iter+1],v[iter+1]))  
     v_norm2=h[iter+1][iter]  
674    
675  # Reorthogonalize if needed     *norm(F(x) <= atol + rtol * norm(F(x0)*
     if v_norm1 + 0.001*v_norm2 == v_norm1:   #Brown/Hindmarsh condition (default)  
      for j in range(iter+1):    
         hr=bilinearform(v[j],v[iter+1])  
             h[j][iter]=h[j][iter]+hr  
             v[iter+1] -= hr*v[j]  
676    
677       v_norm2=math.sqrt(bilinearform(v[iter+1], v[iter+1]))       where *x0* is the initial guess.
      h[iter+1][iter]=v_norm2  
678    
679  #   watch out for happy breakdown     :param defect: object defining the function *F*. ``defect.norm`` defines the
680          if not v_norm2 == 0:                    *norm* used in the stopping criterion.
681           v[iter+1]=v[iter+1]/h[iter+1][iter]     :type defect: `Defect`
682       :param x: initial guess for the solution, ``x`` is altered.
683       :type x: any object type allowing basic operations such as
684                ``numpy.ndarray``, `Data`
685       :param iter_max: maximum number of iteration steps
686       :type iter_max: positive ``int``
687       :param sub_iter_max: maximum number of inner iteration steps
688       :type sub_iter_max: positive ``int``
689       :param atol: absolute tolerance for the solution
690       :type atol: positive ``float``
691       :param rtol: relative tolerance for the solution
692       :type rtol: positive ``float``
693       :param gamma: tolerance safety factor for inner iteration
694       :type gamma: positive ``float``, less than 1
695       :param subtol_max: upper bound for inner tolerance
696       :type subtol_max: positive ``float``, less than 1
697       :return: an approximation of the solution with the desired accuracy
698       :rtype: same type as the initial guess
699       """
700       lmaxit=iter_max
701       if atol<0: raise ValueError,"atol needs to be non-negative."
702       if rtol<0: raise ValueError,"rtol needs to be non-negative."
703       if rtol+atol<=0: raise ValueError,"rtol or atol needs to be non-negative."
704       if gamma<=0 or gamma>=1: raise ValueError,"tolerance safety factor for inner iteration (gamma =%s) needs to be positive and less than 1."%gamma
705       if subtol_max<=0 or subtol_max>=1: raise ValueError,"upper bound for inner tolerance for inner iteration (subtol_max =%s) needs to be positive and less than 1."%subtol_max
706    
707       F=defect(x)
708       fnrm=defect.norm(F)
709       stop_tol=atol + rtol*fnrm
710       subtol=subtol_max
711       if verbose: print "NewtonGMRES: initial residual = %e."%fnrm
712       if verbose: print "             tolerance = %e."%subtol
713       iter=1
714       #
715       # main iteration loop
716       #
717       while not fnrm<=stop_tol:
718                if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
719                #
720            #   adjust subtol_
721            #
722                if iter > 1:
723               rat=fnrm/fnrmo
724                   subtol_old=subtol
725               subtol=gamma*rat**2
726               if gamma*subtol_old**2 > .1: subtol=max(subtol,gamma*subtol_old**2)
727               subtol=max(min(subtol,subtol_max), .5*stop_tol/fnrm)
728            #
729            # calculate newton increment xc
730                #     if iter_max in __FDGMRES is reached MaxIterReached is thrown
731                #     if iter_restart -1 is returned as sub_iter
732                #     if  atol is reached sub_iter returns the numer of steps performed to get there
733                #
734                #
735                if verbose: print "             subiteration (GMRES) is called with relative tolerance %e."%subtol
736                try:
737                   xc, sub_iter=__FDGMRES(F, defect, x, subtol*fnrm, iter_max=iter_max-iter, iter_restart=sub_iter_max)
738                except MaxIterReached:
739                   raise MaxIterReached,"maximum number of %s steps reached."%iter_max
740                if sub_iter<0:
741                   iter+=sub_iter_max
742                else:
743                   iter+=sub_iter
744                # ====
745            x+=xc
746                F=defect(x)
747            iter+=1
748                fnrmo, fnrm=fnrm, defect.norm(F)
749                if verbose: print "             step %s: residual %e."%(iter,fnrm)
750       if verbose: print "NewtonGMRES: completed after %s steps."%iter
751       return x
752    
753  #   Form and store the information for the new Givens rotation  def __givapp(c,s,vin):
754      if iter > 0 :      """
755          hhat=numarray.zeros(iter+1,numarray.Float64)      Applies a sequence of Givens rotations (c,s) recursively to the vector
756          for i in range(iter+1) : hhat[i]=h[i][iter]      ``vin``
         hhat=givapp(c[0:iter],s[0:iter],hhat);  
             for i in range(iter+1) : h[i][iter]=hhat[i]  
757    
758      mu=math.sqrt(h[iter][iter]*h[iter][iter]+h[iter+1][iter]*h[iter+1][iter])      :warning: ``vin`` is altered.
759        """
760        vrot=vin
761        if isinstance(c,float):
762            vrot=[c*vrot[0]-s*vrot[1],s*vrot[0]+c*vrot[1]]
763        else:
764            for i in range(len(c)):
765                w1=c[i]*vrot[i]-s[i]*vrot[i+1]
766            w2=s[i]*vrot[i]+c[i]*vrot[i+1]
767                vrot[i]=w1
768                vrot[i+1]=w2
769        return vrot
770    
771      if mu!=0 :  def __FDGMRES(F0, defect, x0, atol, iter_max=100, iter_restart=20):
772          c[iter]=h[iter][iter]/mu     h=numpy.zeros((iter_restart,iter_restart),numpy.float64)
773          s[iter]=-h[iter+1][iter]/mu     c=numpy.zeros(iter_restart,numpy.float64)
774          h[iter][iter]=c[iter]*h[iter][iter]-s[iter]*h[iter+1][iter]     s=numpy.zeros(iter_restart,numpy.float64)
775          h[iter+1][iter]=0.0     g=numpy.zeros(iter_restart,numpy.float64)
776          g[iter:iter+2]=givapp(c[iter],s[iter],g[iter:iter+2])     v=[]
777    
778  # Update the residual norm     rho=defect.norm(F0)
779                     if rho<=0.: return x0*0
         rho=abs(g[iter+1])  
     iter+=1  
780    
781  # At this point either iter > iter_max or rho < tol.     v.append(-F0/rho)
782  # It's time to compute x and leave.             g[0]=rho
783       iter=0
784       while rho > atol and iter<iter_restart-1:
785            if iter  >= iter_max:
786                raise MaxIterReached,"maximum number of %s steps reached."%iter_max
787    
788            p=defect.derivative(F0,x0,v[iter], v_is_normalised=True)
789            v.append(p)
790    
791            v_norm1=defect.norm(v[iter+1])
792    
793            # Modified Gram-Schmidt
794            for j in range(iter+1):
795                h[j,iter]=defect.bilinearform(v[j],v[iter+1])
796                v[iter+1]-=h[j,iter]*v[j]
797    
798            h[iter+1,iter]=defect.norm(v[iter+1])
799            v_norm2=h[iter+1,iter]
800    
801            # Reorthogonalize if needed
802            if v_norm1 + 0.001*v_norm2 == v_norm1:   #Brown/Hindmarsh condition (default)
803                for j in range(iter+1):
804                    hr=defect.bilinearform(v[j],v[iter+1])
805                    h[j,iter]=h[j,iter]+hr
806                    v[iter+1] -= hr*v[j]
807    
808                v_norm2=defect.norm(v[iter+1])
809                h[iter+1,iter]=v_norm2
810            #   watch out for happy breakdown
811            if not v_norm2 == 0:
812                v[iter+1]=v[iter+1]/h[iter+1,iter]
813    
814            #   Form and store the information for the new Givens rotation
815            if iter > 0 :
816                hhat=numpy.zeros(iter+1,numpy.float64)
817                for i in range(iter+1) : hhat[i]=h[i,iter]
818                hhat=__givapp(c[0:iter],s[0:iter],hhat);
819                for i in range(iter+1) : h[i,iter]=hhat[i]
820    
821            mu=math.sqrt(h[iter,iter]*h[iter,iter]+h[iter+1,iter]*h[iter+1,iter])
822    
823            if mu!=0 :
824                c[iter]=h[iter,iter]/mu
825                s[iter]=-h[iter+1,iter]/mu
826                h[iter,iter]=c[iter]*h[iter,iter]-s[iter]*h[iter+1,iter]
827                h[iter+1,iter]=0.0
828                gg=__givapp(c[iter],s[iter],[g[iter],g[iter+1]])
829                g[iter]=gg[0]
830                g[iter+1]=gg[1]
831    
832     if iter > 0 :          # Update the residual norm
833       y=numarray.zeros(iter,numarray.Float64)              rho=abs(g[iter+1])
834       y[iter-1] = g[iter-1] / h[iter-1][iter-1]          iter+=1
835       if iter > 1 :    
836          i=iter-2       # At this point either iter > iter_max or rho < tol.
837       # It's time to compute x and leave.
838       if iter > 0 :
839         y=numpy.zeros(iter,numpy.float64)
840         y[iter-1] = g[iter-1] / h[iter-1,iter-1]
841         if iter > 1 :
842            i=iter-2
843          while i>=0 :          while i>=0 :
844            y[i] = ( g[i] - numarray.dot(h[i][i+1:iter], y[i+1:iter])) / h[i][i]            y[i] = ( g[i] - numpy.dot(h[i,i+1:iter], y[i+1:iter])) / h[i,i]
845            i=i-1            i=i-1
846       xhat=v[iter-1]*y[iter-1]       xhat=v[iter-1]*y[iter-1]
847       for i in range(iter-1):       for i in range(iter-1):
848      xhat += v[i]*y[i]      xhat += v[i]*y[i]
849     else : xhat=v[0]     else :
850          xhat=v[0] * 0
851    
852     x += xhat     if iter<iter_restart-1:
853     if iter<iter_restart-1:        stopped=iter
854        stopped=True     else:
855     else:        stopped=-1
       stopped=False  
856    
857     return x,stopped     return xhat,stopped
858    
859    def GMRES(r, Aprod, x, bilinearform, atol=0, rtol=1.e-8, iter_max=100, iter_restart=20, verbose=False,P_R=None):
860       """
861       Solver for
862    
863  ######################################################     *Ax=b*
 def dirder(x, w, bilinearform, Aprod, Msolve, f0, b ):  
 ######################################################  
864    
865  # DIRDER estimates the directional derivative of a function F.     with a general operator A (more details required!).
866       It uses the generalized minimum residual method (GMRES).
867    
868       The iteration is terminated if
869    
870  # Hardwired difference increment.     *|r| <= atol+rtol*|r0|*
871  #  
872    epsnew = 1.0e-07     where *r0* is the initial residual and *|.|* is the energy norm. In fact
873  #  
874  #  Scale the step.     *|r| = sqrt( bilinearform(r,r))*
875  #  
876    norm_w=math.sqrt(bilinearform(w,w))     :param r: initial residual *r=b-Ax*. ``r`` is altered.
877    if norm_w== 0.0:     :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
878      return x/x     :param x: an initial guess for the solution
879       :type x: same like ``r``
880    epsnew = epsnew / norm_w     :param Aprod: returns the value Ax
881       :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
882                    argument ``x``. The returned object needs to be of the same
883                    type like argument ``r``.
884       :param bilinearform: inner product ``<x,r>``
885       :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
886                           type like argument ``x`` and ``r``. The returned value is
887                           a ``float``.
888       :param atol: absolute tolerance
889       :type atol: non-negative ``float``
890       :param rtol: relative tolerance
891       :type rtol: non-negative ``float``
892       :param iter_max: maximum number of iteration steps
893       :type iter_max: ``int``
894       :param iter_restart: in order to save memory the orthogonalization process
895                            is terminated after ``iter_restart`` steps and the
896                            iteration is restarted.
897       :type iter_restart: ``int``
898       :return: the solution approximation and the corresponding residual
899       :rtype: ``tuple``
900       :warning: ``r`` and ``x`` are altered.
901       """
902       m=iter_restart
903       restarted=False
904       iter=0
905       if rtol>0:
906          r_dot_r = bilinearform(r, r)
907          if r_dot_r<0: raise NegativeNorm,"negative norm."
908          atol2=atol+rtol*math.sqrt(r_dot_r)
909          if verbose: print "GMRES: norm of right hand side = %e (absolute tolerance = %e)"%(math.sqrt(r_dot_r), atol2)
910       else:
911          atol2=atol
912          if verbose: print "GMRES: absolute tolerance = %e"%atol2
913       if atol2<=0:
914          raise ValueError,"Non-positive tolarance."
915    
916    if norm_w > 0.0:     while True:
917      epsnew = epsnew * math.sqrt(bilinearform(x,x))        if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached"%iter_max
918  #        if restarted:
919  #  DEL and F1 could share the same space if storage           r2 = r-Aprod(x-x2)
920  #  is more important than clarity.        else:
921  #           r2=1*r
922          x2=x*1.
923          x,stopped=_GMRESm(r2, Aprod, x, bilinearform, atol2, iter_max=iter_max-iter, iter_restart=m, verbose=verbose,P_R=P_R)
924          iter+=iter_restart
925          if stopped: break
926          if verbose: print "GMRES: restart."
927          restarted=True
928       if verbose: print "GMRES: tolerance has been reached."
929       return x
930    
931    DEL = x + epsnew * w  def _GMRESm(r, Aprod, x, bilinearform, atol, iter_max=100, iter_restart=20, verbose=False, P_R=None):
932    f1 = -Msolve(Aprod(DEL))     iter=0
   z = ( f1 - f0 ) / epsnew  
   return z  
   
