/[escript]/trunk/escript/py_src/pdetools.py
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revision 1105 by gross, Thu Apr 19 01:10:49 2007 UTC revision 2793 by gross, Tue Dec 1 06:10:10 2009 UTC
# Line 1  Line 1 
1  # $Id$  
2    ########################################################
3    #
4    # Copyright (c) 2003-2009 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) 2003-2009 by University of Queensland
15    Earth Systems Science Computational Center (ESSCC)
16    http://www.uq.edu.au/esscc
17    Primary Business: Queensland, Australia"""
18    __license__="""Licensed under the Open Software License version 3.0
19    http://www.opensource.org/licenses/osl-3.0.php"""
20    __url__="https://launchpad.net/escript-finley"
21    
22  """  """
23  Provides some tools related to PDEs.  Provides some tools related to PDEs.
24    
25  Currently includes:  Currently includes:
26      - Projector - to project a discontinuous      - Projector - to project a discontinuous function onto a continuous function
27      - Locator - to trace values in data objects at a certain location      - Locator - to trace values in data objects at a certain location
28      - TimeIntegrationManager - to handel extraplotion in time      - TimeIntegrationManager - to handle extrapolation in time
29      - SaddlePointProblem - solver for Saddle point problems using the inexact uszawa scheme      - SaddlePointProblem - solver for Saddle point problems using the inexact uszawa scheme
30    
31  @var __author__: name of author  :var __author__: name of author
32  @var __copyright__: copyrights  :var __copyright__: copyrights
33  @var __license__: licence agreement  :var __license__: licence agreement
34  @var __url__: url entry point on documentation  :var __url__: url entry point on documentation
35  @var __version__: version  :var __version__: version
36  @var __date__: date of the version  :var __date__: date of the version
37  """  """
38    
39  __author__="Lutz Gross, l.gross@uq.edu.au"  __author__="Lutz Gross, l.gross@uq.edu.au"
 __copyright__="""  Copyright (c) 2006 by ACcESS MNRF  
                     http://www.access.edu.au  
                 Primary Business: Queensland, Australia"""  
 __license__="""Licensed under the Open Software License version 3.0  
              http://www.opensource.org/licenses/osl-3.0.php"""  
 __url__="http://www.iservo.edu.au/esys"  
 __version__="$Revision$"  
 __date__="$Date$"  
40    
41    
42  import escript  import escript
43  import linearPDEs  import linearPDEs
44  import numarray  import numpy
45  import util  import util
46    import math
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 47  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 68  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 77  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 94  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 109  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    def __del__(self):      """
150      return      Returns the solver options of the PDE solver.
151        
152        :rtype: `linearPDEs.SolverOptions`
153        """
154        return self.__pde.getSolverOptions()
155    
156    def __call__(self, input_data):    def __call__(self, input_data):
157      """      """
158      Projects input_data onto a continuous function      Projects ``input_data`` onto a continuous function.
159    
160      @param input_data: The input_data to be projected.      :param input_data: the data to be projected
161      """      """
162      out=escript.Data(0.,input_data.getShape(),self.__pde.getFunctionSpaceForSolution())      out=escript.Data(0.,input_data.getShape(),self.__pde.getFunctionSpaceForSolution())
163        self.__pde.setValue(Y = escript.Data(), Y_reduced = escript.Data())
164      if input_data.getRank()==0:      if input_data.getRank()==0:
165          self.__pde.setValue(Y = input_data)          self.__pde.setValue(Y = input_data)
166          out=self.__pde.getSolution()          out=self.__pde.getSolution()
# Line 171  class Projector: Line 190  class Projector:
190    
191  class NoPDE:  class NoPDE:
192       """       """
193       solves the following problem for u:       Solves the following problem for u:
194    
195       M{kronecker[i,j]*D[j]*u[j]=Y[i]}       *kronecker[i,j]*D[j]*u[j]=Y[i]*
196    
197       with constraint       with constraint
198    
199       M{u[j]=r[j]}  where M{q[j]>0}       *u[j]=r[j]*  where *q[j]>0*
200    
201       where D, Y, r and q are given functions of rank 1.       where *D*, *Y*, *r* and *q* are given functions of rank 1.
202    
203       In the case of scalars this takes the form       In the case of scalars this takes the form
204    
205       M{D*u=Y}       *D*u=Y*
206    
207       with constraint       with constraint
208    
209       M{u=r}  where M{q>0}       *u=r* where *q>0*
210    
211       where D, Y, r and q are given scalar functions.       where *D*, *Y*, *r* and *q* are given scalar functions.
