/[escript]/trunk/escript/py_src/pdetools.py
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revision 782 by bcumming, Tue Jul 18 00:47:47 2006 UTC revision 2948 by gross, Thu Feb 25 04:54:30 2010 UTC
# Line 1  Line 1 
1  # $Id$  
2    ########################################################
3    #
4    # Copyright (c) 2003-2010 by University of Queensland
5    # Earth Systems Science Computational Center (ESSCC)
6    # http://www.uq.edu.au/esscc
7    #
8    # Primary Business: Queensland, Australia
9    # Licensed under the Open Software License version 3.0
10    # http://www.opensource.org/licenses/osl-3.0.php
11    #
12    ########################################################
13    
14    __copyright__="""Copyright (c) 2003-2010 by University of Queensland
15    Earth Systems Science Computational Center (ESSCC)
16    http://www.uq.edu.au/esscc
17    Primary Business: Queensland, Australia"""
18    __license__="""Licensed under the Open Software License version 3.0
19    http://www.opensource.org/licenses/osl-3.0.php"""
20    __url__="https://launchpad.net/escript-finley"
21    
22  """  """
23  Provides some tools related to PDEs.  Provides some tools related to PDEs.
24    
25  Currently includes:  Currently includes:
26      - Projector - to project a discontinuous      - Projector - to project a discontinuous function onto a continuous function
27      - Locator - to trace values in data objects at a certain location      - Locator - to trace values in data objects at a certain location
28      - TimeIntegrationManager - to handel extraplotion in time      - TimeIntegrationManager - to handle extrapolation in time
29        - SaddlePointProblem - solver for Saddle point problems using the inexact uszawa scheme
30    
31  @var __author__: name of author  :var __author__: name of author
32  @var __copyright__: copyrights  :var __copyright__: copyrights
33  @var __license__: licence agreement  :var __license__: licence agreement
34  @var __url__: url entry point on documentation  :var __url__: url entry point on documentation
35  @var __version__: version  :var __version__: version
36  @var __date__: date of the version  :var __date__: date of the version
37  """  """
38    
39  __author__="Lutz Gross, l.gross@uq.edu.au"  __author__="Lutz Gross, l.gross@uq.edu.au"
 __copyright__="""  Copyright (c) 2006 by ACcESS MNRF  
                     http://www.access.edu.au  
                 Primary Business: Queensland, Australia"""  
 __license__="""Licensed under the Open Software License version 3.0  
              http://www.opensource.org/licenses/osl-3.0.php"""  
 __url__="http://www.iservo.edu.au/esys"  
 __version__="$Revision$"  
 __date__="$Date$"  
40    
41    
42  import escript  import escript
43  import linearPDEs  import linearPDEs
44  import numarray  import numpy
45  import util  import util
46    import math
47    
48  class TimeIntegrationManager:  class TimeIntegrationManager:
49    """    """
50    a simple mechanism to manage time dependend values.    A simple mechanism to manage time dependend values.
51    
52    typical usage is::    Typical usage is::
53    
54       dt=0.1 # time increment       dt=0.1 # time increment
55       tm=TimeIntegrationManager(inital_value,p=1)       tm=TimeIntegrationManager(inital_value,p=1)
# Line 46  class TimeIntegrationManager: Line 59  class TimeIntegrationManager:
59           tm.checkin(dt,v)           tm.checkin(dt,v)
60           t+=dt           t+=dt
61    
62    @note: currently only p=1 is supported.    :note: currently only p=1 is supported.
63    """    """
64    def __init__(self,*inital_values,**kwargs):    def __init__(self,*inital_values,**kwargs):
65       """       """
66       sets up the value manager where inital_value is the initial value and p is order used for extrapolation       Sets up the value manager where ``inital_values`` are the initial values
67         and p is the order used for extrapolation.
68       """       """
69       if kwargs.has_key("p"):       if kwargs.has_key("p"):
70              self.__p=kwargs["p"]              self.__p=kwargs["p"]
# Line 67  class TimeIntegrationManager: Line 81  class TimeIntegrationManager:
81    
82    def getTime(self):    def getTime(self):
83        return self.__t        return self.__t
84    
85    def getValue(self):    def getValue(self):
86        out=self.__v_mem[0]        out=self.__v_mem[0]
87        if len(out)==1:        if len(out)==1:
# Line 76  class TimeIntegrationManager: Line 91  class TimeIntegrationManager:
91    
92    def checkin(self,dt,*values):    def checkin(self,dt,*values):
93        """        """
94        adds new values to the manager. the p+1 last value get lost        Adds new values to the manager. The p+1 last values are lost.
95        """        """
96        o=min(self.__order+1,self.__p)        o=min(self.__order+1,self.__p)
97        self.__order=min(self.__order+1,self.__p)        self.__order=min(self.__order+1,self.__p)
# Line 93  class TimeIntegrationManager: Line 108  class TimeIntegrationManager:
108    
109    def extrapolate(self,dt):    def extrapolate(self,dt):
110        """        """
111        extrapolates to dt forward in time.        Extrapolates to ``dt`` forward in time.
112        """        """
113        if self.__order==0:        if self.__order==0:
114           out=self.__v_mem[0]           out=self.__v_mem[0]
# Line 108  class TimeIntegrationManager: Line 123  class TimeIntegrationManager:
123           return out[0]           return out[0]
124        else:        else:
125           return out           return out
126    
127    
128  class Projector:  class Projector:
129    """    """
130    The Projector is a factory which projects a discontiuous function onto a    The Projector is a factory which projects a discontinuous function onto a
131    continuous function on the a given domain.    continuous function on a given domain.
132    """    """
133    def __init__(self, domain, reduce = True, fast=True):    def __init__(self, domain, reduce=True, fast=True):
134      """      """
135      Create a continuous function space projector for a domain.      Creates a continuous function space projector for a domain.
136    
137      @param domain: Domain of the projection.      :param domain: Domain of the projection.
138      @param reduce: Flag to reduce projection order (default is True)      :param reduce: Flag to reduce projection order
139      @param fast: Flag to use a fast method based on matrix lumping (default is true)      :param fast: Flag to use a fast method based on matrix lumping
140      """      """
141      self.__pde = linearPDEs.LinearPDE(domain)      self.__pde = linearPDEs.LinearPDE(domain)
142      if fast:      if fast:
143        self.__pde.setSolverMethod(linearPDEs.LinearPDE.LUMPING)          self.__pde.getSolverOptions().setSolverMethod(linearPDEs.SolverOptions.LUMPING)
144      self.__pde.setSymmetryOn()      self.__pde.setSymmetryOn()
145      self.__pde.setReducedOrderTo(reduce)      self.__pde.setReducedOrderTo(reduce)
146      self.__pde.setValue(D = 1.)      self.__pde.setValue(D = 1.)
147      return      return
148      def getSolverOptions(self):
149        """
150        Returns the solver options of the PDE solver.
151        
152        :rtype: `linearPDEs.SolverOptions`
153        """
154        return self.__pde.getSolverOptions()
155    
156    def __del__(self):    def getValue(self, input_data):
157      return      """
158        Projects ``input_data`` onto a continuous function.
159    
160        :param input_data: the data to be projected
161        """
162        return self(input_data)
163    
164    def __call__(self, input_data):    def __call__(self, input_data):
165      """      """
166      Projects input_data onto a continuous function      Projects ``input_data`` onto a continuous function.
167    
168      @param input_data: The input_data to be projected.      :param input_data: the data to be projected
169      """      """
170      out=escript.Data(0.,input_data.getShape(),self.__pde.getFunctionSpaceForSolution())      out=escript.Data(0.,input_data.getShape(),self.__pde.getFunctionSpaceForSolution())
171        self.__pde.setValue(Y = escript.Data(), Y_reduced = escript.Data())
172      if input_data.getRank()==0:      if input_data.getRank()==0:
173          self.__pde.setValue(Y = input_data)          self.__pde.setValue(Y = input_data)
174          out=self.__pde.getSolution()          out=self.__pde.getSolution()
# Line 170  class Projector: Line 198  class Projector:
198    
199  class NoPDE:  class NoPDE:
200       """       """
201       solves the following problem for u:       Solves the following problem for u:
202    
203       M{kronecker[i,j]*D[j]*u[j]=Y[i]}       *kronecker[i,j]*D[j]*u[j]=Y[i]*
204    
205       with constraint       with constraint
206    
207       M{u[j]=r[j]}  where M{q[j]>0}       *u[j]=r[j]*  where *q[j]>0*
208    
209       where D, Y, r and q are given functions of rank 1.       where *D*, *Y*, *r* and *q* are given functions of rank 1.
210    
211       In the case of scalars this takes the form       In the case of scalars this takes the form
212    
213       M{D*u=Y}       *D*u=Y*
214    
215       with constraint       with constraint
216    
217       M{u=r}  where M{q>0}       *u=r* where *q>0*
218    
219       where D, Y, r and q are given scalar functions.       where *D*, *Y*, *r* and *q* are given scalar functions.
