/[escript]/trunk/escript/py_src/pdetools.py
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revision 2548 by jfenwick, Mon Jul 20 06:20:06 2009 UTC revision 3981 by jfenwick, Fri Sep 21 02:47:54 2012 UTC
# Line 1  Line 1 
1    
2  ########################################################  ##############################################################################
3  #  #
4  # Copyright (c) 2003-2009 by University of Queensland  # Copyright (c) 2003-2012 by University of Queensland
5  # Earth Systems Science Computational Center (ESSCC)  # http://www.uq.edu.au
 # http://www.uq.edu.au/esscc  
6  #  #
7  # Primary Business: Queensland, Australia  # Primary Business: Queensland, Australia
8  # Licensed under the Open Software License version 3.0  # Licensed under the Open Software License version 3.0
9  # http://www.opensource.org/licenses/osl-3.0.php  # http://www.opensource.org/licenses/osl-3.0.php
10  #  #
11  ########################################################  # Development until 2012 by Earth Systems Science Computational Center (ESSCC)
12    # Development since 2012 by School of Earth Sciences
13    #
14    ##############################################################################
15    
16  __copyright__="""Copyright (c) 2003-2008 by University of Queensland  __copyright__="""Copyright (c) 2003-2012 by University of Queensland
17  Earth Systems Science Computational Center (ESSCC)  http://www.uq.edu.au
 http://www.uq.edu.au/esscc  
18  Primary Business: Queensland, Australia"""  Primary Business: Queensland, Australia"""
19  __license__="""Licensed under the Open Software License version 3.0  __license__="""Licensed under the Open Software License version 3.0
20  http://www.opensource.org/licenses/osl-3.0.php"""  http://www.opensource.org/licenses/osl-3.0.php"""
# Line 28  Currently includes: Line 29  Currently includes:
29      - TimeIntegrationManager - to handle extrapolation in time      - TimeIntegrationManager - to handle extrapolation in time
30      - SaddlePointProblem - solver for Saddle point problems using the inexact uszawa scheme      - SaddlePointProblem - solver for Saddle point problems using the inexact uszawa scheme
31    
32  @var __author__: name of author  :var __author__: name of author
33  @var __copyright__: copyrights  :var __copyright__: copyrights
34  @var __license__: licence agreement  :var __license__: licence agreement
35  @var __url__: url entry point on documentation  :var __url__: url entry point on documentation
36  @var __version__: version  :var __version__: version
37  @var __date__: date of the version  :var __date__: date of the version
38  """  """
39    
40  __author__="Lutz Gross, l.gross@uq.edu.au"  __author__="Lutz Gross, l.gross@uq.edu.au"
41    
42    
43  import escript  from . import escript
44  import linearPDEs  from . import linearPDEs
45  import numpy  import numpy
46  import util  from . import util
47  import math  import math
48    
 ##### Added by Artak  
 # from Numeric import zeros,Int,Float64  
 ###################################  
   
   
49  class TimeIntegrationManager:  class TimeIntegrationManager:
50    """    """
51    A simple mechanism to manage time dependend values.    A simple mechanism to manage time dependend values.
# Line 64  class TimeIntegrationManager: Line 60  class TimeIntegrationManager:
60           tm.checkin(dt,v)           tm.checkin(dt,v)
61           t+=dt           t+=dt
62    
63    @note: currently only p=1 is supported.    :note: currently only p=1 is supported.
64    """    """
65    def __init__(self,*inital_values,**kwargs):    def __init__(self,*inital_values,**kwargs):
66       """       """
67       Sets up the value manager where C{inital_values} are the initial values       Sets up the value manager where ``inital_values`` are the initial values
68       and p is the order used for extrapolation.       and p is the order used for extrapolation.
69       """       """
70       if kwargs.has_key("p"):       if "p" in kwargs:
71              self.__p=kwargs["p"]              self.__p=kwargs["p"]
72       else:       else:
73              self.__p=1              self.__p=1
74       if kwargs.has_key("time"):       if "time" in kwargs:
75              self.__t=kwargs["time"]              self.__t=kwargs["time"]
76       else:       else:
77              self.__t=0.              self.__t=0.
# Line 113  class TimeIntegrationManager: Line 109  class TimeIntegrationManager:
109    
110    def extrapolate(self,dt):    def extrapolate(self,dt):
111        """        """
112        Extrapolates to C{dt} forward in time.        Extrapolates to ``dt`` forward in time.
113        """        """
114        if self.__order==0:        if self.__order==0:
115           out=self.__v_mem[0]           out=self.__v_mem[0]
# Line 139  class Projector: Line 135  class Projector:
135      """      """
136      Creates a continuous function space projector for a domain.      Creates a continuous function space projector for a domain.
137    
138      @param domain: Domain of the projection.      :param domain: Domain of the projection.
139      @param reduce: Flag to reduce projection order      :param reduce: Flag to reduce projection order
140      @param fast: Flag to use a fast method based on matrix lumping      :param fast: Flag to use a fast method based on matrix lumping
141      """      """
142      self.__pde = linearPDEs.LinearPDE(domain)      self.__pde = linearPDEs.LinearPDE(domain)
143      if fast:      if fast:
# Line 151  class Projector: Line 147  class Projector:
147      self.__pde.setValue(D = 1.)      self.__pde.setValue(D = 1.)
148      return      return
149    def getSolverOptions(self):    def getSolverOptions(self):
150      """      """
151      Returns the solver options of the PDE solver.      Returns the solver options of the PDE solver.
152            
153      @rtype: L{linearPDEs.SolverOptions}      :rtype: `linearPDEs.SolverOptions`
154      """      """
155        return self.__pde.getSolverOptions()
156    
157      def getValue(self, input_data):
158        """
159        Projects ``input_data`` onto a continuous function.
160    
161        :param input_data: the data to be projected
162        """
163        return self(input_data)
164    
165    def __call__(self, input_data):    def __call__(self, input_data):
166      """      """
167      Projects C{input_data} onto a continuous function.      Projects ``input_data`` onto a continuous function.
168    
169      @param input_data: the data to be projected      :param input_data: the data to be projected
170      """      """
171      out=escript.Data(0.,input_data.getShape(),self.__pde.getFunctionSpaceForSolution())      out=escript.Data(0.,input_data.getShape(),self.__pde.getFunctionSpaceForSolution())
172      self.__pde.setValue(Y = escript.Data(), Y_reduced = escript.Data())      self.__pde.setValue(Y = escript.Data(), Y_reduced = escript.Data())
# Line 196  class NoPDE: Line 201  class NoPDE:
201       """       """
202       Solves the following problem for u:       Solves the following problem for u:
203    
204       M{kronecker[i,j]*D[j]*u[j]=Y[i]}       *kronecker[i,j]*D[j]*u[j]=Y[i]*
205    
206       with constraint       with constraint
207    
208       M{u[j]=r[j]}  where M{q[j]>0}       *u[j]=r[j]*  where *q[j]>0*
209    
210       where M{D}, M{Y}, M{r} and M{q} are given functions of rank 1.       where *D*, *Y*, *r* and *q* are given functions of rank 1.
211    
212       In the case of scalars this takes the form       In the case of scalars this takes the form
213    
214       M{D*u=Y}       *D*u=Y*
215    
216       with constraint       with constraint
217    
218       M{u=r} where M{q>0}       *u=r* where *q>0*
219    
220       where M{D}, M{Y}, M{r} and M{q} are given scalar functions.       where *D*, *Y*, *r* and *q* are given scalar functions.
221    
222       The constraint overwrites any other condition.       The constraint overwrites any other condition.
223    
224       @note: This class is similar to the L{linearPDEs.LinearPDE} class with       :note: This class is similar to the `linearPDEs.LinearPDE` class with
225              A=B=C=X=0 but has the intention that all input parameters are given              A=B=C=X=0 but has the intention that all input parameters are given
226              in L{Solution} or L{ReducedSolution}.              in `Solution` or `ReducedSolution`.
227       """       """
228       # The whole thing is a bit strange and I blame Rob Woodcock (CSIRO) for       # The whole thing is a bit strange and I blame Rob Woodcock (CSIRO) for
229       # this.       # this.
# Line 226  class NoPDE: Line 231  class NoPDE:
231           """           """
232           Initializes the problem.           Initializes the problem.
233    
234           @param domain: domain of the PDE           :param domain: domain of the PDE
235           @type domain: L{Domain}           :type domain: `Domain`
236           @param D: coefficient of the solution           :param D: coefficient of the solution
237           @type D: C{float}, C{int}, C{numpy.ndarray}, L{Data}           :type D: ``float``, ``int``, ``numpy.ndarray``, `Data`
238           @param Y: right hand side           :param Y: right hand side
239           @type Y: C{float}, C{int}, C{numpy.ndarray}, L{Data}           :type Y: ``float``, ``int``, ``numpy.ndarray``, `Data`
240           @param q: location of constraints           :param q: location of constraints
241           @type q: C{float}, C{int}, C{numpy.ndarray}, L{Data}           :type q: ``float``, ``int``, ``numpy.ndarray``, `Data`
242           @param r: value of solution at locations of constraints           :param r: value of solution at locations of constraints
243           @type r: C{float}, C{int}, C{numpy.ndarray}, L{Data}           :type r: ``float``, ``int``, ``numpy.ndarray``, `Data`
244           """           """
245           self.__domain=domain           self.__domain=domain
246           self.__D=D           self.__D=D
# Line 247  class NoPDE: Line 252  class NoPDE:
252    
253       def setReducedOn(self):       def setReducedOn(self):
254           """           """
255           Sets the L{FunctionSpace} of the solution to L{ReducedSolution}.           Sets the `FunctionSpace` of the solution to `ReducedSolution`.
256           """           """
257           self.__function_space=escript.ReducedSolution(self.__domain)           self.__function_space=escript.ReducedSolution(self.__domain)
258           self.__u=None           self.__u=None
259    
260       def setReducedOff(self):       def setReducedOff(self):
261           """           """
262           Sets the L{FunctionSpace} of the solution to L{Solution}.           Sets the `FunctionSpace` of the solution to `Solution`.
263           """           """
264           self.__function_space=escript.Solution(self.__domain)           self.__function_space=escript.Solution(self.__domain)
265           self.__u=None           self.__u=None
# Line 263  class NoPDE: Line 268  class NoPDE:
268           """           """
269           Assigns values to the parameters.           Assigns values to the parameters.
270    
271           @param D: coefficient of the solution           :param D: coefficient of the solution
272           @type D: C{float}, C{int}, C{numpy.ndarray}, L{Data}           :type D: ``float``, ``int``, ``numpy.ndarray``, `Data`
273           @param Y: right hand side           :param Y: right hand side
274           @type Y: C{float}, C{int}, C{numpy.ndarray}, L{Data}           :type Y: ``float``, ``int``, ``numpy.ndarray``, `Data`
275           @param q: location of constraints           :param q: location of constraints
276           @type q: C{float}, C{int}, C{numpy.ndarray}, L{Data}           :type q: ``float``, ``int``, ``numpy.ndarray``, `Data`
277           @param r: value of solution at locations of constraints           :param r: value of solution at locations of constraints
278           @type r: C{float}, C{int}, C{numpy.ndarray}, L{Data}           :type r: ``float``, ``int``, ``numpy.ndarray``, `Data`
279           """           """
280           if not D==None:           if not D==None:
281              self.__D=D              self.__D=D
# Line 289  class NoPDE: Line 294  class NoPDE:
294           """           """
295           Returns the solution.           Returns the solution.
296    
297           @return: the solution of the problem           :return: the solution of the problem
298           @rtype: L{Data} object in the L{FunctionSpace} L{Solution} or           :rtype: `Data` object in the `FunctionSpace` `Solution` or
299                   L{ReducedSolution}                   `ReducedSolution`
300           """           """
301           if self.__u==None:           if self.__u==None:
302              if self.__D==None:              if self.__D==None:
303                 raise ValueError,"coefficient D is undefined"                 raise ValueError("coefficient D is undefined")
304              D=escript.Data(self.__D,self.__function_space)              D=escript.Data(self.__D,self.__function_space)
305              if D.getRank()>1:              if D.getRank()>1:
306                 raise ValueError,"coefficient D must have rank 0 or 1"                 raise ValueError("coefficient D must have rank 0 or 1")
307              if self.__Y==None:              if self.__Y==None:
308                 self.__u=escript.Data(0.,D.getShape(),self.__function_space)                 self.__u=escript.Data(0.,D.getShape(),self.__function_space)
309              else:              else:
310                 self.__u=util.quotient(self.__Y,D)                 self.__u=1./D*self.__Y
311              if not self.__q==None:              if not self.__q==None:
312                  q=util.wherePositive(escript.Data(self.__q,self.__function_space))                  q=util.wherePositive(escript.Data(self.__q,self.__function_space))
313                  self.__u*=(1.-q)                  self.__u*=(1.-q)
# Line 324  class Locator: Line 329  class Locator:
329         or FunctionSpace for the sample point which is closest to the given         or FunctionSpace for the sample point which is closest to the given
330         point x.         point x.