 ######################################################  
 def FDGMRES(f0, Aprod, Msolve, bilinearform, stoppingcriterium, xc=None, x=None, iter_max=100, iter_restart=20,TOL=None):  
 ######################################################  
    b=-f0  
    b_dot_b = bilinearform(b, b)  
    if b_dot_b<0: raise NegativeNorm,"negative norm."  
    norm_b=math.sqrt(b_dot_b)  
933    
934     r=b     h=numpy.zeros((iter_restart+1,iter_restart),numpy.float64)
935       c=numpy.zeros(iter_restart,numpy.float64)
936       s=numpy.zeros(iter_restart,numpy.float64)
937       g=numpy.zeros(iter_restart+1,numpy.float64)
938       v=[]
939    
    if x==None:  
       x=0*f0  
    else:  
       r=-dirder(xc,x,bilinearform,Aprod,Msolve,f0,b)-f0    
         
940     r_dot_r = bilinearform(r, r)     r_dot_r = bilinearform(r, r)
941     if r_dot_r<0: raise NegativeNorm,"negative norm."     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)  
    v=[]  
   
942     rho=math.sqrt(r_dot_r)     rho=math.sqrt(r_dot_r)
943      
944     v.append(r/rho)     v.append(r/rho)
945     g[0]=rho     g[0]=rho
    iter=0  
946    
947     while not (stoppingcriterium(rho,norm_b,solver="FDGMRES",TOL=TOL) or iter==iter_restart-1):     if verbose: print "GMRES: initial residual %e (absolute tolerance = %e)"%(rho,atol)
948       while not (rho<=atol or iter==iter_restart):
949    
950      if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max      if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
951    
952                if P_R!=None:
953          p=dirder(xc, v[iter], bilinearform,Aprod,Msolve,f0,b)              p=Aprod(P_R(v[iter]))
954            else:
955            p=Aprod(v[iter])
956      v.append(p)      v.append(p)
957    
958      v_norm1=math.sqrt(bilinearform(v[iter+1], v[iter+1]))        v_norm1=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
959    
960  # Modified Gram-Schmidt  # Modified Gram-Schmidt
961      for j in range(iter+1):      for j in range(iter+1):
962        h[j][iter]=bilinearform(v[j],v[iter+1])          h[j,iter]=bilinearform(v[j],v[iter+1])
963        v[iter+1]+=(-1.)*h[j][iter]*v[j]        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]  
964    
965        h[iter+1,iter]=math.sqrt(bilinearform(v[iter+1],v[iter+1]))
966        v_norm2=h[iter+1,iter]
967    
968  # Reorthogonalize if needed  # Reorthogonalize if needed
969      if v_norm1 + 0.001*v_norm2 == v_norm1:   #Brown/Hindmarsh condition (default)      if v_norm1 + 0.001*v_norm2 == v_norm1:   #Brown/Hindmarsh condition (default)
970       for j in range(iter+1):       for j in range(iter+1):
971          hr=bilinearform(v[j],v[iter+1])          hr=bilinearform(v[j],v[iter+1])
972              h[j][iter]=h[j][iter]+hr #vhat              h[j,iter]=h[j,iter]+hr
973              v[iter+1] +=(-1.)*hr*v[j]              v[iter+1] -= hr*v[j]
974    
975       v_norm2=math.sqrt(bilinearform(v[iter+1], v[iter+1]))         v_norm2=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
976       h[iter+1][iter]=v_norm2       h[iter+1,iter]=v_norm2
977    
978  #   watch out for happy breakdown  #   watch out for happy breakdown
979          if v_norm2 != 0:          if not v_norm2 == 0:
980           v[iter+1]=v[iter+1]/h[iter+1][iter]           v[iter+1]=v[iter+1]/h[iter+1,iter]
981    
982  #   Form and store the information for the new Givens rotation  #   Form and store the information for the new Givens rotation
983      if iter > 0 :      if iter > 0: h[:iter+1,iter]=__givapp(c[:iter],s[:iter],h[:iter+1,iter])
984          hhat=[]      mu=math.sqrt(h[iter,iter]*h[iter,iter]+h[iter+1,iter]*h[iter+1,iter])
         for i in range(iter+1) : hhat.append(h[i][iter])  
         hhat=givapp(c[0:iter],s[0:iter],hhat);  
             for i in range(iter+1) : h[i][iter]=hhat[i]  
985    
     mu=math.sqrt(h[iter][iter]*h[iter][iter]+h[iter+1][iter]*h[iter+1][iter])  
986      if mu!=0 :      if mu!=0 :
987          c[iter]=h[iter][iter]/mu          c[iter]=h[iter,iter]/mu
988          s[iter]=-h[iter+1][iter]/mu          s[iter]=-h[iter+1,iter]/mu
989          h[iter][iter]=c[iter]*h[iter][iter]-s[iter]*h[iter+1][iter]          h[iter,iter]=c[iter]*h[iter,iter]-s[iter]*h[iter+1,iter]
990          h[iter+1][iter]=0.0          h[iter+1,iter]=0.0
991          g[iter:iter+2]=givapp(c[iter],s[iter],g[iter:iter+2])                  gg=__givapp(c[iter],s[iter],[g[iter],g[iter+1]])
992                    g[iter]=gg[0]
993                    g[iter+1]=gg[1]
994  # Update the residual norm  # Update the residual norm
995    
996          rho=abs(g[iter+1])          rho=abs(g[iter+1])
997            if verbose: print "GMRES: iteration step %s: residual %e"%(iter,rho)
998      iter+=1      iter+=1
999    
1000     if iter > 0 :  # At this point either iter > iter_max or rho < tol.
1001       y=numarray.zeros(iter,numarray.Float64)      # It's time to compute x and leave.
1002       y[iter-1] = g[iter-1] / h[iter-1][iter-1]  
1003       if iter > 1 :       if verbose: print "GMRES: iteration stopped after %s step."%iter
1004          i=iter-2       if iter > 0 :
1005         y=numpy.zeros(iter,numpy.float64)
1006         y[iter-1] = g[iter-1] / h[iter-1,iter-1]
1007         if iter > 1 :
1008            i=iter-2
1009          while i>=0 :          while i>=0 :
1010            y[i] = ( g[i] - numarray.dot(h[i][i+1:iter], y[i+1:iter])) / h[i][i]            y[i] = ( g[i] - numpy.dot(h[i,i+1:iter], y[i+1:iter])) / h[i,i]
1011            i=i-1            i=i-1
1012       xhat=v[iter-1]*y[iter-1]       xhat=v[iter-1]*y[iter-1]
1013       for i in range(iter-1):       for i in range(iter-1):
1014      xhat += v[i]*y[i]      xhat += v[i]*y[i]
1015     else : xhat=v[0]     else:
1016             xhat=v[0] * 0
1017     x += xhat     if P_R!=None:
1018     if iter<iter_restart-1:        x += P_R(xhat)
1019        stopped=True     else:
1020     else:        x += xhat
1021       if iter<iter_restart-1:
1022          stopped=True
1023       else:
1024        stopped=False        stopped=False
1025    
1026     return x,stopped     return x,stopped
1027    
1028  #################################################  def MINRES(r, Aprod, x, Msolve, bilinearform, atol=0, rtol=1.e-8, iter_max=100):
1029  def MINRES(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100):      """
1030  #################################################      Solver for
1031      #  
1032      #  minres solves the system of linear equations Ax = b      *Ax=b*
1033      #  where A is a symmetric matrix (possibly indefinite or singular)  
1034      #  and b is a given vector.      with a symmetric and positive definite operator A (more details required!).
1035      #        It uses the minimum residual method (MINRES) with preconditioner M
1036      #  "A" may be a dense or sparse matrix (preferably sparse!)      providing an approximation of A.
1037      #  or the name of a function such that  
1038      #               y = A(x)      The iteration is terminated if
1039      #  returns the product y = Ax for any given vector x.  
1040      #      *|r| <= atol+rtol*|r0|*
1041      #  "M" defines a positive-definite preconditioner M = C C'.  
1042      #  "M" may be a dense or sparse matrix (preferably sparse!)      where *r0* is the initial residual and *|.|* is the energy norm. In fact
1043      #  or the name of a function such that  
1044      #  solves the system My = x for any given vector x.      *|r| = sqrt( bilinearform(Msolve(r),r))*
1045      #  
1046      #      For details on the preconditioned conjugate gradient method see the book:
1047        
1048        I{Templates for the Solution of Linear Systems by R. Barrett, M. Berry,
1049        T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo,
1050        C. Romine, and H. van der Vorst}.
1051    
1052        :param r: initial residual *r=b-Ax*. ``r`` is altered.
1053        :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1054        :param x: an initial guess for the solution
1055        :type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1056        :param Aprod: returns the value Ax
1057        :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
1058                     argument ``x``. The returned object needs to be of the same
1059                     type like argument ``r``.
1060        :param Msolve: solves Mx=r
1061        :type Msolve: function ``Msolve(r)`` where ``r`` is of the same type like
1062                      argument ``r``. The returned object needs to be of the same
1063                      type like argument ``x``.
1064        :param bilinearform: inner product ``<x,r>``
1065        :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
1066                            type like argument ``x`` and ``r`` is. The returned value
1067                            is a ``float``.
1068        :param atol: absolute tolerance
1069        :type atol: non-negative ``float``
1070        :param rtol: relative tolerance
1071        :type rtol: non-negative ``float``
1072        :param iter_max: maximum number of iteration steps
1073        :type iter_max: ``int``
1074        :return: the solution approximation and the corresponding residual
1075        :rtype: ``tuple``
1076        :warning: ``r`` and ``x`` are altered.
1077        """
1078      #------------------------------------------------------------------      #------------------------------------------------------------------
1079      # Set up y and v for the first Lanczos vector v1.      # Set up y and v for the first Lanczos vector v1.
1080      # y  =  beta1 P' v1,  where  P = C**(-1).      # y  =  beta1 P' v1,  where  P = C**(-1).
1081      # v is really P' v1.      # v is really P' v1.
1082      #------------------------------------------------------------------      #------------------------------------------------------------------
1083      if x==None:      r1    = r
1084        x=0*b      y = Msolve(r)
1085      else:      beta1 = bilinearform(y,r)
       b += (-1)*Aprod(x)  
1086    
     r1    = b  
     y = Msolve(b)  
     beta1 = bilinearform(y,b)  
   