220    
221       The constraint is overwriting any other condition.       The constraint overwrites any other condition.
222    
223       @note: This class is similar to the L{linearPDEs.LinearPDE} class with A=B=C=X=0 but has the intention       :note: This class is similar to the `linearPDEs.LinearPDE` class with
224              that all input parameter are given in L{Solution} or L{ReducedSolution}. The whole              A=B=C=X=0 but has the intention that all input parameters are given
225              thing is a bit strange and I blame Robert.Woodcock@csiro.au for this.              in `Solution` or `ReducedSolution`.
226       """       """
227         # The whole thing is a bit strange and I blame Rob Woodcock (CSIRO) for
228         # this.
229       def __init__(self,domain,D=None,Y=None,q=None,r=None):       def __init__(self,domain,D=None,Y=None,q=None,r=None):
230           """           """
231           initialize the problem           Initializes the problem.
232    
233           @param domain: domain of the PDE.           :param domain: domain of the PDE
234           @type domain: L{Domain}           :type domain: `Domain`
235           @param D: coefficient of the solution.           :param D: coefficient of the solution
236           @type D: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type D: ``float``, ``int``, ``numpy.ndarray``, `Data`
237           @param Y: right hand side           :param Y: right hand side
238           @type Y: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type Y: ``float``, ``int``, ``numpy.ndarray``, `Data`
239           @param q: location of constraints           :param q: location of constraints
240           @type q: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type q: ``float``, ``int``, ``numpy.ndarray``, `Data`
241           @param r: value of solution at locations of constraints           :param r: value of solution at locations of constraints
242           @type r: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type r: ``float``, ``int``, ``numpy.ndarray``, `Data`
243           """           """
244           self.__domain=domain           self.__domain=domain
245           self.__D=D           self.__D=D
# Line 218  class NoPDE: Line 248  class NoPDE:
248           self.__r=r           self.__r=r
249           self.__u=None           self.__u=None
250           self.__function_space=escript.Solution(self.__domain)           self.__function_space=escript.Solution(self.__domain)
251    
252       def setReducedOn(self):       def setReducedOn(self):
253           """           """
254           sets the L{FunctionSpace} of the solution to L{ReducedSolution}           Sets the `FunctionSpace` of the solution to `ReducedSolution`.
255           """           """
256           self.__function_space=escript.ReducedSolution(self.__domain)           self.__function_space=escript.ReducedSolution(self.__domain)
257           self.__u=None           self.__u=None
258    
259       def setReducedOff(self):       def setReducedOff(self):
260           """           """
261           sets the L{FunctionSpace} of the solution to L{Solution}           Sets the `FunctionSpace` of the solution to `Solution`.
262           """           """
263           self.__function_space=escript.Solution(self.__domain)           self.__function_space=escript.Solution(self.__domain)
264           self.__u=None           self.__u=None
265            
266       def setValue(self,D=None,Y=None,q=None,r=None):       def setValue(self,D=None,Y=None,q=None,r=None):
267           """           """
268           assigns values to the parameters.           Assigns values to the parameters.
269    
270           @param D: coefficient of the solution.           :param D: coefficient of the solution
271           @type D: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type D: ``float``, ``int``, ``numpy.ndarray``, `Data`
272           @param Y: right hand side           :param Y: right hand side
273           @type Y: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type Y: ``float``, ``int``, ``numpy.ndarray``, `Data`
274           @param q: location of constraints           :param q: location of constraints
275           @type q: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type q: ``float``, ``int``, ``numpy.ndarray``, `Data`
276           @param r: value of solution at locations of constraints           :param r: value of solution at locations of constraints
277           @type r: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type r: ``float``, ``int``, ``numpy.ndarray``, `Data`
278           """           """
279           if not D==None:           if not D==None:
280              self.__D=D              self.__D=D
# Line 260  class NoPDE: Line 291  class NoPDE:
291    
292       def getSolution(self):       def getSolution(self):
293           """           """
294           returns the solution           Returns the solution.
295            
296           @return: the solution of the problem           :return: the solution of the problem
297           @rtype: L{Data} object in the L{FunctionSpace} L{Solution} or L{ReducedSolution}.           :rtype: `Data` object in the `FunctionSpace` `Solution` or
298                     `ReducedSolution`
299           """           """
300           if self.__u==None:           if self.__u==None:
301              if self.__D==None:              if self.__D==None:
# Line 280  class NoPDE: Line 312  class NoPDE:
312                  self.__u*=(1.-q)                  self.__u*=(1.-q)
313                  if not self.__r==None: self.__u+=q*self.__r                  if not self.__r==None: self.__u+=q*self.__r
314           return self.__u           return self.__u
315                
316  class Locator:  class Locator:
317       """       """
318       Locator provides access to the values of data objects at a given       Locator provides access to the values of data objects at a given spatial
319       spatial coordinate x.         coordinate x.
320        
321       In fact, a Locator object finds the sample in the set of samples of a       In fact, a Locator object finds the sample in the set of samples of a
322       given function space or domain where which is closest to the given       given function space or domain which is closest to the given point x.
      point x.  
323       """       """
324    
325       def __init__(self,where,x=numarray.zeros((3,))):       def __init__(self,where,x=numpy.zeros((3,))):
326         """         """
327         Initializes a Locator to access values in Data objects on the Doamin         Initializes a Locator to access values in Data objects on the Doamin
328         or FunctionSpace where for the sample point which         or FunctionSpace for the sample point which is closest to the given
329         closest to the given point x.         point x.
330    
331           :param where: function space
332           :type where: `escript.FunctionSpace`
333           :param x: location(s) of the Locator
334           :type x: ``numpy.ndarray`` or ``list`` of ``numpy.ndarray``
335         """         """
336         if isinstance(where,escript.FunctionSpace):         if isinstance(where,escript.FunctionSpace):
337            self.__function_space=where            self.__function_space=where
338         else:         else:
339            self.__function_space=escript.ContinuousFunction(where)            self.__function_space=escript.ContinuousFunction(where)
340         self.__id=util.length(self.__function_space.getX()-x[:self.__function_space.getDim()]).mindp()         iterative=False
341           if isinstance(x, list):
342               if len(x)==0:
343                  raise ValueError, "At least one point must be given."
344               try:
345                 iter(x[0])
346                 iterative=True
347               except TypeError:
348                 iterative=False
349           if iterative:
350               self.__id=[]
351               xxx=self.__function_space.getX()
352               for p in x:
353                  self.__id.append(util.length(xxx-p[:self.__function_space.getDim()]).minGlobalDataPoint())
354           else:
355               self.__id=util.length(xxx-x[:self.__function_space.getDim()]).minGlobalDataPoint()
356    
357       def __str__(self):       def __str__(self):
358         """         """
359         Returns the coordinates of the Locator as a string.         Returns the coordinates of the Locator as a string.
360         """         """
361         return "<Locator %s>"%str(self.getX())         x=self.getX()
362           if isinstance(x,list):
363              out="["
364              first=True
365              for xx in x:
366                if not first:
367                    out+=","
368                else:
369                    first=False
370                out+=str(xx)
371              out+="]>"
372           else:
373              out=str(x)
374           return out
375    
376         def getX(self):
377            """
378            Returns the exact coordinates of the Locator.
379            """
380            return self(self.getFunctionSpace().getX())
381    
382       def getFunctionSpace(self):       def getFunctionSpace(self):
383          """          """
384      Returns the function space of the Locator.          Returns the function space of the Locator.
385      """          """
386          return self.__function_space          return self.__function_space
387    
388       def getId(self):       def getId(self,item=None):
389          """          """
390      Returns the identifier of the location.          Returns the identifier of the location.
     """  
         return self.__id  
   
      def getX(self):  
391          """          """
392      Returns the exact coordinates of the Locator.          if item == None:
393      """             return self.__id
394          return self(self.getFunctionSpace().getX())          else:
395               if isinstance(self.__id,list):
396                  return self.__id[item]
397               else:
398                  return self.__id
399    
400    
401       def __call__(self,data):       def __call__(self,data):
402          """          """
403      Returns the value of data at the Locator of a Data object otherwise          Returns the value of data at the Locator of a Data object.
404      the object is returned.          """
     """  
405          return self.getValue(data)          return self.getValue(data)
406    
407       def getValue(self,data):       def getValue(self,data):
408          """          """
409      Returns the value of data at the Locator if data is a Data object          Returns the value of ``data`` at the Locator if ``data`` is a `Data`
410      otherwise the object is returned.          object otherwise the object is returned.
411      """          """
412          if isinstance(data,escript.Data):          if isinstance(data,escript.Data):
413             if data.getFunctionSpace()==self.getFunctionSpace():             dat=util.interpolate(data,self.getFunctionSpace())
414               out=data.convertToNumArrayFromDPNo(self.getId()[0],self.getId()[1])             id=self.getId()
415               #out=data.convertToNumArrayFromDPNo(self.getId()[0],self.getId()[1],self.getId()[2])             r=data.getRank()
416             else:             if isinstance(id,list):
417               out=data.interpolate(self.getFunctionSpace()).convertToNumArrayFromDPNo(self.getId()[0],self.getId()[1])                 out=[]
418               #out=data.interpolate(self.getFunctionSpace()).convertToNumArrayFromDPNo(self.getId()[0],self.getId()[1],self.getId()[2])                 for i in id:
419             if data.getRank()==0:                    o=numpy.array(dat.getTupleForGlobalDataPoint(*i))
420                return out[0]                    if data.getRank()==0:
421                         out.append(o[0])
422                      else:
423                         out.append(o)
424                   return out
425             else:             else:
426                return out               out=numpy.array(dat.getTupleForGlobalDataPoint(*id))
427                 if data.getRank()==0:
428                    return out[0]
429                 else:
430                    return out
431          else:          else:
432             return data             return data
433    
434  # vim: expandtab shiftwidth=4:  
435    def getInfLocator(arg):
436        """
437        Return a Locator for a point with the inf value over all arg.
438        """
439        if not isinstance(arg, escript.Data):
440        raise TypeError,"getInfLocator: Unknown argument type."