331    
332         @param where: function space         :param where: function space
333         @type where: L{escript.FunctionSpace}         :type where: `escript.FunctionSpace`
334         @param x: location(s) of the Locator         :param x: location(s) of the Locator
335         @type x: C{numpy.ndarray} or C{list} of C{numpy.ndarray}         :type x: ``numpy.ndarray`` or ``list`` of ``numpy.ndarray``
336         """         """
337         if isinstance(where,escript.FunctionSpace):         if isinstance(where,escript.FunctionSpace):
338            self.__function_space=where            self.__function_space=where
# Line 336  class Locator: Line 341  class Locator:
341         iterative=False         iterative=False
342         if isinstance(x, list):         if isinstance(x, list):
343             if len(x)==0:             if len(x)==0:
344                raise "ValueError", "At least one point must be given."                raise ValueError("At least one point must be given.")
345             try:             try:
346               iter(x[0])               iter(x[0])
347               iterative=True               iterative=True
348             except TypeError:             except TypeError:
349               iterative=False               iterative=False
350           xxx=self.__function_space.getX()
351         if iterative:         if iterative:
352             self.__id=[]             self.__id=[]
353             for p in x:             for p in x:
354                self.__id.append(util.length(self.__function_space.getX()-p[:self.__function_space.getDim()]).minGlobalDataPoint())                self.__id.append(util.length(xxx-p[:self.__function_space.getDim()]).minGlobalDataPoint())
355         else:         else:
356             self.__id=util.length(self.__function_space.getX()-x[:self.__function_space.getDim()]).minGlobalDataPoint()             self.__id=util.length(xxx-x[:self.__function_space.getDim()]).minGlobalDataPoint()
357    
358       def __str__(self):       def __str__(self):
359         """         """
360         Returns the coordinates of the Locator as a string.         Returns the coordinates of the Locator as a string.
361         """         """
362         x=self.getX()         x=self.getX()
363         if instance(x,list):         if isinstance(x,list):
364            out="["            out="["
365            first=True            first=True
366            for xx in x:            for xx in x:
# Line 401  class Locator: Line 407  class Locator:
407    
408       def getValue(self,data):       def getValue(self,data):
409          """          """
410          Returns the value of C{data} at the Locator if C{data} is a L{Data}          Returns the value of ``data`` at the Locator if ``data`` is a `Data`
411          object otherwise the object is returned.          object otherwise the object is returned.
412          """          """
413          if isinstance(data,escript.Data):          if isinstance(data,escript.Data):
# Line 425  class Locator: Line 431  class Locator:
431                  return out                  return out
432          else:          else:
433             return data             return data
434              
435         def setValue(self, data, v):
436          """
437          Sets the value of the ``data`` at the Locator.
438          """
439          if isinstance(data, escript.Data):
440             if data.getFunctionSpace()!=self.getFunctionSpace():
441               raise TypeError, "setValue: FunctionSpace of Locator and Data object must match."
442             data.expand()  
443             id=self.getId()
444             if isinstance(id, list):
445              for i in id:
446               data._setTupleForGlobalDataPoint(i[1], i[0], v)
447             else:
448               data._setTupleForGlobalDataPoint(id[1], id[0], v)
449          else:
450               raise TypeError, "setValue: Invalid argument type."
451    
452    
453    def getInfLocator(arg):
454        """
455        Return a Locator for a point with the inf value over all arg.
456        """
457        if not isinstance(arg, escript.Data):
458           raise TypeError("getInfLocator: Unknown argument type.")
459        a_inf=util.inf(arg)
460        loc=util.length(arg-a_inf).minGlobalDataPoint() # This gives us the location but not coords
461        x=arg.getFunctionSpace().getX()
462        x_min=x.getTupleForGlobalDataPoint(*loc)
463        return Locator(arg.getFunctionSpace(),x_min)
464    
465    def getSupLocator(arg):
466        """
467        Return a Locator for a point with the sup value over all arg.
468        """
469        if not isinstance(arg, escript.Data):
470           raise TypeError("getInfLocator: Unknown argument type.")
471        a_inf=util.sup(arg)
472        loc=util.length(arg-a_inf).minGlobalDataPoint() # This gives us the location but not coords
473        x=arg.getFunctionSpace().getX()
474        x_min=x.getTupleForGlobalDataPoint(*loc)
475        return Locator(arg.getFunctionSpace(),x_min)
476        
477    
478  class SolverSchemeException(Exception):  class SolverSchemeException(Exception):
479     """     """
# Line 467  def PCG(r, Aprod, x, Msolve, bilinearfor Line 516  def PCG(r, Aprod, x, Msolve, bilinearfor
516     """     """
517     Solver for     Solver for
518    
519     M{Ax=b}     *Ax=b*
520    
521     with a symmetric and positive definite operator A (more details required!).     with a symmetric and positive definite operator A (more details required!).
522     It uses the conjugate gradient method with preconditioner M providing an     It uses the conjugate gradient method with preconditioner M providing an
# Line 475  def PCG(r, Aprod, x, Msolve, bilinearfor Line 524  def PCG(r, Aprod, x, Msolve, bilinearfor
524    
525     The iteration is terminated if     The iteration is terminated if
526    
527     M{|r| <= atol+rtol*|r0|}     *|r| <= atol+rtol*|r0|*
528    
529     where M{r0} is the initial residual and M{|.|} is the energy norm. In fact     where *r0* is the initial residual and *|.|* is the energy norm. In fact
530    
531     M{|r| = sqrt( bilinearform(Msolve(r),r))}     *|r| = sqrt( bilinearform(Msolve(r),r))*
532    
533     For details on the preconditioned conjugate gradient method see the book:     For details on the preconditioned conjugate gradient method see the book:
534    
# Line 487  def PCG(r, Aprod, x, Msolve, bilinearfor Line 536  def PCG(r, Aprod, x, Msolve, bilinearfor
536     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,
537     C. Romine, and H. van der Vorst}.     C. Romine, and H. van der Vorst}.
538    
539     @param r: initial residual M{r=b-Ax}. C{r} is altered.     :param r: initial residual *r=b-Ax*. ``r`` is altered.
540     @type r: 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)
541     @param x: an initial guess for the solution     :param x: an initial guess for the solution
542     @type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)     :type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
543     @param Aprod: returns the value Ax     :param Aprod: returns the value Ax
544     @type Aprod: function C{Aprod(x)} where C{x} is of the same object like     :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
545                  argument C{x}. The returned object needs to be of the same type                  argument ``x``. The returned object needs to be of the same type
546                  like argument C{r}.                  like argument ``r``.
547     @param Msolve: solves Mx=r     :param Msolve: solves Mx=r
548     @type Msolve: function C{Msolve(r)} where C{r} is of the same type like     :type Msolve: function ``Msolve(r)`` where ``r`` is of the same type like
549                   argument C{r}. The returned object needs to be of the same                   argument ``r``. The returned object needs to be of the same
550                   type like argument C{x}.                   type like argument ``x``.
551     @param bilinearform: inner product C{<x,r>}     :param bilinearform: inner product ``<x,r>``
552     @type bilinearform: function C{bilinearform(x,r)} where C{x} is of the same     :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
553                         type like argument C{x} and C{r} is. The returned value                         type like argument ``x`` and ``r`` is. The returned value
554                         is a C{float}.                         is a ``float``.
555     @param atol: absolute tolerance     :param atol: absolute tolerance
556     @type atol: non-negative C{float}     :type atol: non-negative ``float``
557     @param rtol: relative tolerance     :param rtol: relative tolerance
558     @type rtol: non-negative C{float}     :type rtol: non-negative ``float``
559     @param iter_max: maximum number of iteration steps     :param iter_max: maximum number of iteration steps
560     @type iter_max: C{int}     :type iter_max: ``int``
561     @return: the solution approximation and the corresponding residual     :return: the solution approximation and the corresponding residual
562     @rtype: C{tuple}     :rtype: ``tuple``
563     @warning: C{r} and C{x} are altered.     :warning: ``r`` and ``x`` are altered.
564     """     """
565     iter=0     iter=0
566     rhat=Msolve(r)     rhat=Msolve(r)
567     d = rhat     d = rhat
568     rhat_dot_r = bilinearform(rhat, r)     rhat_dot_r = bilinearform(rhat, r)
569     if rhat_dot_r<0: raise NegativeNorm,"negative norm."     if rhat_dot_r<0: raise NegativeNorm("negative norm.")
570     norm_r0=math.sqrt(rhat_dot_r)     norm_r0=math.sqrt(rhat_dot_r)
571     atol2=atol+rtol*norm_r0     atol2=atol+rtol*norm_r0
572     if atol2<=0:     if atol2<=0:
573        raise ValueError,"Non-positive tolarance."        raise ValueError("Non-positive tolarance.")
574     atol2=max(atol2, 100. * util.EPSILON * norm_r0)     atol2=max(atol2, 100. * util.EPSILON * norm_r0)
575    
576     if verbose: print "PCG: initial residual norm = %e (absolute tolerance = %e)"%(norm_r0, atol2)     if verbose: print(("PCG: initial residual norm = %e (absolute tolerance = %e)"%(norm_r0, atol2)))
577    
578    
579     while not math.sqrt(rhat_dot_r) <= atol2:     while not math.sqrt(rhat_dot_r) <= atol2:
580         iter+=1         iter+=1
581         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)
582    
583         q=Aprod(d)         q=Aprod(d)
584         alpha = rhat_dot_r / bilinearform(d, q)         alpha = rhat_dot_r / bilinearform(d, q)
585         x += alpha * d         x += alpha * d
586         if isinstance(q,ArithmeticTuple):         if isinstance(q,ArithmeticTuple):
587         r += q * (-alpha)      # Doing it the other way calls the float64.__mul__ not AT.__rmul__            r += q * (-alpha)      # Doing it the other way calls the float64.__mul__ not AT.__rmul__
588         else:         else:
589             r += (-alpha) * q             r += (-alpha) * q
590         rhat=Msolve(r)         rhat=Msolve(r)
# Line 545  def PCG(r, Aprod, x, Msolve, bilinearfor Line 594  def PCG(r, Aprod, x, Msolve, bilinearfor
594         d=rhat         d=rhat
595    
596         rhat_dot_r = rhat_dot_r_new         rhat_dot_r = rhat_dot_r_new
597         if rhat_dot_r<0: raise NegativeNorm,"negative norm."         if rhat_dot_r<0: raise NegativeNorm("negative norm.")