1087      if beta1< 0: raise NegativeNorm,"negative norm."      if beta1< 0: raise NegativeNorm,"negative norm."
1088    
1089      #  If b = 0 exactly, stop with x = 0.      #  If r = 0 exactly, stop with x
1090      if beta1==0: return x*0.      if beta1==0: return x
1091    
1092      if beta1> 0:      if beta1> 0: beta1  = math.sqrt(beta1)
       beta1  = math.sqrt(beta1)        
1093    
1094      #------------------------------------------------------------------      #------------------------------------------------------------------
1095      # Initialize quantities.      # Initialize quantities.
# Line 892  def MINRES(b, Aprod, Msolve, bilinearfor Line 1109  def MINRES(b, Aprod, Msolve, bilinearfor
1109      ynorm2 = 0      ynorm2 = 0
1110      cs     = -1      cs     = -1
1111      sn     = 0      sn     = 0
1112      w      = b*0.      w      = r*0.
1113      w2     = b*0.      w2     = r*0.
1114      r2     = r1      r2     = r1
1115      eps    = 0.0001      eps    = 0.0001
1116    
1117      #---------------------------------------------------------------------      #---------------------------------------------------------------------
1118      # Main iteration loop.      # Main iteration loop.
1119      # --------------------------------------------------------------------      # --------------------------------------------------------------------
1120      while not stoppingcriterium(rnorm,Anorm*ynorm,'MINRES'):    #  checks ||r|| < (||A|| ||x||) * TOL      while not rnorm<=atol+rtol*Anorm*ynorm:    #  checks ||r|| < (||A|| ||x||) * TOL
1121    
1122      if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max      if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
1123          iter    = iter  +  1          iter    = iter  +  1
# Line 919  def MINRES(b, Aprod, Msolve, bilinearfor Line 1136  def MINRES(b, Aprod, Msolve, bilinearfor
1136          #-----------------------------------------------------------------          #-----------------------------------------------------------------
1137          s = 1/beta                 # Normalize previous vector (in y).          s = 1/beta                 # Normalize previous vector (in y).
1138          v = s*y                    # v = vk if P = I          v = s*y                    # v = vk if P = I
1139        
1140          y      = Aprod(v)          y      = Aprod(v)
1141        
1142          if iter >= 2:          if iter >= 2:
1143            y = y - (beta/oldb)*r1            y = y - (beta/oldb)*r1
1144    
1145          alfa   = bilinearform(v,y)              # alphak          alfa   = bilinearform(v,y)              # alphak
1146          y      += (- alfa/beta)*r2          y      += (- alfa/beta)*r2
1147          r1     = r2          r1     = r2
1148          r2     = y          r2     = y
1149          y = Msolve(r2)          y = Msolve(r2)
# Line 936  def MINRES(b, Aprod, Msolve, bilinearfor Line 1153  def MINRES(b, Aprod, Msolve, bilinearfor
1153    
1154          beta   = math.sqrt( beta )          beta   = math.sqrt( beta )
1155          tnorm2 = tnorm2 + alfa*alfa + oldb*oldb + beta*beta          tnorm2 = tnorm2 + alfa*alfa + oldb*oldb + beta*beta
1156            
1157          if iter==1:                 # Initialize a few things.          if iter==1:                 # Initialize a few things.
1158            gmax   = abs( alfa )      # alpha1            gmax   = abs( alfa )      # alpha1
1159            gmin   = gmax             # alpha1            gmin   = gmax             # alpha1
# Line 944  def MINRES(b, Aprod, Msolve, bilinearfor Line 1161  def MINRES(b, Aprod, Msolve, bilinearfor
1161          # Apply previous rotation Qk-1 to get          # Apply previous rotation Qk-1 to get
1162          #   [deltak epslnk+1] = [cs  sn][dbark    0   ]          #   [deltak epslnk+1] = [cs  sn][dbark    0   ]
1163          #   [gbar k dbar k+1]   [sn -cs][alfak betak+1].          #   [gbar k dbar k+1]   [sn -cs][alfak betak+1].
1164        
1165          oldeps = epsln          oldeps = epsln
1166          delta  = cs * dbar  +  sn * alfa  # delta1 = 0         deltak          delta  = cs * dbar  +  sn * alfa  # delta1 = 0         deltak
1167          gbar   = sn * dbar  -  cs * alfa  # gbar 1 = alfa1     gbar k          gbar   = sn * dbar  -  cs * alfa  # gbar 1 = alfa1     gbar k
# Line 954  def MINRES(b, Aprod, Msolve, bilinearfor Line 1171  def MINRES(b, Aprod, Msolve, bilinearfor
1171          # Compute the next plane rotation Qk          # Compute the next plane rotation Qk
1172    
1173          gamma  = math.sqrt(gbar*gbar+beta*beta)  # gammak          gamma  = math.sqrt(gbar*gbar+beta*beta)  # gammak
1174          gamma  = max(gamma,eps)          gamma  = max(gamma,eps)
1175          cs     = gbar / gamma             # ck          cs     = gbar / gamma             # ck
1176          sn     = beta / gamma             # sk          sn     = beta / gamma             # sk
1177          phi    = cs * phibar              # phik          phi    = cs * phibar              # phik
# Line 962  def MINRES(b, Aprod, Msolve, bilinearfor Line 1179  def MINRES(b, Aprod, Msolve, bilinearfor
1179    
1180          # Update  x.          # Update  x.
1181    
1182          denom = 1/gamma          denom = 1/gamma
1183          w1    = w2          w1    = w2
1184          w2    = w          w2    = w
1185          w     = (v - oldeps*w1 - delta*w2) * denom          w     = (v - oldeps*w1 - delta*w2) * denom
1186          x     +=  phi*w          x     +=  phi*w
1187    
# Line 979  def MINRES(b, Aprod, Msolve, bilinearfor Line 1196  def MINRES(b, Aprod, Msolve, bilinearfor
1196    
1197          # Estimate various norms and test for convergence.          # Estimate various norms and test for convergence.
1198    
1199          Anorm  = math.sqrt( tnorm2 )          Anorm  = math.sqrt( tnorm2 )
1200          ynorm  = math.sqrt( ynorm2 )          ynorm  = math.sqrt( ynorm2 )
1201    
1202          rnorm  = phibar          rnorm  = phibar
1203    
1204      return x      return x
1205    
1206  ######################################      def TFQMR(r, Aprod, x, bilinearform, atol=0, rtol=1.e-8, iter_max=100):
1207  def NewtonGMRES(b, Aprod, Msolve, bilinearform, stoppingcriterium,x=None, iter_max=100,iter_restart=20,atol=1.e-2,rtol=1.e-4):    """
1208  #####################################    Solver for
     gamma=.9  
     lmaxit=100  
     etamax=.5  
   