441        a_inf=util.inf(arg)
442        loc=util.length(arg-a_inf).minGlobalDataPoint() # This gives us the location but not coords
443        x=arg.getFunctionSpace().getX()
444        x_min=x.getTupleForGlobalDataPoint(*loc)
445        return Locator(arg.getFunctionSpace(),x_min)
446    
447    def getSupLocator(arg):
448        """
449        Return a Locator for a point with the sup value over all arg.
450        """
451        if not isinstance(arg, escript.Data):
452        raise TypeError,"getInfLocator: Unknown argument type."
453        a_inf=util.sup(arg)
454        loc=util.length(arg-a_inf).minGlobalDataPoint() # This gives us the location but not coords
455        x=arg.getFunctionSpace().getX()
456        x_min=x.getTupleForGlobalDataPoint(*loc)
457        return Locator(arg.getFunctionSpace(),x_min)
458        
459    
460    class SolverSchemeException(Exception):
461       """
462       This is a generic exception thrown by solvers.
463       """
464       pass
465    
466    class IndefinitePreconditioner(SolverSchemeException):
467       """
468       Exception thrown if the preconditioner is not positive definite.
469       """
470       pass
471    
472    class MaxIterReached(SolverSchemeException):
473       """
474       Exception thrown if the maximum number of iteration steps is reached.
475       """
476       pass
477    
478    class CorrectionFailed(SolverSchemeException):
479       """
480       Exception thrown if no convergence has been achieved in the solution
481       correction scheme.
482       """
483       pass
484    
485    class IterationBreakDown(SolverSchemeException):
486       """
487       Exception thrown if the iteration scheme encountered an incurable breakdown.
488       """
489       pass
490    
491    class NegativeNorm(SolverSchemeException):
492       """
493       Exception thrown if a norm calculation returns a negative norm.
494       """
495       pass
496    
497    def PCG(r, Aprod, x, Msolve, bilinearform, atol=0, rtol=1.e-8, iter_max=100, initial_guess=True, verbose=False):
498       """
499       Solver for
500    
501       *Ax=b*
502    
503       with a symmetric and positive definite operator A (more details required!).
504       It uses the conjugate gradient method with preconditioner M providing an
505       approximation of A.
506    
507       The iteration is terminated if
508    
509       *|r| <= atol+rtol*|r0|*
510    
511       where *r0* is the initial residual and *|.|* is the energy norm. In fact
512    
513       *|r| = sqrt( bilinearform(Msolve(r),r))*
514    
515       For details on the preconditioned conjugate gradient method see the book:
516    
517       I{Templates for the Solution of Linear Systems by R. Barrett, M. Berry,
518       T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo,
519       C. Romine, and H. van der Vorst}.
520    
521       :param r: initial residual *r=b-Ax*. ``r`` is altered.
522       :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
523       :param x: an initial guess for the solution
524       :type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
525       :param Aprod: returns the value Ax
526       :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
527                    argument ``x``. The returned object needs to be of the same type
528                    like argument ``r``.
529       :param Msolve: solves Mx=r
530       :type Msolve: function ``Msolve(r)`` where ``r`` is of the same type like
531                     argument ``r``. The returned object needs to be of the same
532                     type like argument ``x``.
533       :param bilinearform: inner product ``<x,r>``
534       :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
535                           type like argument ``x`` and ``r`` is. The returned value
536                           is a ``float``.
537       :param atol: absolute tolerance
538       :type atol: non-negative ``float``
539       :param rtol: relative tolerance
540       :type rtol: non-negative ``float``
541       :param iter_max: maximum number of iteration steps
542       :type iter_max: ``int``
543       :return: the solution approximation and the corresponding residual
544       :rtype: ``tuple``
545       :warning: ``r`` and ``x`` are altered.
546       """
547       iter=0
548       rhat=Msolve(r)
549       d = rhat
550       rhat_dot_r = bilinearform(rhat, r)
551       if rhat_dot_r<0: raise NegativeNorm,"negative norm."
552       norm_r0=math.sqrt(rhat_dot_r)
553       atol2=atol+rtol*norm_r0
554       if atol2<=0:
555          raise ValueError,"Non-positive tolarance."
556       atol2=max(atol2, 100. * util.EPSILON * norm_r0)
557    
558       if verbose: print "PCG: initial residual norm = %e (absolute tolerance = %e)"%(norm_r0, atol2)
559    
560    
561       while not math.sqrt(rhat_dot_r) <= atol2:
562           iter+=1
563           if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
564    
565           q=Aprod(d)
566           alpha = rhat_dot_r / bilinearform(d, q)
567           x += alpha * d
568           if isinstance(q,ArithmeticTuple):
569           r += q * (-alpha)      # Doing it the other way calls the float64.__mul__ not AT.__rmul__
570           else:
571               r += (-alpha) * q
572           rhat=Msolve(r)
573           rhat_dot_r_new = bilinearform(rhat, r)
574           beta = rhat_dot_r_new / rhat_dot_r
575           rhat+=beta * d
576           d=rhat
577    
578           rhat_dot_r = rhat_dot_r_new
579           if rhat_dot_r<0: raise NegativeNorm,"negative norm."
580           if verbose: print "PCG: iteration step %s: residual norm = %e"%(iter, math.sqrt(rhat_dot_r))
581       if verbose: print "PCG: tolerance reached after %s steps."%iter
582       return x,r,math.sqrt(rhat_dot_r)
583    
584    class Defect(object):
585        """
586        Defines a non-linear defect F(x) of a variable x.
587        """
588        def __init__(self):
589            """
590            Initializes defect.
591            """
592            self.setDerivativeIncrementLength()
593    
594        def bilinearform(self, x0, x1):
595            """
596            Returns the inner product of x0 and x1
597    
598            :param x0: value for x0
599            :param x1: value for x1
600            :return: the inner product of x0 and x1
601            :rtype: ``float``
602            """
603            return 0
604    
605        def norm(self,x):
606            """
607            Returns the norm of argument ``x``.
608    
609            :param x: a value
610            :return: norm of argument x
611            :rtype: ``float``
612            :note: by default ``sqrt(self.bilinearform(x,x)`` is returned.
613            """
614            s=self.bilinearform(x,x)
615            if s<0: raise NegativeNorm,"negative norm."
616            return math.sqrt(s)
617    
618        def eval(self,x):
619            """
620            Returns the value F of a given ``x``.
621    
622            :param x: value for which the defect ``F`` is evaluated
623            :return: value of the defect at ``x``
624            """
625            return 0
626    
627        def __call__(self,x):
628            return self.eval(x)
629    
630        def setDerivativeIncrementLength(self,inc=1000.*math.sqrt(util.EPSILON)):
631            """
632            Sets the relative length of the increment used to approximate the
633            derivative of the defect. The increment is inc*norm(x)/norm(v)*v in the
634            direction of v with x as a starting point.
635    
636            :param inc: relative increment length
637            :type inc: positive ``float``
638            """
639            if inc<=0: raise ValueError,"positive increment required."
640            self.__inc=inc
641    
642        def getDerivativeIncrementLength(self):
643            """
644            Returns the relative increment length used to approximate the
645            derivative of the defect.
646            :return: value of the defect at ``x``
647            :rtype: positive ``float``
648            """
649            return self.__inc
650    
651        def derivative(self, F0, x0, v, v_is_normalised=True):
652            """
653            Returns the directional derivative at ``x0`` in the direction of ``v``.
654    
655            :param F0: value of this defect at x0
656            :param x0: value at which derivative is calculated
657            :param v: direction
658            :param v_is_normalised: True to indicate that ``v`` is nomalized
659                                    (self.norm(v)=0)
660            :return: derivative of this defect at x0 in the direction of ``v``
661            :note: by default numerical evaluation (self.eval(x0+eps*v)-F0)/eps is
662                   used but this method maybe overwritten to use exact evaluation.
663            """
664            normx=self.norm(x0)
665            if normx>0:
666                 epsnew = self.getDerivativeIncrementLength() * normx
667            else:
668                 epsnew = self.getDerivativeIncrementLength()
669            if not v_is_normalised:
670                normv=self.norm(v)
671                if normv<=0:
672                   return F0*0
673                else:
674                   epsnew /= normv
675            F1=self.eval(x0 + epsnew * v)
676            return (F1-F0)/epsnew
677    
678    ######################################
679    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):
680       """
681       Solves a non-linear problem *F(x)=0* for unknown *x* using the stopping
682       criterion:
683    
684       *norm(F(x) <= atol + rtol * norm(F(x0)*
685    
686       where *x0* is the initial guess.
687    
688       :param defect: object defining the function *F*. ``defect.norm`` defines the
689                      *norm* used in the stopping criterion.