212    
213       The constraint is overwriting any other condition.       The constraint overwrites any other condition.
214    
215       @note: This class is similar to the L{linearPDEs.LinearPDE} class with A=B=C=X=0 but has the intention       :note: This class is similar to the `linearPDEs.LinearPDE` class with
216              that all input parameter are given in L{Solution} or L{ReducedSolution}. The whole              A=B=C=X=0 but has the intention that all input parameters are given
217              thing is a bit strange and I blame Robert.Woodcock@csiro.au for this.              in `Solution` or `ReducedSolution`.
218       """       """
219         # The whole thing is a bit strange and I blame Rob Woodcock (CSIRO) for
220         # this.
221       def __init__(self,domain,D=None,Y=None,q=None,r=None):       def __init__(self,domain,D=None,Y=None,q=None,r=None):
222           """           """
223           initialize the problem           Initializes the problem.
224    
225           @param domain: domain of the PDE.           :param domain: domain of the PDE
226           @type domain: L{Domain}           :type domain: `Domain`
227           @param D: coefficient of the solution.           :param D: coefficient of the solution
228           @type D: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type D: ``float``, ``int``, ``numpy.ndarray``, `Data`
229           @param Y: right hand side           :param Y: right hand side
230           @type Y: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type Y: ``float``, ``int``, ``numpy.ndarray``, `Data`
231           @param q: location of constraints           :param q: location of constraints
232           @type q: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type q: ``float``, ``int``, ``numpy.ndarray``, `Data`
233           @param r: value of solution at locations of constraints           :param r: value of solution at locations of constraints
234           @type r: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type r: ``float``, ``int``, ``numpy.ndarray``, `Data`
235           """           """
236           self.__domain=domain           self.__domain=domain
237           self.__D=D           self.__D=D
# Line 219  class NoPDE: Line 240  class NoPDE:
240           self.__r=r           self.__r=r
241           self.__u=None           self.__u=None
242           self.__function_space=escript.Solution(self.__domain)           self.__function_space=escript.Solution(self.__domain)
243    
244       def setReducedOn(self):       def setReducedOn(self):
245           """           """
246           sets the L{FunctionSpace} of the solution to L{ReducedSolution}           Sets the `FunctionSpace` of the solution to `ReducedSolution`.
247           """           """
248           self.__function_space=escript.ReducedSolution(self.__domain)           self.__function_space=escript.ReducedSolution(self.__domain)
249           self.__u=None           self.__u=None
250    
251       def setReducedOff(self):       def setReducedOff(self):
252           """           """
253           sets the L{FunctionSpace} of the solution to L{Solution}           Sets the `FunctionSpace` of the solution to `Solution`.
254           """           """
255           self.__function_space=escript.Solution(self.__domain)           self.__function_space=escript.Solution(self.__domain)
256           self.__u=None           self.__u=None
257            
258       def setValue(self,D=None,Y=None,q=None,r=None):       def setValue(self,D=None,Y=None,q=None,r=None):
259           """           """
260           assigns values to the parameters.           Assigns values to the parameters.
261    
262           @param D: coefficient of the solution.           :param D: coefficient of the solution
263           @type D: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type D: ``float``, ``int``, ``numpy.ndarray``, `Data`
264           @param Y: right hand side           :param Y: right hand side
265           @type Y: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type Y: ``float``, ``int``, ``numpy.ndarray``, `Data`
266           @param q: location of constraints           :param q: location of constraints
267           @type q: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type q: ``float``, ``int``, ``numpy.ndarray``, `Data`
268           @param r: value of solution at locations of constraints           :param r: value of solution at locations of constraints
269           @type r: C{float}, C{int}, L{numarray.NumArray}, L{Data}           :type r: ``float``, ``int``, ``numpy.ndarray``, `Data`
270           """           """
271           if not D==None:           if not D==None:
272              self.__D=D              self.__D=D
# Line 261  class NoPDE: Line 283  class NoPDE:
283    
284       def getSolution(self):       def getSolution(self):
285           """           """
286           returns the solution           Returns the solution.
287            
288           @return: the solution of the problem           :return: the solution of the problem
289           @rtype: L{Data} object in the L{FunctionSpace} L{Solution} or L{ReducedSolution}.           :rtype: `Data` object in the `FunctionSpace` `Solution` or
290                     `ReducedSolution`
291           """           """
292           if self.__u==None:           if self.__u==None:
293              if self.__D==None:              if self.__D==None:
# Line 281  class NoPDE: Line 304  class NoPDE:
304                  self.__u*=(1.-q)                  self.__u*=(1.-q)
305                  if not self.__r==None: self.__u+=q*self.__r                  if not self.__r==None: self.__u+=q*self.__r
306           return self.__u           return self.__u
307                
308  class Locator:  class Locator:
309       """       """
310       Locator provides access to the values of data objects at a given       Locator provides access to the values of data objects at a given spatial
311       spatial coordinate x.         coordinate x.