     n = 1 #len(x)  
     iter=0  
       
     # evaluate f at the initial iterate  
     # compute the stop tolerance  
     #  
     r=b  
     if x==None:  
       x=0*b  
     else:  
       r += (-1)*Aprod(x)  
   
     f0=-Msolve(r)  
     fnrm=math.sqrt(bilinearform(f0,f0))/math.sqrt(n)  
     fnrmo=1  
     stop_tol=atol + rtol*fnrm  
     #  
     # main iteration loop  
     #  
     while not stoppingcriterium(fnrm*1,stop_tol,'NewtonGMRES',TOL=1.):  
   
             if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max  
         #  
         # keep track of the ratio (rat = fnrm/frnmo)  
         # of successive residual norms and  
         # the iteration counter (iter)  
         #  
         #rat=fnrm/fnrmo  
         fnrmo=fnrm  
         iter+=1  
         #  
             # compute the step using a GMRES(m) routine especially designed for this purpose  
         #  
             initer=0  
             while True:  
                xc,stopped=FDGMRES(f0*1, Aprod, Msolve, bilinearform, stoppingcriterium, xc=x*1, iter_max=lmaxit-initer, iter_restart=iter_restart, TOL=etamax)  
                if stopped: break  
                initer+=iter_restart  
         x+=xc  
         f0=-Msolve(Aprod(x))  
         fnrm=math.sqrt(bilinearform(f0,f0))/math.sqrt(n)  
         rat=fnrm/fnrmo  
   
   
     #   adjust eta  
     #  
         if etamax > 0:  
             etaold=etamax  
             etanew=gamma*rat*rat  
             if gamma*etaold*etaold > .1 :  
                 etanew=max(etanew,gamma*etaold*etaold)  
             etamax=min(etanew,etamax)  
             etamax=max(etamax,.5*stop_tol/fnrm)  
     return x  
1209    
1210  def TFQMR(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100):    *Ax=b*
1211    
1212  # TFQMR solver for linear systems    with a general operator A (more details required!).
1213  #    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  
1214    
1215    r=Msolve(r)    The iteration is terminated if
1216    
1217      *|r| <= atol+rtol*|r0|*
1218    
1219      where *r0* is the initial residual and *|.|* is the energy norm. In fact
1220    
1221      *|r| = sqrt( bilinearform(r,r))*
1222    
1223      :param r: initial residual *r=b-Ax*. ``r`` is altered.
1224      :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1225      :param x: an initial guess for the solution
1226      :type x: same like ``r``
1227      :param Aprod: returns the value Ax
1228      :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
1229                   argument ``x``. The returned object needs to be of the same type
1230                   like argument ``r``.
1231      :param bilinearform: inner product ``<x,r>``
1232      :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
1233                          type like argument ``x`` and ``r``. The returned value is
1234                          a ``float``.
1235      :param atol: absolute tolerance
1236      :type atol: non-negative ``float``
1237      :param rtol: relative tolerance
1238      :type rtol: non-negative ``float``
1239      :param iter_max: maximum number of iteration steps
1240      :type iter_max: ``int``
1241      :rtype: ``tuple``
1242      :warning: ``r`` and ``x`` are altered.
1243      """
1244    u1=0    u1=0
1245    u2=0    u2=0
1246    y1=0    y1=0
1247    y2=0    y2=0
1248    
1249    w = r    w = r
1250    y1 = r    y1 = r
1251    iter = 0    iter = 0
1252    d = 0    d = 0
1253        v = Aprod(y1)
   v = Msolve(Aprod(y1))  
1254    u1 = v    u1 = v
1255      
1256    theta = 0.0;    theta = 0.0;
1257    eta = 0.0;    eta = 0.0;
1258    tau = math.sqrt(bilinearform(r,r))    rho=bilinearform(r,r)
1259    error = [ error, tau ]    if rho < 0: raise NegativeNorm,"negative norm."
1260    rho = tau * tau    tau = math.sqrt(rho)
1261    m=1    norm_r0=tau
1262  #    while tau>atol+rtol*norm_r0:
 #  TFQMR iteration  
 #  
 #  while ( iter < kmax-1 ):  
     
   while not stoppingcriterium(tau*math.sqrt ( m + 1 ),norm_b,'TFQMR'):  
1263      if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max      if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
1264    
1265      sigma = bilinearform(r,v)      sigma = bilinearform(r,v)
1266        if sigma == 0.0: raise IterationBreakDown,'TFQMR breakdown, sigma=0'
     if ( sigma == 0.0 ):  
       raise 'TFQMR breakdown, sigma=0'  
       
1267    
1268      alpha = rho / sigma      alpha = rho / sigma
1269    
# Line 1108  def TFQMR(b, Aprod, Msolve, bilinearform Line 1273  def TFQMR(b, Aprod, Msolve, bilinearform
1273  #  #
1274        if ( j == 1 ):        if ( j == 1 ):
1275          y2 = y1 - alpha * v          y2 = y1 - alpha * v
1276          u2 = Msolve(Aprod(y2))          u2 = Aprod(y2)
1277    
1278        m = 2 * (iter+1) - 2 + (j+1)        m = 2 * (iter+1) - 2 + (j+1)
1279        if j==0:        if j==0:
1280           w = w - alpha * u1           w = w - alpha * u1
1281           d = y1 + ( theta * theta * eta / alpha ) * d           d = y1 + ( theta * theta * eta / alpha ) * d
1282        if j==1:        if j==1:
# Line 1126  def TFQMR(b, Aprod, Msolve, bilinearform Line 1291  def TFQMR(b, Aprod, Msolve, bilinearform
1291  #  #
1292  #  Try to terminate the iteration at each pass through the loop  #  Try to terminate the iteration at each pass through the loop
1293  #  #
1294       # if ( tau * math.sqrt ( m + 1 ) <= errtol ):      if rho == 0.0: raise IterationBreakDown,'TFQMR breakdown, rho=0'
      #   error = [ error, tau ]  
      #   total_iters = iter  
      #   break  
         
   
     if ( rho == 0.0 ):  
       raise 'TFQMR breakdown, rho=0'  
       
1295    
1296      rhon = bilinearform(r,w)      rhon = bilinearform(r,w)
1297      beta = rhon / rho;      beta = rhon / rho;
1298      rho = rhon;      rho = rhon;
1299      y1 = w + beta * y2;      y1 = w + beta * y2;
1300      u1 = Msolve(Aprod(y1))      u1 = Aprod(y1)
1301      v = u1 + beta * ( u2 + beta * v )      v = u1 + beta * ( u2 + beta * v )
1302      error = [ error, tau ]  
1303      total_iters = iter      iter += 1
       