598         if verbose: print "PCG: iteration step %s: residual norm = %e"%(iter, math.sqrt(rhat_dot_r))         if verbose: print(("PCG: iteration step %s: residual norm = %e"%(iter, math.sqrt(rhat_dot_r))))
599     if verbose: print "PCG: tolerance reached after %s steps."%iter     if verbose: print(("PCG: tolerance reached after %s steps."%iter))
600     return x,r,math.sqrt(rhat_dot_r)     return x,r,math.sqrt(rhat_dot_r)
601    
602  class Defect(object):  class Defect(object):
# Line 564  class Defect(object): Line 613  class Defect(object):
613          """          """
614          Returns the inner product of x0 and x1          Returns the inner product of x0 and x1
615    
616          @param x0: value for x0          :param x0: value for x0
617          @param x1: value for x1          :param x1: value for x1
618          @return: the inner product of x0 and x1          :return: the inner product of x0 and x1
619          @rtype: C{float}          :rtype: ``float``
620          """          """
621          return 0          return 0
622    
623      def norm(self,x):      def norm(self,x):
624          """          """
625          Returns the norm of argument C{x}.          Returns the norm of argument ``x``.
626    
627          @param x: a value          :param x: a value
628          @return: norm of argument x          :return: norm of argument x
629          @rtype: C{float}          :rtype: ``float``
630          @note: by default C{sqrt(self.bilinearform(x,x)} is returned.          :note: by default ``sqrt(self.bilinearform(x,x)`` is returned.
631          """          """
632          s=self.bilinearform(x,x)          s=self.bilinearform(x,x)
633          if s<0: raise NegativeNorm,"negative norm."          if s<0: raise NegativeNorm("negative norm.")
634          return math.sqrt(s)          return math.sqrt(s)
635    
636      def eval(self,x):      def eval(self,x):
637          """          """
638          Returns the value F of a given C{x}.          Returns the value F of a given ``x``.
639    
640          @param x: value for which the defect C{F} is evaluated          :param x: value for which the defect ``F`` is evaluated
641          @return: value of the defect at C{x}          :return: value of the defect at ``x``
642          """          """
643          return 0          return 0
644    
645      def __call__(self,x):      def __call__(self,x):
646          return self.eval(x)          return self.eval(x)
647    
648      def setDerivativeIncrementLength(self,inc=math.sqrt(util.EPSILON)):      def setDerivativeIncrementLength(self,inc=1000.*math.sqrt(util.EPSILON)):
649          """          """
650          Sets the relative length of the increment used to approximate the          Sets the relative length of the increment used to approximate the
651          derivative of the defect. The increment is inc*norm(x)/norm(v)*v in the          derivative of the defect. The increment is inc*norm(x)/norm(v)*v in the
652          direction of v with x as a starting point.          direction of v with x as a starting point.
653    
654          @param inc: relative increment length          :param inc: relative increment length
655          @type inc: positive C{float}          :type inc: positive ``float``
656          """          """
657          if inc<=0: raise ValueError,"positive increment required."          if inc<=0: raise ValueError("positive increment required.")
658          self.__inc=inc          self.__inc=inc
659    
660      def getDerivativeIncrementLength(self):      def getDerivativeIncrementLength(self):
661          """          """
662          Returns the relative increment length used to approximate the          Returns the relative increment length used to approximate the
663          derivative of the defect.          derivative of the defect.
664          @return: value of the defect at C{x}          :return: value of the defect at ``x``
665          @rtype: positive C{float}          :rtype: positive ``float``
666          """          """
667          return self.__inc          return self.__inc
668    
669      def derivative(self, F0, x0, v, v_is_normalised=True):      def derivative(self, F0, x0, v, v_is_normalised=True):
670          """          """
671          Returns the directional derivative at C{x0} in the direction of C{v}.          Returns the directional derivative at ``x0`` in the direction of ``v``.
672    
673          @param F0: value of this defect at x0          :param F0: value of this defect at x0
674          @param x0: value at which derivative is calculated          :param x0: value at which derivative is calculated
675          @param v: direction          :param v: direction
676          @param v_is_normalised: True to indicate that C{v} is nomalized          :param v_is_normalised: True to indicate that ``v`` is nomalized
677                                  (self.norm(v)=0)                                  (self.norm(v)=0)
678          @return: derivative of this defect at x0 in the direction of C{v}          :return: derivative of this defect at x0 in the direction of ``v``
679          @note: by default numerical evaluation (self.eval(x0+eps*v)-F0)/eps is          :note: by default numerical evaluation (self.eval(x0+eps*v)-F0)/eps is
680                 used but this method maybe overwritten to use exact evaluation.                 used but this method maybe overwritten to use exact evaluation.
681          """          """
682          normx=self.norm(x0)          normx=self.norm(x0)
# Line 645  class Defect(object): Line 694  class Defect(object):
694          return (F1-F0)/epsnew          return (F1-F0)/epsnew
695    
696  ######################################  ######################################
697  def NewtonGMRES(defect, x, iter_max=100, sub_iter_max=20, atol=0,rtol=1.e-4, sub_tol_max=0.5, gamma=0.9, verbose=False):  def NewtonGMRES(defect, x, iter_max=100, sub_iter_max=20, atol=0,rtol=1.e-4, subtol_max=0.5, gamma=0.9, verbose=False):
698     """     """
699     Solves a non-linear problem M{F(x)=0} for unknown M{x} using the stopping     Solves a non-linear problem *F(x)=0* for unknown *x* using the stopping
700     criterion:     criterion:
701    
702     M{norm(F(x) <= atol + rtol * norm(F(x0)}     *norm(F(x) <= atol + rtol * norm(F(x0)*
703    
704     where M{x0} is the initial guess.     where *x0* is the initial guess.
705    
706     @param defect: object defining the function M{F}. C{defect.norm} defines the     :param defect: object defining the function *F*. ``defect.norm`` defines the
707                    M{norm} used in the stopping criterion.                    *norm* used in the stopping criterion.
708     @type defect: L{Defect}     :type defect: `Defect`
709     @param x: initial guess for the solution, C{x} is altered.     :param x: initial guess for the solution, ``x`` is altered.
710     @type x: any object type allowing basic operations such as     :type x: any object type allowing basic operations such as
711              C{numpy.ndarray}, L{Data}              ``numpy.ndarray``, `Data`
712     @param iter_max: maximum number of iteration steps     :param iter_max: maximum number of iteration steps
713     @type iter_max: positive C{int}     :type iter_max: positive ``int``
714     @param sub_iter_max: maximum number of inner iteration steps     :param sub_iter_max: maximum number of inner iteration steps
715     @type sub_iter_max: positive C{int}     :type sub_iter_max: positive ``int``
716     @param atol: absolute tolerance for the solution     :param atol: absolute tolerance for the solution
717     @type atol: positive C{float}     :type atol: positive ``float``
718     @param rtol: relative tolerance for the solution     :param rtol: relative tolerance for the solution
719     @type rtol: positive C{float}     :type rtol: positive ``float``
720     @param gamma: tolerance safety factor for inner iteration     :param gamma: tolerance safety factor for inner iteration
721     @type gamma: positive C{float}, less than 1     :type gamma: positive ``float``, less than 1
722     @param sub_tol_max: upper bound for inner tolerance     :param subtol_max: upper bound for inner tolerance
723     @type sub_tol_max: positive C{float}, less than 1     :type subtol_max: positive ``float``, less than 1
724     @return: an approximation of the solution with the desired accuracy     :return: an approximation of the solution with the desired accuracy
725     @rtype: same type as the initial guess     :rtype: same type as the initial guess
726     """     """
727     lmaxit=iter_max     lmaxit=iter_max
728     if atol<0: raise ValueError,"atol needs to be non-negative."     if atol<0: raise ValueError("atol needs to be non-negative.")
729     if rtol<0: raise ValueError,"rtol needs to be non-negative."     if rtol<0: raise ValueError("rtol needs to be non-negative.")
730     if rtol+atol<=0: raise ValueError,"rtol or atol needs to be non-negative."     if rtol+atol<=0: raise ValueError("rtol or atol needs to be non-negative.")
731     if gamma<=0 or gamma>=1: raise ValueError,"tolerance safety factor for inner iteration (gamma =%s) needs to be positive and less than 1."%gamma     if gamma<=0 or gamma>=1: raise ValueError("tolerance safety factor for inner iteration (gamma =%s) needs to be positive and less than 1."%gamma)
732     if sub_tol_max<=0 or sub_tol_max>=1: raise ValueError,"upper bound for inner tolerance for inner iteration (sub_tol_max =%s) needs to be positive and less than 1."%sub_tol_max     if subtol_max<=0 or subtol_max>=1: raise ValueError("upper bound for inner tolerance for inner iteration (subtol_max =%s) needs to be positive and less than 1."%subtol_max)
733    
734     F=defect(x)     F=defect(x)
735     fnrm=defect.norm(F)     fnrm=defect.norm(F)
736     stop_tol=atol + rtol*fnrm     stop_tol=atol + rtol*fnrm
737     sub_tol=sub_tol_max     subtol=subtol_max
738     if verbose: print "NewtonGMRES: initial residual = %e."%fnrm     if verbose: print(("NewtonGMRES: initial residual = %e."%fnrm))
739     if verbose: print "             tolerance = %e."%sub_tol     if verbose: print(("             tolerance = %e."%subtol))
740     iter=1     iter=1
741     #     #
742     # main iteration loop     # main iteration loop
743     #     #
744     while not fnrm<=stop_tol:     while not fnrm<=stop_tol:
745              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)
746              #              #
747          #   adjust sub_tol_          #   adjust subtol_
748          #          #
749              if iter > 1:              if iter > 1:
750             rat=fnrm/fnrmo                 rat=fnrm/fnrmo
751                 sub_tol_old=sub_tol                 subtol_old=subtol
752             sub_tol=gamma*rat**2                 subtol=gamma*rat**2
753             if gamma*sub_tol_old**2 > .1: sub_tol=max(sub_tol,gamma*sub_tol_old**2)                 if gamma*subtol_old**2 > .1: subtol=max(subtol,gamma*subtol_old**2)
754             sub_tol=max(min(sub_tol,sub_tol_max), .5*stop_tol/fnrm)                 subtol=max(min(subtol,subtol_max), .5*stop_tol/fnrm)
755          #          #
756          # calculate newton increment xc          # calculate newton increment xc
757              #     if iter_max in __FDGMRES is reached MaxIterReached is thrown              #     if iter_max in __FDGMRES is reached MaxIterReached is thrown
# Line 710  def NewtonGMRES(defect, x, iter_max=100, Line 759  def NewtonGMRES(defect, x, iter_max=100,
759              #     if  atol is reached sub_iter returns the numer of steps performed to get there              #     if  atol is reached sub_iter returns the numer of steps performed to get there
760              #              #
761              #              #
762              if verbose: print "             subiteration (GMRES) is called with relative tolerance %e."%sub_tol              if verbose: print(("             subiteration (GMRES) is called with relative tolerance %e."%subtol))
763              try:              try:
764                 xc, sub_iter=__FDGMRES(F, defect, x, sub_tol*fnrm, iter_max=iter_max-iter, iter_restart=sub_iter_max)                 xc, sub_iter=__FDGMRES(F, defect, x, subtol*fnrm, iter_max=iter_max-iter, iter_restart=sub_iter_max)
765              except MaxIterReached:              except MaxIterReached:
766                 raise MaxIterReached,"maximum number of %s steps reached."%iter_max                 raise MaxIterReached("maximum number of %s steps reached."%iter_max)
767              if sub_iter<0:              if sub_iter<0:
768                 iter+=sub_iter_max                 iter+=sub_iter_max
769              else:              else:
770                 iter+=sub_iter                 iter+=sub_iter
771              # ====              # ====
772          x+=xc              x+=xc
773              F=defect(x)              F=defect(x)
774          iter+=1              iter+=1
775              fnrmo, fnrm=fnrm, defect.norm(F)              fnrmo, fnrm=fnrm, defect.norm(F)
776              if verbose: print "             step %s: residual %e."%(iter,fnrm)              if verbose: print(("             step %s: residual %e."%(iter,fnrm)))
777     if verbose: print "NewtonGMRES: completed after %s steps."%iter     if verbose: print(("NewtonGMRES: completed after %s steps."%iter))
778     return x     return x
779    
780  def __givapp(c,s,vin):  def __givapp(c,s,vin):
781      """      """
782      Applies a sequence of Givens rotations (c,s) recursively to the vector      Applies a sequence of Givens rotations (c,s) recursively to the vector
783      C{vin}      ``vin``
784    
785      @warning: C{vin} is altered.      :warning: ``vin`` is altered.