312        
313       In fact, a Locator object finds the sample in the set of samples of a       In fact, a Locator object finds the sample in the set of samples of a
314       given function space or domain where which is closest to the given       given function space or domain which is closest to the given point x.
      point x.  
315       """       """
316    
317       def __init__(self,where,x=numarray.zeros((3,))):       def __init__(self,where,x=numpy.zeros((3,))):
318         """         """
319         Initializes a Locator to access values in Data objects on the Doamin         Initializes a Locator to access values in Data objects on the Doamin
320         or FunctionSpace where for the sample point which         or FunctionSpace for the sample point which is closest to the given
321         closest to the given point x.         point x.
322    
323         @param where: function space         :param where: function space
324         @type where: L{escript.FunctionSpace}         :type where: `escript.FunctionSpace`
325         @param x: coefficient of the solution.         :param x: location(s) of the Locator
326         @type x: L{numarray.NumArray} or C{list} of L{numarray.NumArray}         :type x: ``numpy.ndarray`` or ``list`` of ``numpy.ndarray``
327         """         """
328         if isinstance(where,escript.FunctionSpace):         if isinstance(where,escript.FunctionSpace):
329            self.__function_space=where            self.__function_space=where
330         else:         else:
331            self.__function_space=escript.ContinuousFunction(where)            self.__function_space=escript.ContinuousFunction(where)
332           iterative=False
333         if isinstance(x, list):         if isinstance(x, list):
334               if len(x)==0:
335                  raise "ValueError", "At least one point must be given."
336               try:
337                 iter(x[0])
338                 iterative=True
339               except TypeError:
340                 iterative=False
341           if iterative:
342             self.__id=[]             self.__id=[]
343             for p in x:             for p in x:
344                self.__id.append(util.length(self.__function_space.getX()-p[:self.__function_space.getDim()]).minGlobalDataPoint())                self.__id.append(util.length(self.__function_space.getX()-p[:self.__function_space.getDim()]).minGlobalDataPoint())
# Line 319  class Locator: Line 350  class Locator:
350         Returns the coordinates of the Locator as a string.         Returns the coordinates of the Locator as a string.
351         """         """
352         x=self.getX()         x=self.getX()
353         if instance(x,list):         if isinstance(x,list):
354            out="["            out="["
355            first=True            first=True
356            for xx in x:            for xx in x:
# Line 335  class Locator: Line 366  class Locator:
366    
367       def getX(self):       def getX(self):
368          """          """
369      Returns the exact coordinates of the Locator.          Returns the exact coordinates of the Locator.
370      """          """
371          return self(self.getFunctionSpace().getX())          return self(self.getFunctionSpace().getX())
372    
373       def getFunctionSpace(self):       def getFunctionSpace(self):
374          """          """
375      Returns the function space of the Locator.          Returns the function space of the Locator.
376      """          """
377          return self.__function_space          return self.__function_space
378    
379       def getId(self,item=None):       def getId(self,item=None):
380          """          """
381      Returns the identifier of the location.          Returns the identifier of the location.
382      """          """
383          if item == None:          if item == None:
384             return self.__id             return self.__id
385          else:          else:
# Line 360  class Locator: Line 391  class Locator:
391    
392       def __call__(self,data):       def __call__(self,data):
393          """          """
394      Returns the value of data at the Locator of a Data object otherwise          Returns the value of data at the Locator of a Data object.
395      the object is returned.          """
     """  
396          return self.getValue(data)          return self.getValue(data)
397    
398       def getValue(self,data):       def getValue(self,data):
399          """          """
400      Returns the value of data at the Locator if data is a Data object          Returns the value of ``data`` at the Locator if ``data`` is a `Data`
401      otherwise the object is returned.          object otherwise the object is returned.