690       :type defect: `Defect`
691       :param x: initial guess for the solution, ``x`` is altered.
692       :type x: any object type allowing basic operations such as
693                ``numpy.ndarray``, `Data`
694       :param iter_max: maximum number of iteration steps
695       :type iter_max: positive ``int``
696       :param sub_iter_max: maximum number of inner iteration steps
697       :type sub_iter_max: positive ``int``
698       :param atol: absolute tolerance for the solution
699       :type atol: positive ``float``
700       :param rtol: relative tolerance for the solution
701       :type rtol: positive ``float``
702       :param gamma: tolerance safety factor for inner iteration
703       :type gamma: positive ``float``, less than 1
704       :param subtol_max: upper bound for inner tolerance
705       :type subtol_max: positive ``float``, less than 1
706       :return: an approximation of the solution with the desired accuracy
707       :rtype: same type as the initial guess
708       """
709       lmaxit=iter_max
710       if atol<0: raise ValueError,"atol needs to be non-negative."
711       if rtol<0: raise ValueError,"rtol needs to be non-negative."
712       if rtol+atol<=0: raise ValueError,"rtol or atol needs to be non-negative."
713       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
714       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
715    
716       F=defect(x)
717       fnrm=defect.norm(F)
718       stop_tol=atol + rtol*fnrm
719       subtol=subtol_max
720       if verbose: print "NewtonGMRES: initial residual = %e."%fnrm
721       if verbose: print "             tolerance = %e."%subtol
722       iter=1
723       #
724       # main iteration loop
725       #
726       while not fnrm<=stop_tol:
727                if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
728                #
729            #   adjust subtol_
730            #
731                if iter > 1:
732               rat=fnrm/fnrmo
733                   subtol_old=subtol
734               subtol=gamma*rat**2
735               if gamma*subtol_old**2 > .1: subtol=max(subtol,gamma*subtol_old**2)
736               subtol=max(min(subtol,subtol_max), .5*stop_tol/fnrm)
737            #
738            # calculate newton increment xc
739                #     if iter_max in __FDGMRES is reached MaxIterReached is thrown
740                #     if iter_restart -1 is returned as sub_iter
741                #     if  atol is reached sub_iter returns the numer of steps performed to get there
742                #
743                #
744                if verbose: print "             subiteration (GMRES) is called with relative tolerance %e."%subtol
745                try:
746                   xc, sub_iter=__FDGMRES(F, defect, x, subtol*fnrm, iter_max=iter_max-iter, iter_restart=sub_iter_max)
747                except MaxIterReached:
748                   raise MaxIterReached,"maximum number of %s steps reached."%iter_max
749                if sub_iter<0:
750                   iter+=sub_iter_max
751                else:
752                   iter+=sub_iter
753                # ====
754            x+=xc
755                F=defect(x)
756            iter+=1
757                fnrmo, fnrm=fnrm, defect.norm(F)
758                if verbose: print "             step %s: residual %e."%(iter,fnrm)
759       if verbose: print "NewtonGMRES: completed after %s steps."%iter
760       return x
761    
762    def __givapp(c,s,vin):
763        """
764        Applies a sequence of Givens rotations (c,s) recursively to the vector
765        ``vin``
766    
767        :warning: ``vin`` is altered.
768        """
769        vrot=vin
770        if isinstance(c,float):
771            vrot=[c*vrot[0]-s*vrot[1],s*vrot[0]+c*vrot[1]]
772        else:
773            for i in range(len(c)):
774                w1=c[i]*vrot[i]-s[i]*vrot[i+1]
775            w2=s[i]*vrot[i]+c[i]*vrot[i+1]
776                vrot[i]=w1
777                vrot[i+1]=w2
778        return vrot
779    
780    def __FDGMRES(F0, defect, x0, atol, iter_max=100, iter_restart=20):
781       h=numpy.zeros((iter_restart,iter_restart),numpy.float64)
782       c=numpy.zeros(iter_restart,numpy.float64)
783       s=numpy.zeros(iter_restart,numpy.float64)
784       g=numpy.zeros(iter_restart,numpy.float64)
785       v=[]
786    
787       rho=defect.norm(F0)
788       if rho<=0.: return x0*0
789    
790       v.append(-F0/rho)
791       g[0]=rho
792       iter=0
793       while rho > atol and iter<iter_restart-1:
794            if iter  >= iter_max:
795                raise MaxIterReached,"maximum number of %s steps reached."%iter_max
796    
797            p=defect.derivative(F0,x0,v[iter], v_is_normalised=True)
798            v.append(p)
799    
800            v_norm1=defect.norm(v[iter+1])
801    
802            # Modified Gram-Schmidt
803            for j in range(iter+1):
804                h[j,iter]=defect.bilinearform(v[j],v[iter+1])
805                v[iter+1]-=h[j,iter]*v[j]
806    
807            h[iter+1,iter]=defect.norm(v[iter+1])
808            v_norm2=h[iter+1,iter]
809    
810            # Reorthogonalize if needed
811            if v_norm1 + 0.001*v_norm2 == v_norm1:   #Brown/Hindmarsh condition (default)
812                for j in range(iter+1):
813                    hr=defect.bilinearform(v[j],v[iter+1])
814                    h[j,iter]=h[j,iter]+hr
815                    v[iter+1] -= hr*v[j]
816    
817                v_norm2=defect.norm(v[iter+1])
818                h[iter+1,iter]=v_norm2
819            #   watch out for happy breakdown
820            if not v_norm2 == 0:
821                v[iter+1]=v[iter+1]/h[iter+1,iter]
822    
823            #   Form and store the information for the new Givens rotation
824            if iter > 0 :
825                hhat=numpy.zeros(iter+1,numpy.float64)
826                for i in range(iter+1) : hhat[i]=h[i,iter]
827                hhat=__givapp(c[0:iter],s[0:iter],hhat);
828                for i in range(iter+1) : h[i,iter]=hhat[i]
829    
830            mu=math.sqrt(h[iter,iter]*h[iter,iter]+h[iter+1,iter]*h[iter+1,iter])
831    
832            if mu!=0 :
833                c[iter]=h[iter,iter]/mu
834                s[iter]=-h[iter+1,iter]/mu
835                h[iter,iter]=c[iter]*h[iter,iter]-s[iter]*h[iter+1,iter]
836                h[iter+1,iter]=0.0
837                gg=__givapp(c[iter],s[iter],[g[iter],g[iter+1]])
838                g[iter]=gg[0]
839                g[iter+1]=gg[1]
840    
841            # Update the residual norm
842            rho=abs(g[iter+1])
843            iter+=1
844    
845       # At this point either iter > iter_max or rho < tol.
846       # It's time to compute x and leave.
847       if iter > 0 :
848         y=numpy.zeros(iter,numpy.float64)
849         y[iter-1] = g[iter-1] / h[iter-1,iter-1]
850         if iter > 1 :
851            i=iter-2
852            while i>=0 :
853              y[i] = ( g[i] - numpy.dot(h[i,i+1:iter], y[i+1:iter])) / h[i,i]
854              i=i-1
855         xhat=v[iter-1]*y[iter-1]
856         for i in range(iter-1):
857        xhat += v[i]*y[i]
858       else :
859          xhat=v[0] * 0
860    
861       if iter<iter_restart-1:
862          stopped=iter
863       else:
864          stopped=-1
865    
866       return xhat,stopped
867    
868    def GMRES(r, Aprod, x, bilinearform, atol=0, rtol=1.e-8, iter_max=100, iter_restart=20, verbose=False,P_R=None):
869       """
870       Solver for
871    
872       *Ax=b*
873    
874       with a general operator A (more details required!).
875       It uses the generalized minimum residual method (GMRES).
876    
877       The iteration is terminated if
878    
879       *|r| <= atol+rtol*|r0|*
880    
881       where *r0* is the initial residual and *|.|* is the energy norm. In fact
882    
883       *|r| = sqrt( bilinearform(r,r))*
884    
885       :param r: initial residual *r=b-Ax*. ``r`` is altered.
886       :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
887       :param x: an initial guess for the solution
888       :type x: same like ``r``
889       :param Aprod: returns the value Ax
890       :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
891                    argument ``x``. The returned object needs to be of the same
892                    type like argument ``r``.
893       :param bilinearform: inner product ``<x,r>``
894       :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
895                           type like argument ``x`` and ``r``. The returned value is
896                           a ``float``.
897       :param atol: absolute tolerance
898       :type atol: non-negative ``float``
899       :param rtol: relative tolerance
900       :type rtol: non-negative ``float``
901       :param iter_max: maximum number of iteration steps
902       :type iter_max: ``int``
903       :param iter_restart: in order to save memory the orthogonalization process
904                            is terminated after ``iter_restart`` steps and the
905                            iteration is restarted.
906       :type iter_restart: ``int``
907       :return: the solution approximation and the corresponding residual
908       :rtype: ``tuple``
909       :warning: ``r`` and ``x`` are altered.
910       """
911       m=iter_restart
912       restarted=False
913       iter=0
914       if rtol>0:
915          r_dot_r = bilinearform(r, r)
916          if r_dot_r<0: raise NegativeNorm,"negative norm."