     iter = iter + 1  
1304    
1305    return x    return x
1306    
# Line 1154  def TFQMR(b, Aprod, Msolve, bilinearform Line 1309  def TFQMR(b, Aprod, Msolve, bilinearform
1309    
1310  class ArithmeticTuple(object):  class ArithmeticTuple(object):
1311     """     """
1312     tuple supporting inplace update x+=y and scaling x=a*y where x,y is an ArithmeticTuple and a is a float.     Tuple supporting inplace update x+=y and scaling x=a*y where ``x,y`` is an
1313       ArithmeticTuple and ``a`` is a float.
1314    
1315     example of usage:     Example of usage::
1316    
1317     from esys.escript import Data         from esys.escript import Data
1318     from numarray import array         from numpy import array
1319     a=Data(...)         a=Data(...)
1320     b=array([1.,4.])         b=array([1.,4.])
1321     x=ArithmeticTuple(a,b)         x=ArithmeticTuple(a,b)
1322     y=5.*x         y=5.*x
1323    
1324     """     """
1325     def __init__(self,*args):     def __init__(self,*args):
1326         """         """
1327         initialize object with elements args.         Initializes object with elements ``args``.
1328    
1329         @param args: tuple of object that support implace add (x+=y) and scaling (x=a*y)         :param args: tuple of objects that support inplace add (x+=y) and
1330                        scaling (x=a*y)
1331         """         """
1332         self.__items=list(args)         self.__items=list(args)
1333    
1334     def __len__(self):     def __len__(self):
1335         """         """
1336         number of items         Returns the number of items.
1337    
1338         @return: number of items         :return: number of items
1339         @rtype: C{int}         :rtype: ``int``
1340         """         """
1341         return len(self.__items)         return len(self.__items)
1342    
1343     def __getitem__(self,index):     def __getitem__(self,index):
1344         """         """
1345         get an item         Returns item at specified position.
1346    
1347         @param index: item to be returned         :param index: index of item to be returned
1348         @type index: C{int}         :type index: ``int``
1349         @return: item with index C{index}         :return: item with index ``index``
1350         """         """
1351         return self.__items.__getitem__(index)         return self.__items.__getitem__(index)
1352    
1353     def __mul__(self,other):     def __mul__(self,other):
1354         """         """
1355         scaling from the right         Scales by ``other`` from the right.
1356    
1357         @param other: scaling factor         :param other: scaling factor
1358         @type other: C{float}         :type other: ``float``
1359         @return: itemwise self*other         :return: itemwise self*other
1360         @rtype: L{ArithmeticTuple}         :rtype: `ArithmeticTuple`
1361         """         """
1362         out=[]         out=[]
1363         other=1.*other         try:
1364         if isinstance(other,float):             l=len(other)
1365      for i in range(len(self)):             if l!=len(self):
1366             out.append(self[i]*other)                 raise ValueError,"length of arguments don't match."
1367         else:             for i in range(l): out.append(self[i]*other[i])
1368          for i in range(len(self)):         except TypeError:
1369             out.append(self[i]*other[i])             for i in range(len(self)): out.append(self[i]*other)
1370         return ArithmeticTuple(*tuple(out))         return ArithmeticTuple(*tuple(out))
1371    
1372     def __rmul__(self,other):     def __rmul__(self,other):
1373         """         """
1374         scaling from the left         Scales by ``other`` from the left.
1375    
1376         @param other: scaling factor         :param other: scaling factor
1377         @type other: C{float}         :type other: ``float``
1378         @return: itemwise other*self         :return: itemwise other*self
1379         @rtype: L{ArithmeticTuple}         :rtype: `ArithmeticTuple`
1380         """         """
1381         out=[]         out=[]
1382         other=1.*other         try:
1383         if isinstance(other,float):             l=len(other)
1384      for i in range(len(self)):             if l!=len(self):
1385             out.append(other*self[i])                 raise ValueError,"length of arguments don't match."
1386         else:             for i in range(l): out.append(other[i]*self[i])
1387          for i in range(len(self)):         except TypeError:
1388             out.append(other[i]*self[i])             for i in range(len(self)): out.append(other*self[i])
1389         return ArithmeticTuple(*tuple(out))         return ArithmeticTuple(*tuple(out))
1390    
 #########################  
 # Added by Artak  
 #########################  
1391     def __div__(self,other):     def __div__(self,other):
1392         """         """
1393         dividing from the right         Scales by (1/``other``) from the right.
1394    
1395         @param other: scaling factor         :param other: scaling factor
1396         @type other: C{float}         :type other: ``float``
1397         @return: itemwise self/other         :return: itemwise self/other
1398         @rtype: L{ArithmeticTuple}         :rtype: `ArithmeticTuple`
1399         """         """
1400         out=[]         return self*(1/other)
        other=1.*other  
        if isinstance(other,float):  
     for i in range(len(self)):  
            out.append(self[i]*(1./other))  
        else:  
         for i in range(len(self)):  
            out.append(self[i]*(1./other[i]))  
        return ArithmeticTuple(*tuple(out))  
1401    
1402     def __rdiv__(self,other):     def __rdiv__(self,other):
1403         """         """
1404         dividing from the left         Scales by (1/``other``) from the left.
1405    
1406         @param other: scaling factor         :param other: scaling factor
1407         @type other: C{float}         :type other: ``float``
1408         @return: itemwise other/self         :return: itemwise other/self
1409         @rtype: L{ArithmeticTuple}         :rtype: `ArithmeticTuple`
1410         """         """
1411         out=[]         out=[]
1412         other=1.*other         try:
1413         if isinstance(other,float):             l=len(other)
1414          for i in range(len(self)):             if l!=len(self):
1415             out.append(other*(1./self[i]))                 raise ValueError,"length of arguments don't match."
1416         else:             for i in range(l): out.append(other[i]/self[i])
1417          for i in range(len(self)):         except TypeError:
1418             out.append(other[i]*(1./self[i]))             for i in range(len(self)): out.append(other/self[i])
1419         return ArithmeticTuple(*tuple(out))         return ArithmeticTuple(*tuple(out))
     
 ##########################################33  
1420    
1421     def __iadd__(self,other):     def __iadd__(self,other):
1422         """         """
1423         in-place add of other to self         Inplace addition of ``other`` to self.
1424    
1425         @param other: increment         :param other: increment
1426         @type other: C{ArithmeticTuple}         :type other: ``ArithmeticTuple``
1427         """         """
1428         if len(self) != len(other):         if len(self) != len(other):
1429             raise ValueError,"tuple length must match."             raise ValueError,"tuple lengths must match."
1430         for i in range(len(self)):         for i in range(len(self)):
1431             self.__items[i]+=other[i]             self.__items[i]+=other[i]
1432         return self         return self
1433    
1434     def __add__(self,other):     def __add__(self,other):
1435         """         """
1436         add other to self         Adds ``other`` to self.
1437    
1438         @param other: increment         :param other: increment
1439         @type other: C{ArithmeticTuple}         :type other: ``ArithmeticTuple``
1440         """         """
1441  #      if not isinstance(other,float):         out=[]
1442         if len(self) != len(other):         try:
1443            raise ValueError,"tuple length must match."             l=len(other)
1444         for i in range(len(self)):             if l!=len(self):
1445            self.__items[i]+=other[i]                 raise ValueError,"length of arguments don't match."
1446  #       else:             for i in range(l): out.append(self[i]+other[i])
1447  #        for i in range(len(self)):         except TypeError:
1448  #           self.__items[i]+=other             for i in range(len(self)): out.append(self[i]+other)
1449           return ArithmeticTuple(*tuple(out))
        return self  
1450    
1451     def __sub__(self,other):     def __sub__(self,other):
1452         """         """
1453         subtract other from self         Subtracts ``other`` from self.
1454    
1455         @param other: increment         :param other: decrement
1456         @type other: C{ArithmeticTuple}         :type other: ``ArithmeticTuple``
1457         """         """
1458         if len(self) != len(other):         out=[]
1459             raise ValueError,"tuple length must match."         try:
1460         for i in range(len(self)):             l=len(other)
1461             self.__items[i]-=other[i]             if l!=len(self):
1462         return self                 raise ValueError,"length of arguments don't match."
1463                 for i in range(l): out.append(self[i]-other[i])
1464           except TypeError:
1465               for i in range(len(self)): out.append(self[i]-other)
1466           return ArithmeticTuple(*tuple(out))
1467    
1468     def __isub__(self,other):     def __isub__(self,other):
1469         """         """
1470         subtract other from self         Inplace subtraction of ``other`` from self.
1471    
1472         @param other: increment         :param other: decrement
1473         @type other: C{ArithmeticTuple}         :type other: ``ArithmeticTuple``
1474         """         """
1475         if len(self) != len(other):         if len(self) != len(other):
1476             raise ValueError,"tuple length must match."             raise ValueError,"tuple length must match."
# Line 1333  class ArithmeticTuple(object): Line 1480  class ArithmeticTuple(object):
1480    
1481     def __neg__(self):     def __neg__(self):
1482         """         """
1483         negate         Negates values.
   