786      """      """
787      vrot=vin      vrot=vin
788      if isinstance(c,float):      if isinstance(c,float):
# Line 741  def __givapp(c,s,vin): Line 790  def __givapp(c,s,vin):
790      else:      else:
791          for i in range(len(c)):          for i in range(len(c)):
792              w1=c[i]*vrot[i]-s[i]*vrot[i+1]              w1=c[i]*vrot[i]-s[i]*vrot[i+1]
793          w2=s[i]*vrot[i]+c[i]*vrot[i+1]              w2=s[i]*vrot[i]+c[i]*vrot[i+1]
794              vrot[i]=w1              vrot[i]=w1
795              vrot[i+1]=w2              vrot[i+1]=w2
796      return vrot      return vrot
# Line 761  def __FDGMRES(F0, defect, x0, atol, iter Line 810  def __FDGMRES(F0, defect, x0, atol, iter
810     iter=0     iter=0
811     while rho > atol and iter<iter_restart-1:     while rho > atol and iter<iter_restart-1:
812          if iter  >= iter_max:          if iter  >= iter_max:
813              raise MaxIterReached,"maximum number of %s steps reached."%iter_max              raise MaxIterReached("maximum number of %s steps reached."%iter_max)
814    
815          p=defect.derivative(F0,x0,v[iter], v_is_normalised=True)          p=defect.derivative(F0,x0,v[iter], v_is_normalised=True)
816          v.append(p)          v.append(p)
# Line 823  def __FDGMRES(F0, defect, x0, atol, iter Line 872  def __FDGMRES(F0, defect, x0, atol, iter
872            i=i-1            i=i-1
873       xhat=v[iter-1]*y[iter-1]       xhat=v[iter-1]*y[iter-1]
874       for i in range(iter-1):       for i in range(iter-1):
875      xhat += v[i]*y[i]         xhat += v[i]*y[i]
876     else :     else :
877        xhat=v[0] * 0        xhat=v[0] * 0
878    
# Line 838  def GMRES(r, Aprod, x, bilinearform, ato Line 887  def GMRES(r, Aprod, x, bilinearform, ato
887     """     """
888     Solver for     Solver for
889    
890     M{Ax=b}     *Ax=b*
891    
892     with a general operator A (more details required!).     with a general operator A (more details required!).
893     It uses the generalized minimum residual method (GMRES).     It uses the generalized minimum residual method (GMRES).
894    
895     The iteration is terminated if     The iteration is terminated if
896    
897     M{|r| <= atol+rtol*|r0|}     *|r| <= atol+rtol*|r0|*
898    
899     where M{r0} is the initial residual and M{|.|} is the energy norm. In fact     where *r0* is the initial residual and *|.|* is the energy norm. In fact
900    
901     M{|r| = sqrt( bilinearform(r,r))}     *|r| = sqrt( bilinearform(r,r))*
902    
903     @param r: initial residual M{r=b-Ax}. C{r} is altered.     :param r: initial residual *r=b-Ax*. ``r`` is altered.
904     @type r: 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)
905     @param x: an initial guess for the solution     :param x: an initial guess for the solution
906     @type x: same like C{r}     :type x: same like ``r``
907     @param Aprod: returns the value Ax     :param Aprod: returns the value Ax
908     @type Aprod: function C{Aprod(x)} where C{x} is of the same object like     :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
909                  argument C{x}. The returned object needs to be of the same                  argument ``x``. The returned object needs to be of the same
910                  type like argument C{r}.                  type like argument ``r``.
911     @param bilinearform: inner product C{<x,r>}     :param bilinearform: inner product ``<x,r>``
912     @type bilinearform: function C{bilinearform(x,r)} where C{x} is of the same     :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
913                         type like argument C{x} and C{r}. The returned value is                         type like argument ``x`` and ``r``. The returned value is
914                         a C{float}.                         a ``float``.
915     @param atol: absolute tolerance     :param atol: absolute tolerance
916     @type atol: non-negative C{float}     :type atol: non-negative ``float``
917     @param rtol: relative tolerance     :param rtol: relative tolerance
918     @type rtol: non-negative C{float}     :type rtol: non-negative ``float``
919     @param iter_max: maximum number of iteration steps     :param iter_max: maximum number of iteration steps
920     @type iter_max: C{int}     :type iter_max: ``int``
921     @param iter_restart: in order to save memory the orthogonalization process     :param iter_restart: in order to save memory the orthogonalization process
922                          is terminated after C{iter_restart} steps and the                          is terminated after ``iter_restart`` steps and the
923                          iteration is restarted.                          iteration is restarted.
924     @type iter_restart: C{int}     :type iter_restart: ``int``
925     @return: the solution approximation and the corresponding residual     :return: the solution approximation and the corresponding residual
926     @rtype: C{tuple}     :rtype: ``tuple``
927     @warning: C{r} and C{x} are altered.     :warning: ``r`` and ``x`` are altered.
928     """     """
929     m=iter_restart     m=iter_restart
930     restarted=False     restarted=False
931     iter=0     iter=0
932     if rtol>0:     if rtol>0:
933        r_dot_r = bilinearform(r, r)        r_dot_r = bilinearform(r, r)
934        if r_dot_r<0: raise NegativeNorm,"negative norm."        if r_dot_r<0: raise NegativeNorm("negative norm.")
935        atol2=atol+rtol*math.sqrt(r_dot_r)        atol2=atol+rtol*math.sqrt(r_dot_r)
936        if verbose: print "GMRES: norm of right hand side = %e (absolute tolerance = %e)"%(math.sqrt(r_dot_r), atol2)        if verbose: print(("GMRES: norm of right hand side = %e (absolute tolerance = %e)"%(math.sqrt(r_dot_r), atol2)))
937     else:     else:
938        atol2=atol        atol2=atol
939        if verbose: print "GMRES: absolute tolerance = %e"%atol2        if verbose: print(("GMRES: absolute tolerance = %e"%atol2))
940     if atol2<=0:     if atol2<=0:
941        raise ValueError,"Non-positive tolarance."        raise ValueError("Non-positive tolarance.")
942    
943     while True:     while True:
944        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)
945        if restarted:        if restarted:
946           r2 = r-Aprod(x-x2)           r2 = r-Aprod(x-x2)
947        else:        else:
# Line 901  def GMRES(r, Aprod, x, bilinearform, ato Line 950  def GMRES(r, Aprod, x, bilinearform, ato
950        x,stopped=_GMRESm(r2, Aprod, x, bilinearform, atol2, iter_max=iter_max-iter, iter_restart=m, verbose=verbose,P_R=P_R)        x,stopped=_GMRESm(r2, Aprod, x, bilinearform, atol2, iter_max=iter_max-iter, iter_restart=m, verbose=verbose,P_R=P_R)
951        iter+=iter_restart        iter+=iter_restart
952        if stopped: break        if stopped: break
953        if verbose: print "GMRES: restart."        if verbose: print("GMRES: restart.")
954        restarted=True        restarted=True
955     if verbose: print "GMRES: tolerance has been reached."     if verbose: print("GMRES: tolerance has been reached.")
956     return x     return x
957    
958  def _GMRESm(r, Aprod, x, bilinearform, atol, iter_max=100, iter_restart=20, verbose=False, P_R=None):  def _GMRESm(r, Aprod, x, bilinearform, atol, iter_max=100, iter_restart=20, verbose=False, P_R=None):
# Line 916  def _GMRESm(r, Aprod, x, bilinearform, a Line 965  def _GMRESm(r, Aprod, x, bilinearform, a
965     v=[]     v=[]
966    
967     r_dot_r = bilinearform(r, r)     r_dot_r = bilinearform(r, r)
968     if r_dot_r<0: raise NegativeNorm,"negative norm."     if r_dot_r<0: raise NegativeNorm("negative norm.")
969     rho=math.sqrt(r_dot_r)     rho=math.sqrt(r_dot_r)
970    
971     v.append(r/rho)     v.append(r/rho)
972     g[0]=rho     g[0]=rho
973    
974     if verbose: print "GMRES: initial residual %e (absolute tolerance = %e)"%(rho,atol)     if verbose: print(("GMRES: initial residual %e (absolute tolerance = %e)"%(rho,atol)))
975     while not (rho<=atol or iter==iter_restart):     while not (rho<=atol or iter==iter_restart):
976    
977      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)
978    
979          if P_R!=None:          if P_R!=None:
980              p=Aprod(P_R(v[iter]))              p=Aprod(P_R(v[iter]))
981          else:          else:
982          p=Aprod(v[iter])              p=Aprod(v[iter])
983      v.append(p)          v.append(p)
984    
985      v_norm1=math.sqrt(bilinearform(v[iter+1], v[iter+1]))          v_norm1=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
986    
987  # Modified Gram-Schmidt  # Modified Gram-Schmidt
988      for j in range(iter+1):          for j in range(iter+1):
989        h[j,iter]=bilinearform(v[j],v[iter+1])            h[j,iter]=bilinearform(v[j],v[iter+1])
990        v[iter+1]-=h[j,iter]*v[j]            v[iter+1]-=h[j,iter]*v[j]
991    
992      h[iter+1,iter]=math.sqrt(bilinearform(v[iter+1],v[iter+1]))          h[iter+1,iter]=math.sqrt(bilinearform(v[iter+1],v[iter+1]))
993      v_norm2=h[iter+1,iter]          v_norm2=h[iter+1,iter]
994    
995  # Reorthogonalize if needed  # Reorthogonalize if needed
996      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)
997       for j in range(iter+1):           for j in range(iter+1):
998          hr=bilinearform(v[j],v[iter+1])              hr=bilinearform(v[j],v[iter+1])
999              h[j,iter]=h[j,iter]+hr              h[j,iter]=h[j,iter]+hr
1000              v[iter+1] -= hr*v[j]              v[iter+1] -= hr*v[j]
1001    
1002       v_norm2=math.sqrt(bilinearform(v[iter+1], v[iter+1]))           v_norm2=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
1003       h[iter+1,iter]=v_norm2           h[iter+1,iter]=v_norm2
1004    
1005  #   watch out for happy breakdown  #   watch out for happy breakdown
1006          if not v_norm2 == 0:          if not v_norm2 == 0:
1007           v[iter+1]=v[iter+1]/h[iter+1,iter]           v[iter+1]=v[iter+1]/h[iter+1,iter]
1008    
1009  #   Form and store the information for the new Givens rotation  #   Form and store the information for the new Givens rotation
1010      if iter > 0: h[:iter+1,iter]=__givapp(c[:iter],s[:iter],h[:iter+1,iter])          if iter > 0: h[:iter+1,iter]=__givapp(c[:iter],s[:iter],h[:iter+1,iter])
1011      mu=math.sqrt(h[iter,iter]*h[iter,iter]+h[iter+1,iter]*h[iter+1,iter])          mu=math.sqrt(h[iter,iter]*h[iter,iter]+h[iter+1,iter]*h[iter+1,iter])
1012    
1013      if mu!=0 :          if mu!=0 :
1014          c[iter]=h[iter,iter]/mu                  c[iter]=h[iter,iter]/mu
1015          s[iter]=-h[iter+1,iter]/mu                  s[iter]=-h[iter+1,iter]/mu
1016          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]
1017          h[iter+1,iter]=0.0                  h[iter+1,iter]=0.0
1018                  gg=__givapp(c[iter],s[iter],[g[iter],g[iter+1]])                  gg=__givapp(c[iter],s[iter],[g[iter],g[iter+1]])
1019                  g[iter]=gg[0]                  g[iter]=gg[0]
1020                  g[iter+1]=gg[1]                  g[iter+1]=gg[1]
1021  # Update the residual norm  # Update the residual norm
1022    
1023          rho=abs(g[iter+1])          rho=abs(g[iter+1])
1024          if verbose: print "GMRES: iteration step %s: residual %e"%(iter,rho)          if verbose: print(("GMRES: iteration step %s: residual %e"%(iter,rho)))
1025      iter+=1          iter+=1
1026    
1027  # At this point either iter > iter_max or rho < tol.  # At this point either iter > iter_max or rho < tol.