917          atol2=atol+rtol*math.sqrt(r_dot_r)
918          if verbose: print "GMRES: norm of right hand side = %e (absolute tolerance = %e)"%(math.sqrt(r_dot_r), atol2)
919       else:
920          atol2=atol
921          if verbose: print "GMRES: absolute tolerance = %e"%atol2
922       if atol2<=0:
923          raise ValueError,"Non-positive tolarance."
924    
925       while True:
926          if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached"%iter_max
927          if restarted:
928             r2 = r-Aprod(x-x2)
929          else:
930             r2=1*r
931          x2=x*1.
932          x,stopped=_GMRESm(r2, Aprod, x, bilinearform, atol2, iter_max=iter_max-iter, iter_restart=m, verbose=verbose,P_R=P_R)
933          iter+=iter_restart
934          if stopped: break
935          if verbose: print "GMRES: restart."
936          restarted=True
937       if verbose: print "GMRES: tolerance has been reached."
938       return x
939    
940    def _GMRESm(r, Aprod, x, bilinearform, atol, iter_max=100, iter_restart=20, verbose=False, P_R=None):
941       iter=0
942    
943       h=numpy.zeros((iter_restart+1,iter_restart),numpy.float64)
944       c=numpy.zeros(iter_restart,numpy.float64)
945       s=numpy.zeros(iter_restart,numpy.float64)
946       g=numpy.zeros(iter_restart+1,numpy.float64)
947       v=[]
948    
949       r_dot_r = bilinearform(r, r)
950       if r_dot_r<0: raise NegativeNorm,"negative norm."
951       rho=math.sqrt(r_dot_r)
952    
953       v.append(r/rho)
954       g[0]=rho
955    
956       if verbose: print "GMRES: initial residual %e (absolute tolerance = %e)"%(rho,atol)
957       while not (rho<=atol or iter==iter_restart):
958    
959        if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
960    
961            if P_R!=None:
962                p=Aprod(P_R(v[iter]))
963            else:
964            p=Aprod(v[iter])
965        v.append(p)
966    
967        v_norm1=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
968    
969    # Modified Gram-Schmidt
970        for j in range(iter+1):
971          h[j,iter]=bilinearform(v[j],v[iter+1])
972          v[iter+1]-=h[j,iter]*v[j]
973    
974        h[iter+1,iter]=math.sqrt(bilinearform(v[iter+1],v[iter+1]))
975        v_norm2=h[iter+1,iter]
976    
977    # Reorthogonalize if needed
978        if v_norm1 + 0.001*v_norm2 == v_norm1:   #Brown/Hindmarsh condition (default)
979         for j in range(iter+1):
980            hr=bilinearform(v[j],v[iter+1])
981                h[j,iter]=h[j,iter]+hr
982                v[iter+1] -= hr*v[j]
983    
984         v_norm2=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
985         h[iter+1,iter]=v_norm2
986    
987    #   watch out for happy breakdown
988            if not v_norm2 == 0:
989             v[iter+1]=v[iter+1]/h[iter+1,iter]
990    
991    #   Form and store the information for the new Givens rotation
992        if iter > 0: h[:iter+1,iter]=__givapp(c[:iter],s[:iter],h[:iter+1,iter])
993        mu=math.sqrt(h[iter,iter]*h[iter,iter]+h[iter+1,iter]*h[iter+1,iter])
994    
995        if mu!=0 :
996            c[iter]=h[iter,iter]/mu
997            s[iter]=-h[iter+1,iter]/mu
998            h[iter,iter]=c[iter]*h[iter,iter]-s[iter]*h[iter+1,iter]
999            h[iter+1,iter]=0.0
1000                    gg=__givapp(c[iter],s[iter],[g[iter],g[iter+1]])
1001                    g[iter]=gg[0]
1002                    g[iter+1]=gg[1]
1003    # Update the residual norm
1004    
1005            rho=abs(g[iter+1])
1006            if verbose: print "GMRES: iteration step %s: residual %e"%(iter,rho)
1007        iter+=1
1008    
1009    # At this point either iter > iter_max or rho < tol.
1010    # It's time to compute x and leave.
1011    
1012       if verbose: print "GMRES: iteration stopped after %s step."%iter
1013       if iter > 0 :
1014         y=numpy.zeros(iter,numpy.float64)
1015         y[iter-1] = g[iter-1] / h[iter-1,iter-1]
1016         if iter > 1 :
1017            i=iter-2
1018            while i>=0 :
1019              y[i] = ( g[i] - numpy.dot(h[i,i+1:iter], y[i+1:iter])) / h[i,i]
1020              i=i-1
1021         xhat=v[iter-1]*y[iter-1]
1022         for i in range(iter-1):
1023        xhat += v[i]*y[i]
1024       else:
1025         xhat=v[0] * 0
1026       if P_R!=None:
1027          x += P_R(xhat)
1028       else:
1029          x += xhat
1030       if iter<iter_restart-1:
1031          stopped=True
1032       else:
1033          stopped=False
1034    
1035       return x,stopped
1036    
1037    def MINRES(r, Aprod, x, Msolve, bilinearform, atol=0, rtol=1.e-8, iter_max=100):
1038        """
1039        Solver for
1040    
1041        *Ax=b*
1042    
1043        with a symmetric and positive definite operator A (more details required!).
1044        It uses the minimum residual method (MINRES) with preconditioner M
1045        providing an approximation of A.
1046    
1047        The iteration is terminated if
1048    
1049        *|r| <= atol+rtol*|r0|*
1050    
1051        where *r0* is the initial residual and *|.|* is the energy norm. In fact
1052    
1053        *|r| = sqrt( bilinearform(Msolve(r),r))*
1054    
1055        For details on the preconditioned conjugate gradient method see the book:
1056    
1057        I{Templates for the Solution of Linear Systems by R. Barrett, M. Berry,
1058        T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo,
1059        C. Romine, and H. van der Vorst}.
1060    
1061        :param r: initial residual *r=b-Ax*. ``r`` is altered.
1062        :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1063        :param x: an initial guess for the solution
1064        :type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1065        :param Aprod: returns the value Ax
1066        :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
1067                     argument ``x``. The returned object needs to be of the same
1068                     type like argument ``r``.
1069        :param Msolve: solves Mx=r
1070        :type Msolve: function ``Msolve(r)`` where ``r`` is of the same type like
1071                      argument ``r``. The returned object needs to be of the same
1072                      type like argument ``x``.
1073        :param bilinearform: inner product ``<x,r>``
1074        :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
1075                            type like argument ``x`` and ``r`` is. The returned value
1076                            is a ``float``.
1077        :param atol: absolute tolerance
1078        :type atol: non-negative ``float``
1079        :param rtol: relative tolerance
1080        :type rtol: non-negative ``float``
1081        :param iter_max: maximum number of iteration steps
1082        :type iter_max: ``int``
1083        :return: the solution approximation and the corresponding residual
1084        :rtype: ``tuple``
1085        :warning: ``r`` and ``x`` are altered.
1086        """
1087        #------------------------------------------------------------------
1088        # Set up y and v for the first Lanczos vector v1.
1089        # y  =  beta1 P' v1,  where  P = C**(-1).
1090        # v is really P' v1.
1091        #------------------------------------------------------------------
1092        r1    = r
1093        y = Msolve(r)
1094        beta1 = bilinearform(y,r)
1095    
1096        if beta1< 0: raise NegativeNorm,"negative norm."
1097    
1098        #  If r = 0 exactly, stop with x
1099        if beta1==0: return x
1100    
1101        if beta1> 0: beta1  = math.sqrt(beta1)
1102    
1103        #------------------------------------------------------------------
1104        # Initialize quantities.
1105        # ------------------------------------------------------------------
1106        iter   = 0
1107        Anorm = 0
1108        ynorm = 0
1109        oldb   = 0
1110        beta   = beta1
1111        dbar   = 0
1112        epsln  = 0
1113        phibar = beta1
1114        rhs1   = beta1
1115        rhs2   = 0
1116        rnorm  = phibar
1117        tnorm2 = 0
1118        ynorm2 = 0
1119        cs     = -1
1120        sn     = 0
1121        w      = r*0.
1122        w2     = r*0.
1123        r2     = r1
1124        eps    = 0.0001
1125    
1126        #---------------------------------------------------------------------
1127        # Main iteration loop.
1128        # --------------------------------------------------------------------
1129        while not rnorm<=atol+rtol*Anorm*ynorm:    #  checks ||r|| < (||A|| ||x||) * TOL
1130    
1131        if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
1132            iter    = iter  +  1
1133    
1134            #-----------------------------------------------------------------
1135            # Obtain quantities for the next Lanczos vector vk+1, k = 1, 2,...
1136            # The general iteration is similar to the case k = 1 with v0 = 0:
1137            #
1138            #   p1      = Operator * v1  -  beta1 * v0,
1139            #   alpha1  = v1'p1,
1140            #   q2      = p2  -  alpha1 * v1,
1141            #   beta2^2 = q2'q2,
1142            #   v2      = (1/beta2) q2.
1143            #
1144            # Again, y = betak P vk,  where  P = C**(-1).
1145            #-----------------------------------------------------------------
1146            s = 1/beta                 # Normalize previous vector (in y).