1484         """         """
1485           out=[]
1486         for i in range(len(self)):         for i in range(len(self)):
1487             self.__items[i]=-self.__items[i]             out.append(-self[i])
1488         return self         return ArithmeticTuple(*tuple(out))
1489    
1490    
1491  class HomogeneousSaddlePointProblem(object):  class HomogeneousSaddlePointProblem(object):
1492        """        """
1493        This provides a framwork for solving homogeneous saddle point problem of the form        This class provides a framework for solving linear homogeneous saddle
1494          point problems of the form::
              Av+B^*p=f  
              Bv    =0  
1495    
1496        for the unknowns v and p and given operators A and B and given right hand side f.            *Av+B^*p=f*
1497        B^* is the adjoint operator of B is the given inner product.            *Bv     =0*
1498    
1499          for the unknowns *v* and *p* and given operators *A* and *B* and
1500          given right hand side *f*. *B^** is the adjoint operator of *B*.
1501          *A* may depend weakly on *v* and *p*.
1502        """        """
1503        def __init__(self,**kwargs):        def __init__(self, **kwargs):
1504        """
1505        initializes the saddle point problem
1506        """
1507            self.resetControlParameters()
1508          self.setTolerance()          self.setTolerance()
1509          self.setToleranceReductionFactor()          self.setAbsoluteTolerance()
1510          def resetControlParameters(self,gamma=0.85, gamma_min=1.e-2,chi_max=0.1, omega_div=0.2, omega_conv=1.1, rtol_min=1.e-7, rtol_max=0.9, chi=1., C_p=1., C_v=1., safety_factor=0.3):
1511             """
1512             sets a control parameter
1513    
1514             :param gamma: ``1/(1-gamma)`` controls the perturbation of the converegence rate due to termination errors in the subproblems.
1515             :type gamma: ``float``
1516             :param gamma_min: minimum value for ``gamma``.
1517             :type gamma_min: ``float``
1518             :param chi_max: maximum tolerable converegence rate.
1519             :type chi_max: ``float``
1520             :param omega_div: reduction fact for ``gamma`` if no convergence is detected.
1521             :type omega_div: ``float``
1522             :param omega_conv: raise fact for ``gamma`` if convergence is detected.
1523             :type omega_conv: ``float``
1524             :param rtol_min: minimum relative tolerance used to calculate presssure and velocity increment.
1525             :type rtol_min: ``float``
1526             :param rtol_max: maximuim relative tolerance used to calculate presssure and velocity increment.
1527             :type rtol_max: ``float``
1528             :param chi: initial convergence measure.
1529             :type chi: ``float``
1530             :param C_p: initial value for constant to adjust pressure tolerance
1531             :type C_p: ``float``
1532             :param C_v: initial value for constant to adjust velocity tolerance
1533             :type C_v: ``float``
1534             :param safety_factor: safety factor for addjustment of pressure and velocity tolerance from stopping criteria
1535             :type safety_factor: ``float``
1536             """
1537             self.setControlParameter(gamma, gamma_min ,chi_max , omega_div , omega_conv, rtol_min , rtol_max, chi,C_p, C_v,safety_factor)
1538    
1539          def setControlParameter(self,gamma=None, gamma_min=None ,chi_max=None, omega_div=None, omega_conv=None, rtol_min=None, rtol_max=None, chi=None, C_p=None, C_v=None, safety_factor=None):
1540             """
1541             sets a control parameter
1542    
1543             :param gamma: ``1/(1-gamma)`` controls the perturbation of the converegence rate due to termination errors in the subproblems.
1544             :type gamma: ``float``
1545             :param gamma_min: minimum value for ``gamma``.
1546             :type gamma_min: ``float``
1547             :param chi_max: maximum tolerable converegence rate.
1548             :type chi_max: ``float``
1549             :param omega_div: reduction fact for ``gamma`` if no convergence is detected.
1550             :type omega_div: ``float``
1551             :param omega_conv: raise fact for ``gamma`` if convergence is detected.
1552             :type omega_conv: ``float``
1553             :param rtol_min: minimum relative tolerance used to calculate presssure and velocity increment.
1554             :type rtol_min: ``float``
1555             :param rtol_max: maximuim relative tolerance used to calculate presssure and velocity increment.
1556             :type rtol_max: ``float``
1557             :param chi: initial convergence measure.
1558             :type chi: ``float``
1559             :param C_p: initial value for constant to adjust pressure tolerance
1560             :type C_p: ``float``
1561             :param C_v: initial value for constant to adjust velocity tolerance
1562             :type C_v: ``float``
1563             :param safety_factor: safety factor for addjustment of pressure and velocity tolerance from stopping criteria
1564             :type safety_factor: ``float``
1565             """
1566             if not gamma == None:
1567                if gamma<=0 or gamma>=1:
1568                   raise ValueError,"Initial gamma needs to be positive and less than 1."
1569             else:
1570                gamma=self.__gamma
1571    
1572             if not gamma_min == None:
1573                if gamma_min<=0 or gamma_min>=1:
1574                   raise ValueError,"gamma_min needs to be positive and less than 1."
1575             else:
1576                gamma_min = self.__gamma_min
1577    
1578             if not chi_max == None:
1579                if chi_max<=0 or chi_max>=1:
1580                   raise ValueError,"chi_max needs to be positive and less than 1."
1581             else:
1582                chi_max = self.__chi_max
1583    
1584             if not omega_div == None:
1585                if omega_div<=0 or omega_div >=1:
1586                   raise ValueError,"omega_div needs to be positive and less than 1."
1587             else:
1588                omega_div=self.__omega_div
1589    
1590             if not omega_conv == None:
1591                if omega_conv<1:
1592                   raise ValueError,"omega_conv needs to be greater or equal to 1."
1593             else:
1594                omega_conv=self.__omega_conv
1595    
1596             if not rtol_min == None:
1597                if rtol_min<=0 or rtol_min>=1:
1598                   raise ValueError,"rtol_min needs to be positive and less than 1."
1599             else:
1600                rtol_min=self.__rtol_min
1601    
1602             if not rtol_max == None:
1603                if rtol_max<=0 or rtol_max>=1:
1604                   raise ValueError,"rtol_max needs to be positive and less than 1."
1605             else:
1606                rtol_max=self.__rtol_max
1607    
1608             if not chi == None:
1609                if chi<=0 or chi>1:
1610                   raise ValueError,"chi needs to be positive and less than 1."
1611             else:
1612                chi=self.__chi
1613    
1614             if not C_p == None:
1615                if C_p<1:
1616                   raise ValueError,"C_p need to be greater or equal to 1."
1617             else:
1618                C_p=self.__C_p
1619    
1620             if not C_v == None:
1621                if C_v<1:
1622                   raise ValueError,"C_v need to be greater or equal to 1."
1623             else:
1624                C_v=self.__C_v
1625    
1626             if not safety_factor == None:
1627                if safety_factor<=0 or safety_factor>1:
1628                   raise ValueError,"safety_factor need to be between zero and one."
1629             else:
1630                safety_factor=self.__safety_factor
1631    
1632             if gamma<gamma_min:
1633                   raise ValueError,"gamma = %e needs to be greater or equal gamma_min = %e."%(gamma,gamma_min)
1634             if rtol_max<=rtol_min:
1635                   raise ValueError,"rtol_max = %e needs to be greater rtol_min = %e."%(rtol_max,rtol_min)
1636                
1637             self.__gamma = gamma
1638             self.__gamma_min = gamma_min
1639             self.__chi_max = chi_max
1640             self.__omega_div = omega_div
1641             self.__omega_conv = omega_conv
1642             self.__rtol_min = rtol_min
1643             self.__rtol_max = rtol_max
1644             self.__chi = chi
1645             self.__C_p = C_p
1646             self.__C_v = C_v
1647             self.__safety_factor = safety_factor
1648    
1649          #=============================================================
1650          def inner_pBv(self,p,Bv):
1651             """
1652             Returns inner product of element p and Bv (overwrite).
1653    
1654             :param p: a pressure increment
1655             :param Bv: a residual
1656             :return: inner product of element p and Bv
1657             :rtype: ``float``
1658             :note: used if PCG is applied.
1659             """
1660             raise NotImplementedError,"no inner product for p and Bv implemented."
1661    
1662          def inner_p(self,p0,p1):
1663             """
1664             Returns inner product of p0 and p1 (overwrite).
1665    
1666             :param p0: a pressure
1667             :param p1: a pressure
1668             :return: inner product of p0 and p1
1669             :rtype: ``float``
1670             """
1671             raise NotImplementedError,"no inner product for p implemented."
1672      
1673          def norm_v(self,v):
1674             """
1675             Returns the norm of v (overwrite).
1676    
1677             :param v: a velovity
1678             :return: norm of v
1679             :rtype: non-negative ``float``
1680             """
1681             raise NotImplementedError,"no norm of v implemented."
1682          def getDV(self, p, v, tol):
1683             """
1684             return a correction to the value for a given v and a given p with accuracy `tol` (overwrite)
1685    
1686        def initialize(self):           :param p: pressure
1687             :param v: pressure
1688             :return: dv given as *dv= A^{-1} (f-A v-B^*p)*
1689             :note: Only *A* may depend on *v* and *p*
1690             """
1691             raise NotImplementedError,"no dv calculation implemented."
1692    
1693            
1694          def Bv(self,v, tol):
1695            """
1696            Returns Bv with accuracy `tol` (overwrite)
1697    
1698            :rtype: equal to the type of p
1699            :note: boundary conditions on p should be zero!
1700          """          """
1701          initialize the problem (overwrite)          raise NotImplementedError, "no operator B implemented."
1702    
1703          def norm_Bv(self,Bv):
1704            """
1705            Returns the norm of Bv (overwrite).
1706    
1707            :rtype: equal to the type of p
1708            :note: boundary conditions on p should be zero!
1709          """          """
1710          pass          raise NotImplementedError, "no norm of Bv implemented."
1711        def B(self,v):  
1712          def solve_AinvBt(self,dp, tol):
1713           """           """
1714           returns Bv (overwrite)           Solves *A dv=B^*dp* with accuracy `tol`
          @rtype: equal to the type of p  
1715    
1716           @note: boundary conditions on p should be zero!           :param dp: a pressure increment
1717             :return: the solution of *A dv=B^*dp*
1718             :note: boundary conditions on dv should be zero! *A* is the operator used in ``getDV`` and must not be altered.
1719           """           """
1720           pass           raise NotImplementedError,"no operator A implemented."
1721    
1722        def inner(self,p0,p1):        def solve_prec(self,Bv, tol):
1723           """           """
1724           returns inner product of two element p0 and p1  (overwrite)           Provides a preconditioner for *(BA^{-1}B^ * )* applied to Bv with accuracy `tol`
           