1028  # It's time to compute x and leave.  # It's time to compute x and leave.
1029    
1030     if verbose: print "GMRES: iteration stopped after %s step."%iter     if verbose: print(("GMRES: iteration stopped after %s step."%iter))
1031     if iter > 0 :     if iter > 0 :
1032       y=numpy.zeros(iter,numpy.float64)       y=numpy.zeros(iter,numpy.float64)
1033       y[iter-1] = g[iter-1] / h[iter-1,iter-1]       y[iter-1] = g[iter-1] / h[iter-1,iter-1]
# Line 989  def _GMRESm(r, Aprod, x, bilinearform, a Line 1038  def _GMRESm(r, Aprod, x, bilinearform, a
1038            i=i-1            i=i-1
1039       xhat=v[iter-1]*y[iter-1]       xhat=v[iter-1]*y[iter-1]
1040       for i in range(iter-1):       for i in range(iter-1):
1041      xhat += v[i]*y[i]         xhat += v[i]*y[i]
1042     else:     else:
1043       xhat=v[0] * 0       xhat=v[0] * 0
1044     if P_R!=None:     if P_R!=None:
# Line 1007  def MINRES(r, Aprod, x, Msolve, bilinear Line 1056  def MINRES(r, Aprod, x, Msolve, bilinear
1056      """      """
1057      Solver for      Solver for
1058    
1059      M{Ax=b}      *Ax=b*
1060    
1061      with a symmetric and positive definite operator A (more details required!).      with a symmetric and positive definite operator A (more details required!).
1062      It uses the minimum residual method (MINRES) with preconditioner M      It uses the minimum residual method (MINRES) with preconditioner M
# Line 1015  def MINRES(r, Aprod, x, Msolve, bilinear Line 1064  def MINRES(r, Aprod, x, Msolve, bilinear
1064    
1065      The iteration is terminated if      The iteration is terminated if
1066    
1067      M{|r| <= atol+rtol*|r0|}      *|r| <= atol+rtol*|r0|*
1068    
1069      where M{r0} is the initial residual and M{|.|} is the energy norm. In fact      where *r0* is the initial residual and *|.|* is the energy norm. In fact
1070    
1071      M{|r| = sqrt( bilinearform(Msolve(r),r))}      *|r| = sqrt( bilinearform(Msolve(r),r))*
1072    
1073      For details on the preconditioned conjugate gradient method see the book:      For details on the preconditioned conjugate gradient method see the book:
1074    
# Line 1027  def MINRES(r, Aprod, x, Msolve, bilinear Line 1076  def MINRES(r, Aprod, x, Msolve, bilinear
1076      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,
1077      C. Romine, and H. van der Vorst}.      C. Romine, and H. van der Vorst}.
1078    
1079      @param r: initial residual M{r=b-Ax}. C{r} is altered.      :param r: initial residual *r=b-Ax*. ``r`` is altered.
1080      @type r: 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)
1081      @param x: an initial guess for the solution      :param x: an initial guess for the solution
1082      @type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)      :type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1083      @param Aprod: returns the value Ax      :param Aprod: returns the value Ax
1084      @type Aprod: function C{Aprod(x)} where C{x} is of the same object like      :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
1085                   argument C{x}. The returned object needs to be of the same                   argument ``x``. The returned object needs to be of the same
1086                   type like argument C{r}.                   type like argument ``r``.
1087      @param Msolve: solves Mx=r      :param Msolve: solves Mx=r
1088      @type Msolve: function C{Msolve(r)} where C{r} is of the same type like      :type Msolve: function ``Msolve(r)`` where ``r`` is of the same type like
1089                    argument C{r}. The returned object needs to be of the same                    argument ``r``. The returned object needs to be of the same
1090                    type like argument C{x}.                    type like argument ``x``.
1091      @param bilinearform: inner product C{<x,r>}      :param bilinearform: inner product ``<x,r>``
1092      @type bilinearform: function C{bilinearform(x,r)} where C{x} is of the same      :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
1093                          type like argument C{x} and C{r} is. The returned value                          type like argument ``x`` and ``r`` is. The returned value
1094                          is a C{float}.                          is a ``float``.
1095      @param atol: absolute tolerance      :param atol: absolute tolerance
1096      @type atol: non-negative C{float}      :type atol: non-negative ``float``
1097      @param rtol: relative tolerance      :param rtol: relative tolerance
1098      @type rtol: non-negative C{float}      :type rtol: non-negative ``float``
1099      @param iter_max: maximum number of iteration steps      :param iter_max: maximum number of iteration steps
1100      @type iter_max: C{int}      :type iter_max: ``int``
1101      @return: the solution approximation and the corresponding residual      :return: the solution approximation and the corresponding residual
1102      @rtype: C{tuple}      :rtype: ``tuple``
1103      @warning: C{r} and C{x} are altered.      :warning: ``r`` and ``x`` are altered.
1104      """      """
1105      #------------------------------------------------------------------      #------------------------------------------------------------------
1106      # Set up y and v for the first Lanczos vector v1.      # Set up y and v for the first Lanczos vector v1.
# Line 1062  def MINRES(r, Aprod, x, Msolve, bilinear Line 1111  def MINRES(r, Aprod, x, Msolve, bilinear
1111      y = Msolve(r)      y = Msolve(r)
1112      beta1 = bilinearform(y,r)      beta1 = bilinearform(y,r)
1113    
1114      if beta1< 0: raise NegativeNorm,"negative norm."      if beta1< 0: raise NegativeNorm("negative norm.")
1115    
1116      #  If r = 0 exactly, stop with x      #  If r = 0 exactly, stop with x
1117      if beta1==0: return x      if beta1==0: return x
# Line 1097  def MINRES(r, Aprod, x, Msolve, bilinear Line 1146  def MINRES(r, Aprod, x, Msolve, bilinear
1146      # --------------------------------------------------------------------      # --------------------------------------------------------------------
1147      while not rnorm<=atol+rtol*Anorm*ynorm:    #  checks ||r|| < (||A|| ||x||) * TOL      while not rnorm<=atol+rtol*Anorm*ynorm:    #  checks ||r|| < (||A|| ||x||) * TOL
1148    
1149      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)
1150          iter    = iter  +  1          iter    = iter  +  1
1151    
1152          #-----------------------------------------------------------------          #-----------------------------------------------------------------
# Line 1127  def MINRES(r, Aprod, x, Msolve, bilinear Line 1176  def MINRES(r, Aprod, x, Msolve, bilinear
1176          y = Msolve(r2)          y = Msolve(r2)
1177          oldb   = beta                           # oldb = betak          oldb   = beta                           # oldb = betak
1178          beta   = bilinearform(y,r2)             # beta = betak+1^2          beta   = bilinearform(y,r2)             # beta = betak+1^2
1179          if beta < 0: raise NegativeNorm,"negative norm."          if beta < 0: raise NegativeNorm("negative norm.")
1180    
1181          beta   = math.sqrt( beta )          beta   = math.sqrt( beta )
1182          tnorm2 = tnorm2 + alfa*alfa + oldb*oldb + beta*beta          tnorm2 = tnorm2 + alfa*alfa + oldb*oldb + beta*beta
# Line 1185  def TFQMR(r, Aprod, x, bilinearform, ato Line 1234  def TFQMR(r, Aprod, x, bilinearform, ato
1234    """    """
1235    Solver for    Solver for
1236    
1237    M{Ax=b}    *Ax=b*
1238    
1239    with a general operator A (more details required!).    with a general operator A (more details required!).
1240    It uses the Transpose-Free Quasi-Minimal Residual method (TFQMR).    It uses the Transpose-Free Quasi-Minimal Residual method (TFQMR).
1241    
1242    The iteration is terminated if    The iteration is terminated if
1243    
1244    M{|r| <= atol+rtol*|r0|}    *|r| <= atol+rtol*|r0|*
1245    
1246    where M{r0} is the initial residual and M{|.|} is the energy norm. In fact    where *r0* is the initial residual and *|.|* is the energy norm. In fact
1247    
1248    M{|r| = sqrt( bilinearform(r,r))}    *|r| = sqrt( bilinearform(r,r))*
1249    
1250    @param r: initial residual M{r=b-Ax}. C{r} is altered.    :param r: initial residual *r=b-Ax*. ``r`` is altered.
1251    @type r: 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)
1252    @param x: an initial guess for the solution    :param x: an initial guess for the solution
1253    @type x: same like C{r}    :type x: same like ``r``
1254    @param Aprod: returns the value Ax    :param Aprod: returns the value Ax
1255    @type Aprod: function C{Aprod(x)} where C{x} is of the same object like    :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
1256                 argument C{x}. The returned object needs to be of the same type                 argument ``x``. The returned object needs to be of the same type
1257                 like argument C{r}.                 like argument ``r``.
1258    @param bilinearform: inner product C{<x,r>}    :param bilinearform: inner product ``<x,r>``
1259    @type bilinearform: function C{bilinearform(x,r)} where C{x} is of the same    :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
1260                        type like argument C{x} and C{r}. The returned value is                        type like argument ``x`` and ``r``. The returned value is
1261                        a C{float}.                        a ``float``.
1262    @param atol: absolute tolerance    :param atol: absolute tolerance
1263    @type atol: non-negative C{float}    :type atol: non-negative ``float``
1264    @param rtol: relative tolerance    :param rtol: relative tolerance
1265    @type rtol: non-negative C{float}    :type rtol: non-negative ``float``
1266    @param iter_max: maximum number of iteration steps    :param iter_max: maximum number of iteration steps
1267    @type iter_max: C{int}    :type iter_max: ``int``
1268    @rtype: C{tuple}    :rtype: ``tuple``
1269    @warning: C{r} and C{x} are altered.    :warning: ``r`` and ``x`` are altered.
1270    """    """
1271    u1=0    u1=0
1272    u2=0    u2=0
# Line 1234  def TFQMR(r, Aprod, x, bilinearform, ato Line 1283  def TFQMR(r, Aprod, x, bilinearform, ato
1283    theta = 0.0;    theta = 0.0;
1284    eta = 0.0;    eta = 0.0;
1285    rho=bilinearform(r,r)    rho=bilinearform(r,r)
1286    if rho < 0: raise NegativeNorm,"negative norm."    if rho < 0: raise NegativeNorm("negative norm.")