1147            v = s*y                    # v = vk if P = I
1148    
1149            y      = Aprod(v)
1150    
1151            if iter >= 2:
1152              y = y - (beta/oldb)*r1
1153    
1154            alfa   = bilinearform(v,y)              # alphak
1155            y      += (- alfa/beta)*r2
1156            r1     = r2
1157            r2     = y
1158            y = Msolve(r2)
1159            oldb   = beta                           # oldb = betak
1160            beta   = bilinearform(y,r2)             # beta = betak+1^2
1161            if beta < 0: raise NegativeNorm,"negative norm."
1162    
1163            beta   = math.sqrt( beta )
1164            tnorm2 = tnorm2 + alfa*alfa + oldb*oldb + beta*beta
1165    
1166            if iter==1:                 # Initialize a few things.
1167              gmax   = abs( alfa )      # alpha1
1168              gmin   = gmax             # alpha1
1169    
1170            # Apply previous rotation Qk-1 to get
1171            #   [deltak epslnk+1] = [cs  sn][dbark    0   ]
1172            #   [gbar k dbar k+1]   [sn -cs][alfak betak+1].
1173    
1174            oldeps = epsln
1175            delta  = cs * dbar  +  sn * alfa  # delta1 = 0         deltak
1176            gbar   = sn * dbar  -  cs * alfa  # gbar 1 = alfa1     gbar k
1177            epsln  =               sn * beta  # epsln2 = 0         epslnk+1
1178            dbar   =            -  cs * beta  # dbar 2 = beta2     dbar k+1
1179    
1180            # Compute the next plane rotation Qk
1181    
1182            gamma  = math.sqrt(gbar*gbar+beta*beta)  # gammak
1183            gamma  = max(gamma,eps)
1184            cs     = gbar / gamma             # ck
1185            sn     = beta / gamma             # sk
1186            phi    = cs * phibar              # phik
1187            phibar = sn * phibar              # phibark+1
1188    
1189            # Update  x.
1190    
1191            denom = 1/gamma
1192            w1    = w2
1193            w2    = w
1194            w     = (v - oldeps*w1 - delta*w2) * denom
1195            x     +=  phi*w
1196    
1197            # Go round again.
1198    
1199            gmax   = max(gmax,gamma)
1200            gmin   = min(gmin,gamma)
1201            z      = rhs1 / gamma
1202            ynorm2 = z*z  +  ynorm2
1203            rhs1   = rhs2 -  delta*z
1204            rhs2   =      -  epsln*z
1205    
1206            # Estimate various norms and test for convergence.
1207    
1208            Anorm  = math.sqrt( tnorm2 )
1209            ynorm  = math.sqrt( ynorm2 )
1210    
1211            rnorm  = phibar
1212    
1213        return x
1214    
1215    def TFQMR(r, Aprod, x, bilinearform, atol=0, rtol=1.e-8, iter_max=100):
1216      """
1217      Solver for
1218    
1219      *Ax=b*
1220    
1221      with a general operator A (more details required!).
1222      It uses the Transpose-Free Quasi-Minimal Residual method (TFQMR).
1223    
1224      The iteration is terminated if
1225    
1226      *|r| <= atol+rtol*|r0|*
1227    
1228      where *r0* is the initial residual and *|.|* is the energy norm. In fact
1229    
1230      *|r| = sqrt( bilinearform(r,r))*
1231    
1232      :param r: initial residual *r=b-Ax*. ``r`` is altered.
1233      :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1234      :param x: an initial guess for the solution
1235      :type x: same like ``r``
1236      :param Aprod: returns the value Ax
1237      :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
1238                   argument ``x``. The returned object needs to be of the same type
1239                   like argument ``r``.
1240      :param bilinearform: inner product ``<x,r>``
1241      :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
1242                          type like argument ``x`` and ``r``. The returned value is
1243                          a ``float``.
1244      :param atol: absolute tolerance
1245      :type atol: non-negative ``float``
1246      :param rtol: relative tolerance
1247      :type rtol: non-negative ``float``
1248      :param iter_max: maximum number of iteration steps
1249      :type iter_max: ``int``
1250      :rtype: ``tuple``
1251      :warning: ``r`` and ``x`` are altered.
1252      """
1253      u1=0
1254      u2=0
1255      y1=0
1256      y2=0
1257    
1258      w = r
1259      y1 = r
1260      iter = 0
1261      d = 0
1262      v = Aprod(y1)
1263      u1 = v
1264    
1265      theta = 0.0;
1266      eta = 0.0;
1267      rho=bilinearform(r,r)
1268      if rho < 0: raise NegativeNorm,"negative norm."
1269      tau = math.sqrt(rho)
1270      norm_r0=tau
1271      while tau>atol+rtol*norm_r0:
1272        if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
1273    
1274        sigma = bilinearform(r,v)
1275        if sigma == 0.0: raise IterationBreakDown,'TFQMR breakdown, sigma=0'
1276    
1277        alpha = rho / sigma
1278    
1279        for j in range(2):
1280    #
1281    #   Compute y2 and u2 only if you have to
1282    #
1283          if ( j == 1 ):
1284            y2 = y1 - alpha * v
1285            u2 = Aprod(y2)
1286    
1287          m = 2 * (iter+1) - 2 + (j+1)
1288          if j==0:
1289             w = w - alpha * u1
1290             d = y1 + ( theta * theta * eta / alpha ) * d
1291          if j==1:
1292             w = w - alpha * u2
1293             d = y2 + ( theta * theta * eta / alpha ) * d
1294    
1295          theta = math.sqrt(bilinearform(w,w))/ tau
1296          c = 1.0 / math.sqrt ( 1.0 + theta * theta )
1297          tau = tau * theta * c
1298          eta = c * c * alpha
1299          x = x + eta * d
1300    #
1301    #  Try to terminate the iteration at each pass through the loop
1302    #
1303        if rho == 0.0: raise IterationBreakDown,'TFQMR breakdown, rho=0'
1304    
1305        rhon = bilinearform(r,w)
1306        beta = rhon / rho;
1307        rho = rhon;
1308        y1 = w + beta * y2;
1309        u1 = Aprod(y1)
1310        v = u1 + beta * ( u2 + beta * v )
1311    
1312        iter += 1
1313    
1314      return x
1315    
1316    
1317    #############################################
1318    
1319    class ArithmeticTuple(object):
1320       """
1321       Tuple supporting inplace update x+=y and scaling x=a*y where ``x,y`` is an
1322       ArithmeticTuple and ``a`` is a float.
1323    
1324       Example of usage::
1325    
1326           from esys.escript import Data
1327           from numpy import array
1328           a=Data(...)
1329           b=array([1.,4.])
1330           x=ArithmeticTuple(a,b)
1331           y=5.*x
1332    
1333       """
1334       def __init__(self,*args):
1335           """
1336           Initializes object with elements ``args``.
1337    
1338           :param args: tuple of objects that support inplace add (x+=y) and
1339                        scaling (x=a*y)
1340           """
1341           self.__items=list(args)
1342    
1343       def __len__(self):
1344           """
1345           Returns the number of items.
1346    
1347           :return: number of items
1348           :rtype: ``int``
1349           """
1350           return len(self.__items)
1351    
1352       def __getitem__(self,index):
1353           """
1354           Returns item at specified position.
1355    
1356           :param index: index of item to be returned
1357           :type index: ``int``
1358           :return: item with index ``index``
1359           """
1360           return self.__items.__getitem__(index)
1361    
1362       def __mul__(self,other):
1363           """
1364           Scales by ``other`` from the right.
1365    
1366           :param other: scaling factor
1367           :type other: ``float``
1368           :return: itemwise self*other
1369           :rtype: `ArithmeticTuple`
1370           """
1371           out=[]
1372           try:
1373               l=len(other)
1374               if l!=len(self):
1375                   raise ValueError,"length of arguments don't match."
1376               for i in range(l): out.append(self[i]*other[i])
1377           except TypeError:
1378               for i in range(len(self)): out.append(self[i]*other)
1379           return ArithmeticTuple(*tuple(out))
1380    
1381       def __rmul__(self,other):
1382           """
1383           Scales by ``other`` from the left.
1384    
1385           :param other: scaling factor
1386           :type other: ``float``
1387           :return: itemwise other*self
1388           :rtype: `ArithmeticTuple`
1389           """
1390           out=[]
1391           try:
1392               l=len(other)
1393               if l!=len(self):
1394                   raise ValueError,"length of arguments don't match."
1395               for i in range(l): out.append(other[i]*self[i])
1396           except TypeError:
1397               for i in range(len(self)): out.append(other*self[i])
1398           return ArithmeticTuple(*tuple(out))
1399    
1400       def __div__(self,other):
1401           """
1402           Scales by (1/``other``) from the right.
1403    
1404           :param other: scaling factor
1405           :type other: ``float``
1406           :return: itemwise self/other
1407           :rtype: `ArithmeticTuple`
1408           """
1409           return self*(1/other)
1410    
1411       def __rdiv__(self,other):
1412           """
1413           Scales by (1/``other``) from the left.
1414    
1415           :param other: scaling factor
1416           :type other: ``float``
1417           :return: itemwise other/self
1418           :rtype: `ArithmeticTuple`
1419           """
1420           out=[]
1421           try:
1422               l=len(other)
1423               if l!=len(self):
1424                   raise ValueError,"length of arguments don't match."