          @type p0: equal to the type of p  
          @type p1: equal to the type of p  
          @rtype: C{float}  
1725    
1726           @rtype: equal to the type of p           :rtype: equal to the type of p
1727             :note: boundary conditions on p should be zero!
1728             """
1729             raise NotImplementedError,"no preconditioner for Schur complement implemented."
1730          #=============================================================
1731          def __Aprod_PCG(self,dp):
1732              dv=self.solve_AinvBt(dp, self.__subtol)
1733              return ArithmeticTuple(dv,self.Bv(dv, self.__subtol))
1734    
1735          def __inner_PCG(self,p,r):
1736             return self.inner_pBv(p,r[1])
1737    
1738          def __Msolve_PCG(self,r):
1739              return self.solve_prec(r[1], self.__subtol)
1740          #=============================================================
1741          def __Aprod_GMRES(self,p):
1742              return self.solve_prec(self.Bv(self.solve_AinvBt(p, self.__subtol), self.__subtol), self.__subtol)
1743          def __inner_GMRES(self,p0,p1):
1744             return self.inner_p(p0,p1)
1745    
1746          #=============================================================
1747          def norm_p(self,p):
1748              """
1749              calculates the norm of ``p``
1750              
1751              :param p: a pressure
1752              :return: the norm of ``p`` using the inner product for pressure
1753              :rtype: ``float``
1754              """
1755              f=self.inner_p(p,p)
1756              if f<0: raise ValueError,"negative pressure norm."
1757              return math.sqrt(f)
1758          
1759          def solve(self,v,p,max_iter=20, verbose=False, usePCG=True, iter_restart=20, max_correction_steps=10):
1760           """           """
1761           pass           Solves the saddle point problem using initial guesses v and p.
1762    
1763        def solve_A(self,u,p):           :param v: initial guess for velocity
1764             :param p: initial guess for pressure
1765             :type v: `Data`
1766             :type p: `Data`
1767             :param usePCG: indicates the usage of the PCG rather than GMRES scheme.
1768             :param max_iter: maximum number of iteration steps per correction
1769                              attempt
1770             :param verbose: if True, shows information on the progress of the
1771                             saddlepoint problem solver.
1772             :param iter_restart: restart the iteration after ``iter_restart`` steps
1773                                  (only used if useUzaw=False)
1774             :type usePCG: ``bool``
1775             :type max_iter: ``int``
1776             :type verbose: ``bool``
1777             :type iter_restart: ``int``
1778             :rtype: ``tuple`` of `Data` objects
1779             :note: typically this method is overwritten by a subclass. It provides a wrapper for the ``_solve`` method.
1780           """           """
1781           solves Av=f-Au-B^*p with accuracy self.getReducedTolerance() (overwrite)           return self._solve(v=v,p=p,max_iter=max_iter,verbose=verbose, usePCG=usePCG, iter_restart=iter_restart, max_correction_steps=max_correction_steps)
1782    
1783           @rtype: equal to the type of v        def _solve(self,v,p,max_iter=20, verbose=False, usePCG=True, iter_restart=20, max_correction_steps=10):
1784           @note: boundary conditions on v should be zero!           """
1785             see `_solve` method.
1786           """           """
1787           pass           self.verbose=verbose
1788             rtol=self.getTolerance()
1789             atol=self.getAbsoluteTolerance()
1790             correction_step=0
1791             converged=False
1792             error=None
1793             chi=None
1794             gamma=self.__gamma
1795             C_p=self.__C_p
1796             C_v=self.__C_v
1797             while not converged:
1798                  if error== None or chi == None:
1799                      gamma_new=gamma/self.__omega_conv
1800                  else:
1801                     if chi < self.__chi_max:
1802                        gamma_new=min(max(gamma*self.__omega_conv,1-chi*error/(self.__safety_factor*ATOL)), 1-chi/self.__chi_max)
1803                     else:
1804                        gamma_new=gamma*self.__omega_div
1805                  gamma=max(gamma_new, self.__gamma_min)
1806                  # calculate velocity for current pressure:
1807                  rtol_v=min(max(gamma/(1.+gamma)/C_v,self.__rtol_min), self.__rtol_max)
1808                  rtol_p=min(max(gamma/C_p, self.__rtol_min), self.__rtol_max)
1809                  self.__subtol=rtol_p**2
1810                  if self.verbose: print "HomogeneousSaddlePointProblem: step %s: gamma = %e, rtol_v= %e, rtol_p=%e"%(correction_step,gamma,rtol_v,rtol_p)
1811                  if self.verbose: print "HomogeneousSaddlePointProblem: subtolerance: %e"%self.__subtol
1812                  # calculate velocity for current pressure: A*dv= F-A*v-B*p
1813                  dv1=self.getDV(p,v,rtol_v)
1814                  v1=v+dv1
1815                  Bv1=self.Bv(v1, self.__subtol)
1816                  norm_Bv1=self.norm_Bv(Bv1)
1817                  norm_dv1=self.norm_v(dv1)
1818                  norm_v1=self.norm_v(v1)
1819                  ATOL=norm_v1*rtol+atol
1820                  if self.verbose: print "HomogeneousSaddlePointProblem: step %s: Bv = %e, dv = %e, v=%e"%(correction_step,norm_Bv1, norm_dv1, norm_v1)
1821                  if not ATOL>0: raise ValueError,"overall absolute tolerance needs to be positive."
1822                  if max(norm_Bv1, norm_dv1) <= ATOL:
1823                      converged=True
1824                      v=v1
1825                  else:
1826                      # now we solve for the pressure increment dp from B*A^{-1}B^* dp = Bv1
1827                      if usePCG:
1828                        dp,r,a_norm=PCG(ArithmeticTuple(v1,Bv1),self.__Aprod_PCG,0*p,self.__Msolve_PCG,self.__inner_PCG,atol=0, rtol=rtol_p,iter_max=max_iter, verbose=self.verbose)
1829                        v2=r[0]
1830                        Bv2=r[1]
1831                      else:
1832                        dp=GMRES(self.solve_prec(Bv1,self.__subtol),self.__Aprod_GMRES, 0*p, self.__inner_GMRES,atol=0, rtol=rtol_p,iter_max=max_iter, iter_restart=iter_restart, verbose=self.verbose)
1833                        dv2=self.solve_AinvBt(dp, self.__subtol)
1834                        v2=v1-dv2
1835                        Bv2=self.Bv(v2, self.__subtol)
1836                      #
1837                      # convergence indicators:
1838                      #
1839                      norm_v2=self.norm_v(v2)
1840                      norm_dv2=self.norm_v(v2-v)
1841                      error_new=max(norm_dv2, norm_Bv1)
1842                      norm_Bv2=self.norm_Bv(Bv2)
1843                      if self.verbose: print "HomogeneousSaddlePointProblem: step %s (part 2): Bv = %e, dv = %e, v=%e"%(correction_step,norm_Bv2, norm_dv2, norm_v2)
1844                      if error !=None:
1845                          chi_new=error_new/error
1846                          if self.verbose: print "HomogeneousSaddlePointProblem: step %s: convergence rate = %e, est. error = %e"%(correction_step,chi_new, error_new)
1847                          if chi != None:
1848                              gamma0=max(gamma, 1-chi/chi_new)
1849                              C_p*=gamma0/gamma
1850                              C_v*=gamma0/gamma*(1+gamma)/(1+gamma0)
1851                          chi=chi_new
1852                      else:
1853                          if self.verbose: print "HomogeneousSaddlePointProblem: step %s: est. error = %e"%(correction_step, error_new)
1854    
1855                      error = error_new
1856                      correction_step+=1
1857                      if correction_step>max_correction_steps:
1858                            raise CorrectionFailed,"Given up after %d correction steps."%correction_step
1859                      v,p=v2,p+dp
1860             if self.verbose: print "HomogeneousSaddlePointProblem: tolerance reached after %s steps."%correction_step
1861         return v,p
1862    
1863        def solve_prec(self,p):        #========================================================================
1864          def setTolerance(self,tolerance=1.e-4):
1865           """           """
1866           provides a preconditioner for BA^{-1}B^* with accuracy self.getReducedTolerance() (overwrite)           Sets the relative tolerance for (v,p).
1867    
1868           @rtype: equal to the type of p           :param tolerance: tolerance to be used
1869             :type tolerance: non-negative ``float``
1870           """           """
1871           pass           if tolerance<0:
1872                 raise ValueError,"tolerance must be positive."
1873             self.__rtol=tolerance
1874    
1875        def stoppingcriterium(self,Bv,v,p):        def getTolerance(self):
1876           """           """
1877           returns a True if iteration is terminated. (overwrite)           Returns the relative tolerance.
1878    
1879           @rtype: C{bool}           :return: relative tolerance
1880             :rtype: ``float``
1881           """           """
1882           pass           return self.__rtol
               
       def __inner(self,p,r):  
          return self.inner(p,r[1])  
1883    
1884        def __inner_p(self,p1,p2):        def setAbsoluteTolerance(self,tolerance=0.):
1885           return self.inner(p1,p2)           """
1886                   Sets the absolute tolerance.
       def __inner_a(self,a1,a2):  
          return self.inner_a(a1,a2)  
1887    
1888        def __inner_a1(self,a1,a2):           :param tolerance: tolerance to be used
1889           return self.inner(a1[1],a2[1])           :type tolerance: non-negative ``float``
1890             """
1891             if tolerance<0:
1892                 raise ValueError,"tolerance must be non-negative."
1893             self.__atol=tolerance
1894    
1895        def __stoppingcriterium(self,norm_r,r,p):        def getAbsoluteTolerance(self):
1896            return self.stoppingcriterium(r[1],r[0],p)           """
1897             Returns the absolute tolerance.
1898    
1899        def __stoppingcriterium2(self,norm_r,norm_b,solver='GMRES',TOL=None):           :return: absolute tolerance
1900            return self.stoppingcriterium2(norm_r,norm_b,solver,TOL)           :rtype: ``float``
1901             """
1902             return self.__atol
1903    
1904        def setTolerance(self,tolerance=1.e-8):  
1905                self.__tol=tolerance  def MaskFromBoundaryTag(domain,*tags):
       def getTolerance(self):  
               return self.__tol  
       def setToleranceReductionFactor(self,reduction=0.01):  
               self.__reduction=reduction  
       def getSubProblemTolerance(self):  
               return self.__reduction*self.getTolerance()  
   
       def solve(self,v,p,max_iter=20, verbose=False, show_details=False, solver='PCG',iter_restart=20):  
               """  
               solves the saddle point problem using initial guesses v and p.  
   