1287    tau = math.sqrt(rho)    tau = math.sqrt(rho)
1288    norm_r0=tau    norm_r0=tau
1289    while tau>atol+rtol*norm_r0:    while tau>atol+rtol*norm_r0:
1290      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)
1291    
1292      sigma = bilinearform(r,v)      sigma = bilinearform(r,v)
1293      if sigma == 0.0: raise IterationBreakDown,'TFQMR breakdown, sigma=0'      if sigma == 0.0: raise IterationBreakDown('TFQMR breakdown, sigma=0')
1294    
1295      alpha = rho / sigma      alpha = rho / sigma
1296    
# Line 1269  def TFQMR(r, Aprod, x, bilinearform, ato Line 1318  def TFQMR(r, Aprod, x, bilinearform, ato
1318  #  #
1319  #  Try to terminate the iteration at each pass through the loop  #  Try to terminate the iteration at each pass through the loop
1320  #  #
1321      if rho == 0.0: raise IterationBreakDown,'TFQMR breakdown, rho=0'      if rho == 0.0: raise IterationBreakDown('TFQMR breakdown, rho=0')
1322    
1323      rhon = bilinearform(r,w)      rhon = bilinearform(r,w)
1324      beta = rhon / rho;      beta = rhon / rho;
# Line 1287  def TFQMR(r, Aprod, x, bilinearform, ato Line 1336  def TFQMR(r, Aprod, x, bilinearform, ato
1336    
1337  class ArithmeticTuple(object):  class ArithmeticTuple(object):
1338     """     """
1339     Tuple supporting inplace update x+=y and scaling x=a*y where C{x,y} is an     Tuple supporting inplace update x+=y and scaling x=a*y where ``x,y`` is an
1340     ArithmeticTuple and C{a} is a float.     ArithmeticTuple and ``a`` is a float.
1341    
1342     Example of usage::     Example of usage::
1343    
# Line 1302  class ArithmeticTuple(object): Line 1351  class ArithmeticTuple(object):
1351     """     """
1352     def __init__(self,*args):     def __init__(self,*args):
1353         """         """
1354         Initializes object with elements C{args}.         Initializes object with elements ``args``.
1355    
1356         @param args: tuple of objects that support inplace add (x+=y) and         :param args: tuple of objects that support inplace add (x+=y) and
1357                      scaling (x=a*y)                      scaling (x=a*y)
1358         """         """
1359         self.__items=list(args)         self.__items=list(args)
# Line 1313  class ArithmeticTuple(object): Line 1362  class ArithmeticTuple(object):
1362         """         """
1363         Returns the number of items.         Returns the number of items.
1364    
1365         @return: number of items         :return: number of items
1366         @rtype: C{int}         :rtype: ``int``
1367         """         """
1368         return len(self.__items)         return len(self.__items)
1369    
# Line 1322  class ArithmeticTuple(object): Line 1371  class ArithmeticTuple(object):
1371         """         """
1372         Returns item at specified position.         Returns item at specified position.
1373    
1374         @param index: index of item to be returned         :param index: index of item to be returned
1375         @type index: C{int}         :type index: ``int``
1376         @return: item with index C{index}         :return: item with index ``index``
1377         """         """
1378         return self.__items.__getitem__(index)         return self.__items.__getitem__(index)
1379    
1380     def __mul__(self,other):     def __mul__(self,other):
1381         """         """
1382         Scales by C{other} from the right.         Scales by ``other`` from the right.
1383    
1384         @param other: scaling factor         :param other: scaling factor
1385         @type other: C{float}         :type other: ``float``
1386         @return: itemwise self*other         :return: itemwise self*other
1387         @rtype: L{ArithmeticTuple}         :rtype: `ArithmeticTuple`
1388         """         """
1389         out=[]         out=[]
1390         try:         try:
1391             l=len(other)             l=len(other)
1392             if l!=len(self):             if l!=len(self):
1393                 raise ValueError,"length of arguments don't match."                 raise ValueError("length of arguments don't match.")
1394             for i in range(l): out.append(self[i]*other[i])             for i in range(l): out.append(self[i]*other[i])
1395         except TypeError:         except TypeError:
1396             for i in range(len(self)): out.append(self[i]*other)             for i in range(len(self)): out.append(self[i]*other)
# Line 1349  class ArithmeticTuple(object): Line 1398  class ArithmeticTuple(object):
1398    
1399     def __rmul__(self,other):     def __rmul__(self,other):
1400         """         """
1401         Scales by C{other} from the left.         Scales by ``other`` from the left.
1402    
1403         @param other: scaling factor         :param other: scaling factor
1404         @type other: C{float}         :type other: ``float``
1405         @return: itemwise other*self         :return: itemwise other*self
1406         @rtype: L{ArithmeticTuple}         :rtype: `ArithmeticTuple`
1407         """         """
1408         out=[]         out=[]
1409         try:         try:
1410             l=len(other)             l=len(other)
1411             if l!=len(self):             if l!=len(self):
1412                 raise ValueError,"length of arguments don't match."                 raise ValueError("length of arguments don't match.")
1413             for i in range(l): out.append(other[i]*self[i])             for i in range(l): out.append(other[i]*self[i])
1414         except TypeError:         except TypeError:
1415             for i in range(len(self)): out.append(other*self[i])             for i in range(len(self)): out.append(other*self[i])
# Line 1368  class ArithmeticTuple(object): Line 1417  class ArithmeticTuple(object):
1417    
1418     def __div__(self,other):     def __div__(self,other):
1419         """         """
1420         Scales by (1/C{other}) from the right.         Scales by (1/``other``) from the right.
1421    
1422         @param other: scaling factor         :param other: scaling factor
1423         @type other: C{float}         :type other: ``float``
1424         @return: itemwise self/other         :return: itemwise self/other
1425         @rtype: L{ArithmeticTuple}         :rtype: `ArithmeticTuple`
1426         """         """
1427         return self*(1/other)         return self*(1/other)
1428    
1429     def __rdiv__(self,other):     def __rdiv__(self,other):
1430         """         """
1431         Scales by (1/C{other}) from the left.         Scales by (1/``other``) from the left.
1432    
1433         @param other: scaling factor         :param other: scaling factor
1434         @type other: C{float}         :type other: ``float``
1435         @return: itemwise other/self         :return: itemwise other/self
1436         @rtype: L{ArithmeticTuple}         :rtype: `ArithmeticTuple`
1437         """         """
1438         out=[]         out=[]
1439         try:         try:
1440             l=len(other)             l=len(other)
1441             if l!=len(self):             if l!=len(self):
1442                 raise ValueError,"length of arguments don't match."                 raise ValueError("length of arguments don't match.")
1443             for i in range(l): out.append(other[i]/self[i])             for i in range(l): out.append(other[i]/self[i])
1444         except TypeError:         except TypeError:
1445             for i in range(len(self)): out.append(other/self[i])             for i in range(len(self)): out.append(other/self[i])
# Line 1398  class ArithmeticTuple(object): Line 1447  class ArithmeticTuple(object):
1447    
1448     def __iadd__(self,other):     def __iadd__(self,other):
1449         """         """
1450         Inplace addition of C{other} to self.         Inplace addition of ``other`` to self.
1451    
1452         @param other: increment         :param other: increment
1453         @type other: C{ArithmeticTuple}         :type other: ``ArithmeticTuple``
1454         """         """
1455         if len(self) != len(other):         if len(self) != len(other):
1456             raise ValueError,"tuple lengths must match."             raise ValueError("tuple lengths must match.")
1457         for i in range(len(self)):         for i in range(len(self)):
1458             self.__items[i]+=other[i]             self.__items[i]+=other[i]
1459         return self         return self
1460    
1461     def __add__(self,other):     def __add__(self,other):
1462         """         """
1463         Adds C{other} to self.         Adds ``other`` to self.
1464    
1465         @param other: increment         :param other: increment
1466         @type other: C{ArithmeticTuple}         :type other: ``ArithmeticTuple``
1467         """         """
1468         out=[]         out=[]
1469         try:         try:
1470             l=len(other)             l=len(other)
1471             if l!=len(self):             if l!=len(self):
1472                 raise ValueError,"length of arguments don't match."                 raise ValueError("length of arguments don't match.")
1473             for i in range(l): out.append(self[i]+other[i])             for i in range(l): out.append(self[i]+other[i])
1474         except TypeError:         except TypeError:
1475             for i in range(len(self)): out.append(self[i]+other)             for i in range(len(self)): out.append(self[i]+other)
# Line 1428  class ArithmeticTuple(object): Line 1477  class ArithmeticTuple(object):
1477    
1478     def __sub__(self,other):     def __sub__(self,other):
1479         """         """
1480         Subtracts C{other} from self.         Subtracts ``other`` from self.
1481    
1482         @param other: decrement         :param other: decrement
1483         @type other: C{ArithmeticTuple}         :type other: ``ArithmeticTuple``
1484         """         """
1485         out=[]         out=[]
1486         try:         try:
1487             l=len(other)             l=len(other)
1488             if l!=len(self):             if l!=len(self):
1489                 raise ValueError,"length of arguments don't match."                 raise ValueError("length of arguments don't match.")
1490             for i in range(l): out.append(self[i]-other[i])             for i in range(l): out.append(self[i]-other[i])
1491         except TypeError:         except TypeError:
1492             for i in range(len(self)): out.append(self[i]-other)             for i in range(len(self)): out.append(self[i]-other)
# Line 1445  class ArithmeticTuple(object): Line 1494  class ArithmeticTuple(object):
1494    
1495     def __isub__(self,other):     def __isub__(self,other):
1496         """         """
1497         Inplace subtraction of C{other} from self.         Inplace subtraction of ``other`` from self.
1498    
1499         @param other: decrement         :param other: decrement
1500         @type other: C{ArithmeticTuple}         :type other: ``ArithmeticTuple``
1501         """         """
1502         if len(self) != len(other):         if len(self) != len(other):
1503             raise ValueError,"tuple length must match."             raise ValueError("tuple length must match.")
1504         for i in range(len(self)):         for i in range(len(self)):
1505             self.__items[i]-=other[i]             self.__items[i]-=other[i]
1506         return self         return self
# Line 1471  class HomogeneousSaddlePointProblem(obje Line 1520  class HomogeneousSaddlePointProblem(obje
1520        This class provides a framework for solving linear homogeneous saddle        This class provides a framework for solving linear homogeneous saddle
1521        point problems of the form::        point problems of the form::
1522    
1523            M{Av+B^*p=f}            *Av+B^*p=f*
1524            M{Bv     =0}            *Bv     =0*
1525    
1526        for the unknowns M{v} and M{p} and given operators M{A} and M{B} and        for the unknowns *v* and *p* and given operators *A* and *B* and
1527        given right hand side M{f}. M{B^*} is the adjoint operator of M{B}.        given right hand side *f*. *B^** is the adjoint operator of *B*.
1528          *A* may depend weakly on *v* and *p*.
1529        """        """
1530        def __init__(self, adaptSubTolerance=True, **kwargs):        def __init__(self, **kwargs):
     """  
     initializes the saddle point problem  
       
     @param adaptSubTolerance: If True the tolerance for subproblem is set automatically.  
     @type adaptSubTolerance: C{bool}  
     """  
         self.setTolerance()  
         self.setAbsoluteTolerance()  
     self.__adaptSubTolerance=adaptSubTolerance  
       #=============================================================  
       def initialize(self):  
1531          """          """
1532          Initializes the problem (overwrite).          initializes the saddle point problem
1533          """          """
1534          pass          self.resetControlParameters()
1535            self.setTolerance()
1536            self.setAbsoluteTolerance()
1537          def resetControlParameters(self, K_p=1., K_v=1., rtol_max=0.01, rtol_min = 1.e-7, chi_max=0.5, reduction_factor=0.3, theta = 0.1):
1538             """
1539             sets a control parameter
1540    
1541             :param K_p: initial value for constant to adjust pressure tolerance
1542             :type K_p: ``float``
1543             :param K_v: initial value for constant to adjust velocity tolerance
1544             :type K_v: ``float``
1545             :param rtol_max: maximuim relative tolerance used to calculate presssure and velocity increment.