1425               for i in range(l): out.append(other[i]/self[i])
1426           except TypeError:
1427               for i in range(len(self)): out.append(other/self[i])
1428           return ArithmeticTuple(*tuple(out))
1429    
1430       def __iadd__(self,other):
1431           """
1432           Inplace addition of ``other`` to self.
1433    
1434           :param other: increment
1435           :type other: ``ArithmeticTuple``
1436           """
1437           if len(self) != len(other):
1438               raise ValueError,"tuple lengths must match."
1439           for i in range(len(self)):
1440               self.__items[i]+=other[i]
1441           return self
1442    
1443       def __add__(self,other):
1444           """
1445           Adds ``other`` to self.
1446    
1447           :param other: increment
1448           :type other: ``ArithmeticTuple``
1449           """
1450           out=[]
1451           try:
1452               l=len(other)
1453               if l!=len(self):
1454                   raise ValueError,"length of arguments don't match."
1455               for i in range(l): out.append(self[i]+other[i])
1456           except TypeError:
1457               for i in range(len(self)): out.append(self[i]+other)
1458           return ArithmeticTuple(*tuple(out))
1459    
1460       def __sub__(self,other):
1461           """
1462           Subtracts ``other`` from self.
1463    
1464           :param other: decrement
1465           :type other: ``ArithmeticTuple``
1466           """
1467           out=[]
1468           try:
1469               l=len(other)
1470               if l!=len(self):
1471                   raise ValueError,"length of arguments don't match."
1472               for i in range(l): out.append(self[i]-other[i])
1473           except TypeError:
1474               for i in range(len(self)): out.append(self[i]-other)
1475           return ArithmeticTuple(*tuple(out))
1476    
1477       def __isub__(self,other):
1478           """
1479           Inplace subtraction of ``other`` from self.
1480    
1481           :param other: decrement
1482           :type other: ``ArithmeticTuple``
1483           """
1484           if len(self) != len(other):
1485               raise ValueError,"tuple length must match."
1486           for i in range(len(self)):
1487               self.__items[i]-=other[i]
1488           return self
1489    
1490       def __neg__(self):
1491           """
1492           Negates values.
1493           """
1494           out=[]
1495           for i in range(len(self)):
1496               out.append(-self[i])
1497           return ArithmeticTuple(*tuple(out))
1498    
1499    
1500    class HomogeneousSaddlePointProblem(object):
1501          """
1502          This class provides a framework for solving linear homogeneous saddle
1503          point problems of the form::
1504    
1505              *Av+B^*p=f*
1506              *Bv     =0*
1507    
1508          for the unknowns *v* and *p* and given operators *A* and *B* and
1509          given right hand side *f*. *B^** is the adjoint operator of *B*.
1510          *A* may depend weakly on *v* and *p*.
1511          """
1512          def __init__(self, **kwargs):
1513        """
1514        initializes the saddle point problem
1515        """
1516            self.resetControlParameters()
1517            self.setTolerance()
1518            self.setAbsoluteTolerance()
1519          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):
1520             """
1521             sets a control parameter
1522    
1523             :param K_p: initial value for constant to adjust pressure tolerance
1524             :type K_p: ``float``
1525             :param K_v: initial value for constant to adjust velocity tolerance
1526             :type K_v: ``float``
1527             :param rtol_max: maximuim relative tolerance used to calculate presssure and velocity increment.
1528             :type rtol_max: ``float``
1529             :param chi_max: maximum tolerable converegence rate.
1530             :type chi_max: ``float``
1531             :param reduction_factor: reduction factor for adjustment factors.
1532             :type reduction_factor: ``float``
1533             """
1534             self.setControlParameter(K_p, K_v, rtol_max, rtol_min, chi_max, reduction_factor, theta)
1535    
1536          def setControlParameter(self,K_p=None, K_v=None, rtol_max=None, rtol_min=None, chi_max=None, reduction_factor=None, theta=None):
1537             """
1538             sets a control parameter
1539    
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             :type reduction_factor: ``float``
1550             """
1551             if not K_p == None:
1552                if K_p<1:
1553                   raise ValueError,"K_p need to be greater or equal to 1."
1554             else:
1555                K_p=self.__K_p
1556    
1557             if not K_v == None:
1558                if K_v<1:
1559                   raise ValueError,"K_v need to be greater or equal to 1."
1560             else:
1561                K_v=self.__K_v
1562    
1563             if not rtol_max == None:
1564                if rtol_max<=0 or rtol_max>=1:
1565                   raise ValueError,"rtol_max needs to be positive and less than 1."
1566             else:
1567                rtol_max=self.__rtol_max
1568    
1569             if not rtol_min == None:
1570                if rtol_min<=0 or rtol_min>=1:
1571                   raise ValueError,"rtol_min needs to be positive and less than 1."
1572             else:
1573                rtol_min=self.__rtol_min
1574    
1575             if not chi_max == None:
1576                if chi_max<=0 or chi_max>=1:
1577                   raise ValueError,"chi_max needs to be positive and less than 1."
1578             else:
1579                chi_max = self.__chi_max
1580    
1581             if not reduction_factor == None:
1582                if reduction_factor<=0 or reduction_factor>1:
1583                   raise ValueError,"reduction_factor need to be between zero and one."
1584             else:
1585                reduction_factor=self.__reduction_factor
1586    
1587             if not theta == None:
1588                if theta<=0 or theta>1:
1589                   raise ValueError,"theta need to be between zero and one."
1590             else:
1591                theta=self.__theta
1592    
1593             if rtol_min>=rtol_max:
1594                 raise ValueError,"rtol_max = %e needs to be greater than rtol_min = %e"%(rtol_max,rtol_min)
1595             self.__chi_max = chi_max
1596             self.__rtol_max = rtol_max
1597             self.__K_p = K_p
1598             self.__K_v = K_v
1599             self.__reduction_factor = reduction_factor
1600             self.__theta = theta
1601             self.__rtol_min=rtol_min
1602    
1603          #=============================================================
1604          def inner_pBv(self,p,Bv):
1605             """
1606             Returns inner product of element p and Bv (overwrite).
1607    
1608             :param p: a pressure increment
1609             :param Bv: a residual
1610             :return: inner product of element p and Bv
1611             :rtype: ``float``
1612             :note: used if PCG is applied.
1613             """
1614             raise NotImplementedError,"no inner product for p and Bv implemented."
1615    
1616          def inner_p(self,p0,p1):
1617             """
1618             Returns inner product of p0 and p1 (overwrite).
1619    
1620             :param p0: a pressure
1621             :param p1: a pressure
1622             :return: inner product of p0 and p1
1623             :rtype: ``float``
1624             """
1625             raise NotImplementedError,"no inner product for p implemented."
1626      
1627          def norm_v(self,v):
1628             """
1629             Returns the norm of v (overwrite).
1630    
1631             :param v: a velovity
1632             :return: norm of v
1633             :rtype: non-negative ``float``
1634             """
1635             raise NotImplementedError,"no norm of v implemented."
1636          def getDV(self, p, v, tol):
1637             """
1638             return a correction to the value for a given v and a given p with accuracy `tol` (overwrite)
1639    
1640             :param p: pressure
1641             :param v: pressure
1642             :return: dv given as *dv= A^{-1} (f-A v-B^*p)*
1643             :note: Only *A* may depend on *v* and *p*
1644             """
1645             raise NotImplementedError,"no dv calculation implemented."
1646    
1647            
1648          def Bv(self,v, tol):
1649            """
1650            Returns Bv with accuracy `tol` (overwrite)
1651    
1652            :rtype: equal to the type of p
1653            :note: boundary conditions on p should be zero!
1654            """
1655            raise NotImplementedError, "no operator B implemented."
1656    
1657          def norm_Bv(self,Bv):
1658            """
1659            Returns the norm of Bv (overwrite).
1660    
1661            :rtype: equal to the type of p
1662            :note: boundary conditions on p should be zero!
1663            """
1664            raise NotImplementedError, "no norm of Bv implemented."
1665    
1666          def solve_AinvBt(self,dp, tol):
1667             """
1668             Solves *A dv=B^*dp* with accuracy `tol`
1669    
1670             :param dp: a pressure increment
1671             :return: the solution of *A dv=B^*dp*
1672             :note: boundary conditions on dv should be zero! *A* is the operator used in ``getDV`` and must not be altered.
1673             """
1674             raise NotImplementedError,"no operator A implemented."
1675    
1676          def solve_prec(self,Bv, tol):
1677             """
1678             Provides a preconditioner for *(BA^{-1}B^ * )* applied to Bv with accuracy `tol`
1679    
1680             :rtype: equal to the type of p
1681             :note: boundary conditions on p should be zero!
1682             """
1683             raise NotImplementedError,"no preconditioner for Schur complement implemented."