               @param max_iter: maximum number of iteration steps.  
               """  
               self.verbose=verbose  
               self.show_details=show_details and self.verbose  
   
               # assume p is known: then v=A^-1(f-B^*p)  
               # which leads to BA^-1B^*p = BA^-1f    
   
           # Az=f is solved as A(z-v)=f-Av (z-v = 0 on fixed_u_mask)        
           self.__z=v+self.solve_A(v,p*0)  
               Bz=self.B(self.__z)  
               #  
           #   solve BA^-1B^*p = Bz  
               #  
               #  
               #  
               self.iter=0  
           if solver=='GMRES':        
                 if self.verbose: print "enter GMRES method (iter_max=%s)"%max_iter  
                 p=GMRES(Bz,self.__Aprod2,self.__Msolve2,self.__inner_p,self.__stoppingcriterium2,iter_max=max_iter, x=p*1.,iter_restart=iter_restart)  
                 # solve Au=f-B^*p  
                 #       A(u-v)=f-B^*p-Av  
                 #       u=v+(u-v)  
         u=v+self.solve_A(v,p)  
   
           if solver=='NewtonGMRES':      
                 if self.verbose: print "enter NewtonGMRES method (iter_max=%s)"%max_iter  
                 p=NewtonGMRES(Bz,self.__Aprod_Newton,self.__Msolve2,self.__inner_p,self.__stoppingcriterium2,iter_max=max_iter, x=p*1.,atol=0,rtol=self.getTolerance())  
                 # solve Au=f-B^*p  
                 #       A(u-v)=f-B^*p-Av  
                 #       u=v+(u-v)  
         u=v+self.solve_A(v,p)  
                   
   
           if solver=='TFQMR':        
                 if self.verbose: print "enter TFQMR method (iter_max=%s)"%max_iter  
                 p=TFQMR(Bz,self.__Aprod2,self.__Msolve2,self.__inner_p,self.__stoppingcriterium2,iter_max=max_iter, x=p*1.)  
                 # solve Au=f-B^*p  
                 #       A(u-v)=f-B^*p-Av  
                 #       u=v+(u-v)  
         u=v+self.solve_A(v,p)  
   
           if solver=='MINRES':        
                 if self.verbose: print "enter MINRES method (iter_max=%s)"%max_iter  
                 p=MINRES(Bz,self.__Aprod2,self.__Msolve2,self.__inner_p,self.__stoppingcriterium2,iter_max=max_iter, x=p*1.)  
                 # solve Au=f-B^*p  
                 #       A(u-v)=f-B^*p-Av  
                 #       u=v+(u-v)  
         u=v+self.solve_A(v,p)  
                 
           if solver=='GMRESC':        
                 if self.verbose: print "enter GMRES coupled method (iter_max=%s)"%max_iter  
                 p0=self.solve_prec1(Bz)  
             #alfa=(1/self.vol)*util.integrate(util.interpolate(p,escript.Function(self.domain)))  
                 #p-=alfa  
                 x=GMRES(ArithmeticTuple(self.__z*1.,p0*1),self.__Anext,self.__Mempty,self.__inner_a,self.__stoppingcriterium2,iter_max=max_iter, x=ArithmeticTuple(v*1,p*1),iter_restart=20)  
                 #x=NewtonGMRES(ArithmeticTuple(self.__z*1.,p0*1),self.__Aprod_Newton2,self.__Mempty,self.__inner_a,self.__stoppingcriterium2,iter_max=max_iter, x=ArithmeticTuple(v*1,p*1),atol=0,rtol=self.getTolerance())  
   
                 # solve Au=f-B^*p  
                 #       A(u-v)=f-B^*p-Av  
                 #       u=v+(u-v)  
             p=x[1]  
         u=v+self.solve_A(v,p)        
         #p=x[1]  
         #u=x[0]  
   
               if solver=='PCG':  
                 #   note that the residual r=Bz-BA^-1B^*p = B(z-A^-1B^*p) = Bv  
                 #  
                 #   with                    Av=Az-B^*p = f - B^*p (v=z on fixed_u_mask)  
                 #                           A(v-z)= f -Az - B^*p (v-z=0 on fixed_u_mask)  
                 if self.verbose: print "enter PCG method (iter_max=%s)"%max_iter  
                 p,r=PCG(ArithmeticTuple(self.__z*1.,Bz),self.__Aprod,self.__Msolve,self.__inner,self.__stoppingcriterium,iter_max=max_iter, x=p)  
             u=r[0]    
                 print "DIFF=",util.integrate(p)  
   
               print "RESULT div(u)=",util.Lsup(self.B(u)),util.Lsup(u)  
   
           return u,p  
   
       def __Msolve(self,r):  
           return self.solve_prec(r[1])  
   
       def __Msolve2(self,r):  
           return self.solve_prec(r*1)  
   
       def __Mempty(self,r):  
           return r  
   
   
       def __Aprod(self,p):  
           # return BA^-1B*p  
           #solve Av =B^*p as Av =f-Az-B^*(-p)  
           v=self.solve_A(self.__z,-p)  
           return ArithmeticTuple(v, self.B(v))  
   
       def __Anext(self,x):  
           # return next v,p  
           #solve Av =-B^*p as Av =f-Az-B^*p  
   
       pc=x[1]  
           v=self.solve_A(self.__z,-pc)  
       p=self.solve_prec1(self.B(v))  
   
           return ArithmeticTuple(v,p)  
   
   
       def __Aprod2(self,p):  
           # return BA^-1B*p  
           #solve Av =B^*p as Av =f-Az-B^*(-p)  
       v=self.solve_A(self.__z,-p)  
           return self.B(v)  
   
       def __Aprod_Newton(self,p):  
           # return BA^-1B*p - Bz  
           #solve Av =-B^*p as Av =f-Az-B^*p  
       v=self.solve_A(self.__z,-p)  
           return self.B(v-self.__z)  
   
       def __Aprod_Newton2(self,x):  
           # return BA^-1B*p - Bz  
           #solve Av =-B^*p as Av =f-Az-B^*p  
           pc=x[1]  
       v=self.solve_A(self.__z,-pc)  
           p=self.solve_prec1(self.B(v-self.__z))  
           return ArithmeticTuple(v,p)  
   
 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:  
   
    M{Q_f (u^{k+1}-u^{k}) = - f(u^{k},p^{k})}  
    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.  
    """  
    def __init__(self,verbose=False,*args):  
        """  
        initializes the problem  
   
        @param verbose: switches on the printing out some information  
        @type verbose: C{bool}  
        @note: this method may be overwritten by a particular saddle point problem  
        """  
        if not isinstance(verbose,bool):  
             raise TypeError("verbose needs to be of type bool.")  
        self.__verbose=verbose  
        self.relaxation=1.  
   
    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  
   
        A_f du = f(u,p)  
   
        with tolerance C{tol} and return du. A_f is Jacobiean of f with respect to u.  
   
        @param u: current approximation of u  
        @type u: L{escript.Data}  
        @param p: current approximation of p  
        @type p: L{escript.Data}  
        @param tol: tolerance expected for du  
        @type tol: C{float}  
        @return: increment du  
        @rtype: L{escript.Data}  
        @note: this method has to be overwritten by a particular saddle point problem  
        """  
        pass  
   
    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  
   
    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  
   
         @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  
 #   def solve(self,u0,p0,tolerance=1.e-6,iter_max=10,self.relaxation=1.):  
 #      tol=1.e-2  
 #      iter=0  
 #      converged=False  
 #      u=u0*1.  
 #      p=p0*1.  
 #      while not converged and iter<iter_max:  
 #          du=self.solve_f(u,p,tol)  
 #          u-=du  
 #          norm_du=util.Lsup(du)  
 #          norm_u=util.Lsup(u)  
 #          
 #          dp=self.relaxation*self.solve_g(u,tol)  
 #          p+=dp  
 #          norm_dp=util.sqrt(self.inner(dp,dp))  
 #          norm_p=util.sqrt(self.inner(p,p))  
 #          print iter,"-th step rel. errror u,p= (%s,%s) (%s,%s)(%s,%s)"%(norm_du,norm_dp,norm_du/norm_u,norm_dp/norm_p,norm_u,norm_p)  
 #          iter+=1  
 #  
 #          converged = (norm_du <= tolerance*norm_u) and  (norm_dp <= tolerance*norm_p)  
 #      if converged:  
 #          print "convergence after %s steps."%iter  
 #      else:  
 #          raise ArithmeticError("no convergence after %s steps."%iter)  
 #  
 #      return u,p  
             
 def MaskFromBoundaryTag(function_space,*tags):  
1906     """     """
1907     create a mask on the given function space which one for samples     Creates a mask on the Solution(domain) function space where the value is
1908     that touch the boundary tagged by tags.     one for samples that touch the boundary tagged by tags.
1909    
1910     usage: m=MaskFromBoundaryTag(Solution(domain),"left", "right")     Usage: m=MaskFromBoundaryTag(domain, "left", "right")
1911    
1912     @param function_space: a given function space     :param domain: domain to be used
1913     @type function_space: L{escript.FunctionSpace}     :type domain: `escript.Domain`
1914     @param tags: boundray tags     :param tags: boundary tags
1915     @type tags: C{str}     :type tags: ``str``
1916     @return: a mask which marks samples used by C{function_space} that are touching the     :return: a mask which marks samples that are touching the boundary tagged
1917              boundary tagged by any of the given tags.              by any of the given tags
1918     @rtype: L{escript.Data} of rank 0     :rtype: `escript.Data` of rank 0
1919     """     """
1920     pde=linearPDEs.LinearPDE(function_space.getDomain(),numEquations=1, numSolutions=1)     pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)
1921     d=escript.Scalar(0.,escript.FunctionOnBoundary(function_space.getDomain()))     d=escript.Scalar(0.,escript.FunctionOnBoundary(domain))
1922     for t in tags: d.setTaggedValue(t,1.)     for t in tags: d.setTaggedValue(t,1.)
1923     pde.setValue(y=d)     pde.setValue(y=d)
1924     out=util.whereNonZero(pde.getRightHandSide())     return util.whereNonZero(pde.getRightHandSide())
    if out.getFunctionSpace() == function_space:  
       return out  
    else:  
       return util.whereNonZero(util.interpolate(out,function_space))  
1925    
1926    def MaskFromTag(domain,*tags):
1927       """
1928       Creates a mask on the Solution(domain) function space where the value is
1929       one for samples that touch regions tagged by tags.
1930    
1931       Usage: m=MaskFromTag(domain, "ham")
1932    
1933       :param domain: domain to be used
1934       :type domain: `escript.Domain`
1935       :param tags: boundary tags
1936       :type tags: ``str``
1937       :return: a mask which marks samples that are touching the boundary tagged
1938                by any of the given tags
1939       :rtype: `escript.Data` of rank 0
1940       """
1941       pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)
1942       d=escript.Scalar(0.,escript.Function(domain))
1943       for t in tags: d.setTaggedValue(t,1.)
1944       pde.setValue(Y=d)
1945       return util.whereNonZero(pde.getRightHandSide())
1946    
1947    

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