1546             :type rtol_max: ``float``
1547             :param chi_max: maximum tolerable converegence rate.
1548             :type chi_max: ``float``
1549             :param reduction_factor: reduction factor for adjustment factors.
1550             :type reduction_factor: ``float``
1551             """
1552             self.setControlParameter(K_p, K_v, rtol_max, rtol_min, chi_max, reduction_factor, theta)
1553    
1554          def setControlParameter(self,K_p=None, K_v=None, rtol_max=None, rtol_min=None, chi_max=None, reduction_factor=None, theta=None):
1555             """
1556             sets a control parameter
1557    
1558    
1559             :param K_p: initial value for constant to adjust pressure tolerance
1560             :type K_p: ``float``
1561             :param K_v: initial value for constant to adjust velocity tolerance
1562             :type K_v: ``float``
1563             :param rtol_max: maximuim relative tolerance used to calculate presssure and velocity increment.
1564             :type rtol_max: ``float``
1565             :param chi_max: maximum tolerable converegence rate.
1566             :type chi_max: ``float``
1567             :type reduction_factor: ``float``
1568             """
1569             if not K_p == None:
1570                if K_p<1:
1571                   raise ValueError("K_p need to be greater or equal to 1.")
1572             else:
1573                K_p=self.__K_p
1574    
1575             if not K_v == None:
1576                if K_v<1:
1577                   raise ValueError("K_v need to be greater or equal to 1.")
1578             else:
1579                K_v=self.__K_v
1580    
1581             if not rtol_max == None:
1582                if rtol_max<=0 or rtol_max>=1:
1583                   raise ValueError("rtol_max needs to be positive and less than 1.")
1584             else:
1585                rtol_max=self.__rtol_max
1586    
1587             if not rtol_min == None:
1588                if rtol_min<=0 or rtol_min>=1:
1589                   raise ValueError("rtol_min needs to be positive and less than 1.")
1590             else:
1591                rtol_min=self.__rtol_min
1592    
1593             if not chi_max == None:
1594                if chi_max<=0 or chi_max>=1:
1595                   raise ValueError("chi_max needs to be positive and less than 1.")
1596             else:
1597                chi_max = self.__chi_max
1598    
1599             if not reduction_factor == None:
1600                if reduction_factor<=0 or reduction_factor>1:
1601                   raise ValueError("reduction_factor need to be between zero and one.")
1602             else:
1603                reduction_factor=self.__reduction_factor
1604    
1605             if not theta == None:
1606                if theta<=0 or theta>1:
1607                   raise ValueError("theta need to be between zero and one.")
1608             else:
1609                theta=self.__theta
1610    
1611             if rtol_min>=rtol_max:
1612                 raise ValueError("rtol_max = %e needs to be greater than rtol_min = %e"%(rtol_max,rtol_min))
1613             self.__chi_max = chi_max
1614             self.__rtol_max = rtol_max
1615             self.__K_p = K_p
1616             self.__K_v = K_v
1617             self.__reduction_factor = reduction_factor
1618             self.__theta = theta
1619             self.__rtol_min=rtol_min
1620    
1621          #=============================================================
1622        def inner_pBv(self,p,Bv):        def inner_pBv(self,p,Bv):
1623           """           """
1624           Returns inner product of element p and Bv (overwrite).           Returns inner product of element p and Bv (overwrite).
1625    
1626           @param p: a pressure increment           :param p: a pressure increment
1627           @param v: a residual           :param Bv: a residual
1628           @return: inner product of element p and Bv           :return: inner product of element p and Bv
1629           @rtype: C{float}           :rtype: ``float``
1630           @note: used if PCG is applied.           :note: used if PCG is applied.
1631           """           """
1632           raise NotImplementedError,"no inner product for p and Bv implemented."           raise NotImplementedError("no inner product for p and Bv implemented.")
1633    
1634        def inner_p(self,p0,p1):        def inner_p(self,p0,p1):
1635           """           """
1636           Returns inner product of p0 and p1 (overwrite).           Returns inner product of p0 and p1 (overwrite).
1637    
1638           @param p0: a pressure           :param p0: a pressure
1639           @param p1: a pressure           :param p1: a pressure
1640           @return: inner product of p0 and p1           :return: inner product of p0 and p1
1641           @rtype: C{float}           :rtype: ``float``
1642           """           """
1643           raise NotImplementedError,"no inner product for p implemented."           raise NotImplementedError("no inner product for p implemented.")
1644        
1645        def norm_v(self,v):        def norm_v(self,v):
1646           """           """
1647           Returns the norm of v (overwrite).           Returns the norm of v (overwrite).
1648    
1649           @param v: a velovity           :param v: a velovity
1650           @return: norm of v           :return: norm of v
1651           @rtype: non-negative C{float}           :rtype: non-negative ``float``
1652           """           """
1653           raise NotImplementedError,"no norm of v implemented."           raise NotImplementedError("no norm of v implemented.")
1654        def getV(self, p, v0):        def getDV(self, p, v, tol):
1655           """           """
1656           return the value for v for a given p (overwrite)           return a correction to the value for a given v and a given p with accuracy `tol` (overwrite)
1657    
1658           @param p: a pressure           :param p: pressure
1659           @param v0: a initial guess for the value v to return.           :param v: pressure
1660           @return: v given as M{v= A^{-1} (f-B^*p)}           :return: dv given as *dv= A^{-1} (f-A v-B^*p)*
1661             :note: Only *A* may depend on *v* and *p*
1662           """           """
1663           raise NotImplementedError,"no v calculation implemented."           raise NotImplementedError("no dv calculation implemented.")
1664    
1665                    
1666        def Bv(self,v):        def Bv(self,v, tol):
1667          """          """
1668          Returns Bv (overwrite).          Returns Bv with accuracy `tol` (overwrite)
1669    
1670          @rtype: equal to the type of p          :rtype: equal to the type of p
1671          @note: boundary conditions on p should be zero!          :note: boundary conditions on p should be zero!
1672          """          """
1673          raise NotImplementedError, "no operator B implemented."          raise NotImplementedError("no operator B implemented.")
1674    
1675        def norm_Bv(self,Bv):        def norm_Bv(self,Bv):
1676          """          """
1677          Returns the norm of Bv (overwrite).          Returns the norm of Bv (overwrite).
1678    
1679          @rtype: equal to the type of p          :rtype: equal to the type of p
1680          @note: boundary conditions on p should be zero!          :note: boundary conditions on p should be zero!
1681          """          """
1682          raise NotImplementedError, "no norm of Bv implemented."          raise NotImplementedError("no norm of Bv implemented.")
1683    
1684        def solve_AinvBt(self,p):        def solve_AinvBt(self,dp, tol):
1685           """           """
1686           Solves M{Av=B^*p} with accuracy L{self.getSubProblemTolerance()}           Solves *A dv=B^*dp* with accuracy `tol`
          (overwrite).  
1687    
1688           @param p: a pressure increment           :param dp: a pressure increment
1689           @return: the solution of M{Av=B^*p}           :return: the solution of *A dv=B^*dp*
1690           @note: boundary conditions on v should be zero!           :note: boundary conditions on dv should be zero! *A* is the operator used in ``getDV`` and must not be altered.
1691           """           """
1692           raise NotImplementedError,"no operator A implemented."           raise NotImplementedError("no operator A implemented.")
1693    
1694        def solve_prec(self,Bv):        def solve_prec(self,Bv, tol):
1695           """           """
1696           Provides a preconditioner for M{BA^{-1}B^*} applied to Bv with accuracy           Provides a preconditioner for *(BA^{-1}B^ * )* applied to Bv with accuracy `tol`
          L{self.getSubProblemTolerance()} (overwrite).  
1697    
1698           @rtype: equal to the type of p           :rtype: equal to the type of p
1699           @note: boundary conditions on p should be zero!           :note: boundary conditions on p should be zero!
          """  
          raise NotImplementedError,"no preconditioner for Schur complement implemented."  
       def setSubProblemTolerance(self):  
1700           """           """
1701       Updates the tolerance for subproblems           raise NotImplementedError("no preconditioner for Schur complement implemented.")
      @note: method is typically the method is overwritten.  
          """  
          pass  
1702        #=============================================================        #=============================================================
1703        def __Aprod_PCG(self,p):        def __Aprod_PCG(self,dp):
1704            dv=self.solve_AinvBt(p)            dv=self.solve_AinvBt(dp, self.__subtol)
1705            return ArithmeticTuple(dv,self.Bv(dv))            return ArithmeticTuple(dv,self.Bv(dv, self.__subtol))
1706    
1707        def __inner_PCG(self,p,r):        def __inner_PCG(self,p,r):
1708           return self.inner_pBv(p,r[1])           return self.inner_pBv(p,r[1])
1709    
1710        def __Msolve_PCG(self,r):        def __Msolve_PCG(self,r):
1711            return self.solve_prec(r[1])            return self.solve_prec(r[1], self.__subtol)
1712        #=============================================================        #=============================================================
1713        def __Aprod_GMRES(self,p):        def __Aprod_GMRES(self,p):
1714            return self.solve_prec(self.Bv(self.solve_AinvBt(p)))            return self.solve_prec(self.Bv(self.solve_AinvBt(p, self.__subtol), self.__subtol), self.__subtol)
1715        def __inner_GMRES(self,p0,p1):        def __inner_GMRES(self,p0,p1):
1716           return self.inner_p(p0,p1)           return self.inner_p(p0,p1)
1717    
1718        #=============================================================        #=============================================================
1719        def norm_p(self,p):        def norm_p(self,p):
1720            """            """
1721            calculates the norm of C{p}            calculates the norm of ``p``
1722                        
1723            @param p: a pressure            :param p: a pressure
1724            @return: the norm of C{p} using the inner product for pressure            :return: the norm of ``p`` using the inner product for pressure
1725            @rtype: C{float}            :rtype: ``float``
1726            """            """
1727            f=self.inner_p(p,p)            f=self.inner_p(p,p)
1728            if f<0: raise ValueError,"negative pressure norm."            if f<0: raise ValueError("negative pressure norm.")
1729            return math.sqrt(f)            return math.sqrt(f)
       def adaptSubTolerance(self):  
       """  
       Returns True if tolerance adaption for subproblem is choosen.  
       """  
           self.__adaptSubTolerance  
1730                
1731        def solve(self,v,p,max_iter=20, verbose=False, usePCG=True, iter_restart=20, max_correction_steps=10):        def solve(self,v,p,max_iter=20, verbose=False, usePCG=True, iter_restart=20, max_correction_steps=10):
1732           """           """
1733           Solves the saddle point problem using initial guesses v and p.           Solves the saddle point problem using initial guesses v and p.
1734    
1735           @param v: initial guess for velocity           :param v: initial guess for velocity
1736           @param p: initial guess for pressure           :param p: initial guess for pressure
1737           @type v: L{Data}           :type v: `Data`
1738           @type p: L{Data}           :type p: `Data`
1739           @param usePCG: indicates the usage of the PCG rather than GMRES scheme.           :param usePCG: indicates the usage of the PCG rather than GMRES scheme.
1740           @param max_iter: maximum number of iteration steps per correction           :param max_iter: maximum number of iteration steps per correction
1741                            attempt                            attempt
1742           @param verbose: if True, shows information on the progress of the           :param verbose: if True, shows information on the progress of the
1743                           saddlepoint problem solver.                           saddlepoint problem solver.