1684          #=============================================================
1685          def __Aprod_PCG(self,dp):
1686              dv=self.solve_AinvBt(dp, self.__subtol)
1687              return ArithmeticTuple(dv,self.Bv(dv, self.__subtol))
1688    
1689          def __inner_PCG(self,p,r):
1690             return self.inner_pBv(p,r[1])
1691    
1692          def __Msolve_PCG(self,r):
1693              return self.solve_prec(r[1], self.__subtol)
1694          #=============================================================
1695          def __Aprod_GMRES(self,p):
1696              return self.solve_prec(self.Bv(self.solve_AinvBt(p, self.__subtol), self.__subtol), self.__subtol)
1697          def __inner_GMRES(self,p0,p1):
1698             return self.inner_p(p0,p1)
1699    
1700          #=============================================================
1701          def norm_p(self,p):
1702              """
1703              calculates the norm of ``p``
1704              
1705              :param p: a pressure
1706              :return: the norm of ``p`` using the inner product for pressure
1707              :rtype: ``float``
1708              """
1709              f=self.inner_p(p,p)
1710              if f<0: raise ValueError,"negative pressure norm."
1711              return math.sqrt(f)
1712          
1713          def solve(self,v,p,max_iter=20, verbose=False, usePCG=True, iter_restart=20, max_correction_steps=10):
1714             """
1715             Solves the saddle point problem using initial guesses v and p.
1716    
1717             :param v: initial guess for velocity
1718             :param p: initial guess for pressure
1719             :type v: `Data`
1720             :type p: `Data`
1721             :param usePCG: indicates the usage of the PCG rather than GMRES scheme.
1722             :param max_iter: maximum number of iteration steps per correction
1723                              attempt
1724             :param verbose: if True, shows information on the progress of the
1725                             saddlepoint problem solver.
1726             :param iter_restart: restart the iteration after ``iter_restart`` steps
1727                                  (only used if useUzaw=False)
1728             :type usePCG: ``bool``
1729             :type max_iter: ``int``
1730             :type verbose: ``bool``
1731             :type iter_restart: ``int``
1732             :rtype: ``tuple`` of `Data` objects
1733             :note: typically this method is overwritten by a subclass. It provides a wrapper for the ``_solve`` method.
1734             """
1735             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)
1736    
1737          def _solve(self,v,p,max_iter=20, verbose=False, usePCG=True, iter_restart=20, max_correction_steps=10):
1738             """
1739             see `_solve` method.
1740             """
1741             self.verbose=verbose
1742             rtol=self.getTolerance()
1743             atol=self.getAbsoluteTolerance()
1744    
1745             K_p=self.__K_p
1746             K_v=self.__K_v
1747             correction_step=0
1748             converged=False
1749             chi=None
1750             eps=None
1751    
1752             if self.verbose: print "HomogeneousSaddlePointProblem: start iteration: rtol= %e, atol=%e"%(rtol, atol)
1753             while not converged:
1754    
1755                 # get tolerance for velecity increment:
1756                 if chi == None:
1757                    rtol_v=self.__rtol_max
1758                 else:
1759                    rtol_v=min(chi/K_v,self.__rtol_max)
1760                 rtol_v=max(rtol_v, self.__rtol_min)
1761                 if self.verbose: print "HomogeneousSaddlePointProblem: step %s: rtol_v= %e"%(correction_step,rtol_v)
1762                 # get velocity increment:
1763                 dv1=self.getDV(p,v,rtol_v)
1764                 v1=v+dv1
1765                 Bv1=self.Bv(v1, rtol_v)
1766                 norm_Bv1=self.norm_Bv(Bv1)
1767                 norm_dv1=self.norm_v(dv1)
1768                 if self.verbose: print "HomogeneousSaddlePointProblem: step %s: norm_Bv1 = %e, norm_dv1 = %e"%(correction_step, norm_Bv1, norm_dv1)
1769                 if norm_dv1*self.__theta < norm_Bv1:
1770                    # get tolerance for pressure increment:
1771                    large_Bv1=True
1772                    if chi == None or eps == None:
1773                       rtol_p=self.__rtol_max
1774                    else:
1775                       rtol_p=min(chi**2*eps/K_p/norm_Bv1, self.__rtol_max)
1776                    self.__subtol=max(rtol_p**2, self.__rtol_min)
1777                    if self.verbose: print "HomogeneousSaddlePointProblem: step %s: rtol_p= %e"%(correction_step,rtol_p)
1778                    # now we solve for the pressure increment dp from B*A^{-1}B^* dp = Bv1
1779                    if usePCG:
1780                        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)
1781                        v2=r[0]
1782                        Bv2=r[1]
1783                    else:
1784                        # don't use!!!!
1785                        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)
1786                        dv2=self.solve_AinvBt(dp, self.__subtol)
1787                        v2=v1-dv2
1788                        Bv2=self.Bv(v2, self.__subtol)
1789                    p2=p+dp
1790                 else:
1791                    large_Bv1=False
1792                    v2=v1
1793                    p2=p
1794                 # update business:
1795                 norm_dv2=self.norm_v(v2-v)
1796                 norm_v2=self.norm_v(v2)
1797                 if self.verbose: print "HomogeneousSaddlePointProblem: step %s: v2 = %e, norm_dv2 = %e"%(correction_step, norm_v2, self.norm_v(v2-v))
1798                 eps, eps_old = max(norm_Bv1, norm_dv2), eps
1799                 if eps_old == None:
1800                      chi, chi_old = None, chi
1801                 else:
1802                      chi, chi_old = min(eps/ eps_old, self.__chi_max), chi
1803                 if eps != None:
1804                     if chi !=None:
1805                        if self.verbose: print "HomogeneousSaddlePointProblem: step %s: convergence rate = %e, correction = %e"%(correction_step,chi, eps)
1806                     else:
1807                        if self.verbose: print "HomogeneousSaddlePointProblem: step %s: correction = %e"%(correction_step, eps)
1808                 if eps <= rtol*norm_v2+atol :
1809                     converged = True
1810                 else:
1811                     if correction_step>=max_correction_steps:
1812                          raise CorrectionFailed,"Given up after %d correction steps."%correction_step
1813                     if chi_old!=None:
1814                        K_p=max(1,self.__reduction_factor*K_p,(chi-chi_old)/chi_old**2*K_p)
1815                        K_v=max(1,self.__reduction_factor*K_v,(chi-chi_old)/chi_old**2*K_p)
1816                        if self.verbose: print "HomogeneousSaddlePointProblem: step %s: new adjustment factor K = %e"%(correction_step,K_p)
1817                 correction_step+=1
1818                 v,p =v2, p2
1819             if self.verbose: print "HomogeneousSaddlePointProblem: tolerance reached after %s steps."%correction_step
1820         return v,p
1821          #========================================================================
1822          def setTolerance(self,tolerance=1.e-4):
1823             """
1824             Sets the relative tolerance for (v,p).
1825    
1826             :param tolerance: tolerance to be used
1827             :type tolerance: non-negative ``float``
1828             """
1829             if tolerance<0:
1830                 raise ValueError,"tolerance must be positive."
1831             self.__rtol=tolerance
1832    
1833          def getTolerance(self):
1834             """
1835             Returns the relative tolerance.
1836    
1837             :return: relative tolerance
1838             :rtype: ``float``
1839             """
1840             return self.__rtol
1841    
1842          def setAbsoluteTolerance(self,tolerance=0.):
1843             """
1844             Sets the absolute tolerance.
1845    
1846             :param tolerance: tolerance to be used
1847             :type tolerance: non-negative ``float``
1848             """
1849             if tolerance<0:
1850                 raise ValueError,"tolerance must be non-negative."
1851             self.__atol=tolerance
1852    
1853          def getAbsoluteTolerance(self):
1854             """
1855             Returns the absolute tolerance.
1856    
1857             :return: absolute tolerance
1858             :rtype: ``float``
1859             """
1860             return self.__atol
1861    
1862    
1863    def MaskFromBoundaryTag(domain,*tags):
1864       """
1865       Creates a mask on the Solution(domain) function space where the value is
1866       one for samples that touch the boundary tagged by tags.
1867    
1868       Usage: m=MaskFromBoundaryTag(domain, "left", "right")
1869    
1870       :param domain: domain to be used
1871       :type domain: `escript.Domain`
1872       :param tags: boundary tags
1873       :type tags: ``str``
1874       :return: a mask which marks samples that are touching the boundary tagged
1875                by any of the given tags
1876       :rtype: `escript.Data` of rank 0
1877       """
1878       pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)
1879       d=escript.Scalar(0.,escript.FunctionOnBoundary(domain))
1880       for t in tags: d.setTaggedValue(t,1.)
1881       pde.setValue(y=d)
1882       return util.whereNonZero(pde.getRightHandSide())
1883    
1884    def MaskFromTag(domain,*tags):
1885       """
1886       Creates a mask on the Solution(domain) function space where the value is
1887       one for samples that touch regions tagged by tags.
1888    
1889       Usage: m=MaskFromTag(domain, "ham")
1890    
1891       :param domain: domain to be used
1892       :type domain: `escript.Domain`
1893       :param tags: boundary tags
1894       :type tags: ``str``
1895       :return: a mask which marks samples that are touching the boundary tagged
1896                by any of the given tags
1897       :rtype: `escript.Data` of rank 0
1898       """
1899       pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)
1900       d=escript.Scalar(0.,escript.Function(domain))
1901       for t in tags: d.setTaggedValue(t,1.)
1902       pde.setValue(Y=d)
1903       return util.whereNonZero(pde.getRightHandSide())
1904    
1905    

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