1744           @param iter_restart: restart the iteration after C{iter_restart} steps           :param iter_restart: restart the iteration after ``iter_restart`` steps
1745                                (only used if useUzaw=False)                                (only used if useUzaw=False)
1746           @type usePCG: C{bool}           :type usePCG: ``bool``
1747           @type max_iter: C{int}           :type max_iter: ``int``
1748           @type verbose: C{bool}           :type verbose: ``bool``
1749           @type iter_restart: C{int}           :type iter_restart: ``int``
1750           @rtype: C{tuple} of L{Data} objects           :rtype: ``tuple`` of `Data` objects
1751             :note: typically this method is overwritten by a subclass. It provides a wrapper for the ``_solve`` method.
1752             """
1753             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)
1754    
1755          def _solve(self,v,p,max_iter=20, verbose=False, usePCG=True, iter_restart=20, max_correction_steps=10):
1756             """
1757             see `_solve` method.
1758           """           """
1759           self.verbose=verbose           self.verbose=verbose
1760           rtol=self.getTolerance()           rtol=self.getTolerance()
1761           atol=self.getAbsoluteTolerance()           atol=self.getAbsoluteTolerance()
1762       if self.adaptSubTolerance(): self.setSubProblemTolerance()  
1763             K_p=self.__K_p
1764             K_v=self.__K_v
1765           correction_step=0           correction_step=0
1766           converged=False           converged=False
1767             chi=None
1768             eps=None
1769    
1770             if self.verbose: print(("HomogeneousSaddlePointProblem: start iteration: rtol= %e, atol=%e"%(rtol, atol)))
1771           while not converged:           while not converged:
               # calculate velocity for current pressure:  
               v=self.getV(p,v)  
               Bv=self.Bv(v)  
               norm_v=self.norm_v(v)  
               norm_Bv=self.norm_Bv(Bv)  
               ATOL=norm_v*rtol+atol  
               if self.verbose: print "HomogeneousSaddlePointProblem: norm v= %e, norm_Bv= %e, tolerance = %e."%(norm_v, norm_Bv,ATOL)  
               if not ATOL>0: raise ValueError,"overall absolute tolerance needs to be positive."  
               if norm_Bv <= ATOL:  
                  converged=True  
               else:  
                  correction_step+=1  
                  if correction_step>max_correction_steps:  
                       raise CorrectionFailed,"Given up after %d correction steps."%correction_step  
                  dp=self.solve_prec(Bv)  
                  if usePCG:  
                    norm2=self.inner_pBv(dp,Bv)  
                    if norm2<0: raise ValueError,"negative PCG norm."  
                    norm2=math.sqrt(norm2)  
                  else:  
                    norm2=self.norm_p(dp)  
                  ATOL_ITER=ATOL/norm_Bv*norm2*0.5  
                  if self.verbose: print "HomogeneousSaddlePointProblem: tolerance for solver: %e"%ATOL_ITER  
                  if usePCG:  
                        p,v0,a_norm=PCG(ArithmeticTuple(v,Bv),self.__Aprod_PCG,p,self.__Msolve_PCG,self.__inner_PCG,atol=ATOL_ITER, rtol=0.,iter_max=max_iter, verbose=self.verbose)  
                  else:  
                        p=GMRES(dp,self.__Aprod_GMRES, p, self.__inner_GMRES,atol=ATOL_ITER, rtol=0.,iter_max=max_iter, iter_restart=iter_restart, verbose=self.verbose)  
          if self.verbose: print "HomogeneousSaddlePointProblem: tolerance reached."  
      return v,p  
1772    
1773                 # get tolerance for velecity increment:
1774                 if chi == None:
1775                    rtol_v=self.__rtol_max
1776                 else:
1777                    rtol_v=min(chi/K_v,self.__rtol_max)
1778                 rtol_v=max(rtol_v, self.__rtol_min)
1779                 if self.verbose: print(("HomogeneousSaddlePointProblem: step %s: rtol_v= %e"%(correction_step,rtol_v)))
1780                 # get velocity increment:
1781                 dv1=self.getDV(p,v,rtol_v)
1782                 v1=v+dv1
1783                 Bv1=self.Bv(v1, rtol_v)
1784                 norm_Bv1=self.norm_Bv(Bv1)
1785                 norm_dv1=self.norm_v(dv1)
1786                 if self.verbose: print(("HomogeneousSaddlePointProblem: step %s: norm_Bv1 = %e, norm_dv1 = %e"%(correction_step, norm_Bv1, norm_dv1)))
1787                 if norm_dv1*self.__theta < norm_Bv1:
1788                    # get tolerance for pressure increment:
1789                    large_Bv1=True
1790                    if chi == None or eps == None:
1791                       rtol_p=self.__rtol_max
1792                    else:
1793                       rtol_p=min(chi**2*eps/K_p/norm_Bv1, self.__rtol_max)
1794                    self.__subtol=max(rtol_p**2, self.__rtol_min)
1795                    if self.verbose: print(("HomogeneousSaddlePointProblem: step %s: rtol_p= %e"%(correction_step,rtol_p)))
1796                    # now we solve for the pressure increment dp from B*A^{-1}B^* dp = Bv1
1797                    if usePCG:
1798                        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)
1799                        v2=r[0]
1800                        Bv2=r[1]
1801                    else:
1802                        # don't use!!!!
1803                        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)
1804                        dv2=self.solve_AinvBt(dp, self.__subtol)
1805                        v2=v1-dv2
1806                        Bv2=self.Bv(v2, self.__subtol)
1807                    p2=p+dp
1808                 else:
1809                    large_Bv1=False
1810                    v2=v1
1811                    p2=p
1812                 # update business:
1813                 norm_dv2=self.norm_v(v2-v)
1814                 norm_v2=self.norm_v(v2)
1815                 if self.verbose: print(("HomogeneousSaddlePointProblem: step %s: v2 = %e, norm_dv2 = %e"%(correction_step, norm_v2, self.norm_v(v2-v))))
1816                 eps, eps_old = max(norm_Bv1, norm_dv2), eps
1817                 if eps_old == None:
1818                      chi, chi_old = None, chi
1819                 else:
1820                      chi, chi_old = min(eps/ eps_old, self.__chi_max), chi
1821                 if eps != None:
1822                     if chi !=None:
1823                        if self.verbose: print(("HomogeneousSaddlePointProblem: step %s: convergence rate = %e, correction = %e"%(correction_step,chi, eps)))
1824                     else:
1825                        if self.verbose: print(("HomogeneousSaddlePointProblem: step %s: correction = %e"%(correction_step, eps)))
1826                 if eps <= rtol*norm_v2+atol :
1827                     converged = True
1828                 else:
1829                     if correction_step>=max_correction_steps:
1830                          raise CorrectionFailed("Given up after %d correction steps."%correction_step)
1831                     if chi_old!=None:
1832                        K_p=max(1,self.__reduction_factor*K_p,(chi-chi_old)/chi_old**2*K_p)
1833                        K_v=max(1,self.__reduction_factor*K_v,(chi-chi_old)/chi_old**2*K_p)
1834                        if self.verbose: print(("HomogeneousSaddlePointProblem: step %s: new adjustment factor K = %e"%(correction_step,K_p)))
1835                 correction_step+=1
1836                 v,p =v2, p2
1837             if self.verbose: print(("HomogeneousSaddlePointProblem: tolerance reached after %s steps."%correction_step))
1838             return v,p
1839        #========================================================================        #========================================================================
1840        def setTolerance(self,tolerance=1.e-4):        def setTolerance(self,tolerance=1.e-4):
1841           """           """
1842           Sets the relative tolerance for (v,p).           Sets the relative tolerance for (v,p).
1843    
1844           @param tolerance: tolerance to be used           :param tolerance: tolerance to be used
1845           @type tolerance: non-negative C{float}           :type tolerance: non-negative ``float``
1846           """           """
1847           if tolerance<0:           if tolerance<0:
1848               raise ValueError,"tolerance must be positive."               raise ValueError("tolerance must be positive.")
1849           self.__rtol=tolerance           self.__rtol=tolerance
1850    
1851        def getTolerance(self):        def getTolerance(self):
1852           """           """
1853           Returns the relative tolerance.           Returns the relative tolerance.
1854    
1855           @return: relative tolerance           :return: relative tolerance
1856           @rtype: C{float}           :rtype: ``float``
1857           """           """
1858           return self.__rtol           return self.__rtol
1859    
# Line 1698  class HomogeneousSaddlePointProblem(obje Line 1861  class HomogeneousSaddlePointProblem(obje
1861           """           """
1862           Sets the absolute tolerance.           Sets the absolute tolerance.
1863    
1864           @param tolerance: tolerance to be used           :param tolerance: tolerance to be used
1865           @type tolerance: non-negative C{float}           :type tolerance: non-negative ``float``
1866           """           """
1867           if tolerance<0:           if tolerance<0:
1868               raise ValueError,"tolerance must be non-negative."               raise ValueError("tolerance must be non-negative.")
1869           self.__atol=tolerance           self.__atol=tolerance
1870    
1871        def getAbsoluteTolerance(self):        def getAbsoluteTolerance(self):
1872           """           """
1873           Returns the absolute tolerance.           Returns the absolute tolerance.
1874    
1875           @return: absolute tolerance           :return: absolute tolerance
1876           @rtype: C{float}           :rtype: ``float``
1877           """           """
1878           return self.__atol           return self.__atol
1879    
       def getSubProblemTolerance(self):  
          """  
          Sets the relative tolerance to solve the subproblem(s).  
   
          @param rtol: relative tolerence  
          @type rtol: positive C{float}  
          """  
          return max(200.*util.EPSILON,self.getTolerance()**2)  
1880    
1881  def MaskFromBoundaryTag(domain,*tags):  def MaskFromBoundaryTag(domain,*tags):
1882     """     """
# Line 1730  def MaskFromBoundaryTag(domain,*tags): Line 1885  def MaskFromBoundaryTag(domain,*tags):
1885    
1886     Usage: m=MaskFromBoundaryTag(domain, "left", "right")     Usage: m=MaskFromBoundaryTag(domain, "left", "right")
1887    
1888     @param domain: domain to be used     :param domain: domain to be used
1889     @type domain: L{escript.Domain}     :type domain: `escript.Domain`
1890     @param tags: boundary tags     :param tags: boundary tags
1891     @type tags: C{str}     :type tags: ``str``
1892     @return: a mask which marks samples that are touching the boundary tagged     :return: a mask which marks samples that are touching the boundary tagged
1893              by any of the given tags              by any of the given tags
1894     @rtype: L{escript.Data} of rank 0     :rtype: `escript.Data` of rank 0
1895     """     """
1896     pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)     pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)
1897     d=escript.Scalar(0.,escript.FunctionOnBoundary(domain))     d=escript.Scalar(0.,escript.FunctionOnBoundary(domain))
# Line 1751  def MaskFromTag(domain,*tags): Line 1906  def MaskFromTag(domain,*tags):
1906    
1907     Usage: m=MaskFromTag(domain, "ham")     Usage: m=MaskFromTag(domain, "ham")
1908    
1909     @param domain: domain to be used     :param domain: domain to be used
1910     @type domain: L{escript.Domain}     :type domain: `escript.Domain`
1911     @param tags: boundary tags     :param tags: boundary tags
1912     @type tags: C{str}     :type tags: ``str``
1913     @return: a mask which marks samples that are touching the boundary tagged     :return: a mask which marks samples that are touching the boundary tagged
1914              by any of the given tags              by any of the given tags
1915     @rtype: L{escript.Data} of rank 0     :rtype: `escript.Data` of rank 0
1916     """     """
1917     pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)     pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)
1918     d=escript.Scalar(0.,escript.Function(domain))     d=escript.Scalar(0.,escript.Function(domain))

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