402      """          """
403          if isinstance(data,escript.Data):          if isinstance(data,escript.Data):
404             if data.getFunctionSpace()==self.getFunctionSpace():             dat=util.interpolate(data,self.getFunctionSpace())
              dat=data  
            else:  
              dat=data.interpolate(self.getFunctionSpace())  
405             id=self.getId()             id=self.getId()
406             r=data.getRank()             r=data.getRank()
407             if isinstance(id,list):             if isinstance(id,list):
408                 out=[]                 out=[]
409                 for i in id:                 for i in id:
410                    o=data.getValueOfGlobalDataPoint(*i)                    o=numpy.array(dat.getTupleForGlobalDataPoint(*i))
411                    if data.getRank()==0:                    if data.getRank()==0:
412                       out.append(o[0])                       out.append(o[0])
413                    else:                    else:
414                       out.append(o)                       out.append(o)
415                 return out                 return out
416             else:             else:
417               out=data.getValueOfGlobalDataPoint(*id)               out=numpy.array(dat.getTupleForGlobalDataPoint(*id))
418               if data.getRank()==0:               if data.getRank()==0:
419                  return out[0]                  return out[0]
420               else:               else:
# Line 395  class Locator: Line 422  class Locator:
422          else:          else:
423             return data             return data
424    
425  class SaddlePointProblem(object):  
426    def getInfLocator(arg):
427        """
428        Return a Locator for a point with the inf value over all arg.
429        """
430        if not isinstance(arg, escript.Data):
431        raise TypeError,"getInfLocator: Unknown argument type."
432        a_inf=util.inf(arg)
433        loc=util.length(arg-a_inf).minGlobalDataPoint() # This gives us the location but not coords
434        x=arg.getFunctionSpace().getX()
435        x_min=x.getTupleForGlobalDataPoint(*loc)
436        return Locator(arg.getFunctionSpace(),x_min)
437    
438    def getSupLocator(arg):
439        """
440        Return a Locator for a point with the sup value over all arg.
441        """
442        if not isinstance(arg, escript.Data):
443        raise TypeError,"getInfLocator: Unknown argument type."
444        a_inf=util.sup(arg)
445        loc=util.length(arg-a_inf).minGlobalDataPoint() # This gives us the location but not coords
446        x=arg.getFunctionSpace().getX()
447        x_min=x.getTupleForGlobalDataPoint(*loc)
448        return Locator(arg.getFunctionSpace(),x_min)
449        
450    
451    class SolverSchemeException(Exception):
452       """
453       This is a generic exception thrown by solvers.
454       """
455       pass
456    
457    class IndefinitePreconditioner(SolverSchemeException):
458       """
459       Exception thrown if the preconditioner is not positive definite.
460       """
461       pass
462    
463    class MaxIterReached(SolverSchemeException):
464     """     """
465     This implements a solver for a saddlepoint problem     Exception thrown if the maximum number of iteration steps is reached.
466       """
467       pass
468    
469    class CorrectionFailed(SolverSchemeException):
470       """
471       Exception thrown if no convergence has been achieved in the solution
472       correction scheme.
473       """
474       pass
475    
476    class IterationBreakDown(SolverSchemeException):
477       """
478       Exception thrown if the iteration scheme encountered an incurable breakdown.
479       """
480       pass
481    
482    class NegativeNorm(SolverSchemeException):
483       """
484       Exception thrown if a norm calculation returns a negative norm.
485       """
486       pass
487    
488    def PCG(r, Aprod, x, Msolve, bilinearform, atol=0, rtol=1.e-8, iter_max=100, initial_guess=True, verbose=False):
489       """
490       Solver for
491    
492       *Ax=b*
493    
494       with a symmetric and positive definite operator A (more details required!).
495       It uses the conjugate gradient method with preconditioner M providing an
496       approximation of A.
497    
498       The iteration is terminated if
499    
500       *|r| <= atol+rtol*|r0|*
501    
502       where *r0* is the initial residual and *|.|* is the energy norm. In fact
503    
504       *|r| = sqrt( bilinearform(Msolve(r),r))*
505    
506       For details on the preconditioned conjugate gradient method see the book:
507    
508       I{Templates for the Solution of Linear Systems by R. Barrett, M. Berry,
509       T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo,
510       C. Romine, and H. van der Vorst}.
511    
512       :param r: initial residual *r=b-Ax*. ``r`` is altered.
513       :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
514       :param x: an initial guess for the solution
515       :type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
516       :param Aprod: returns the value Ax
517       :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
518                    argument ``x``. The returned object needs to be of the same type
519                    like argument ``r``.
520       :param Msolve: solves Mx=r
521       :type Msolve: function ``Msolve(r)`` where ``r`` is of the same type like
522                     argument ``r``. The returned object needs to be of the same
523                     type like argument ``x``.
524       :param bilinearform: inner product ``<x,r>``
525       :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
526                           type like argument ``x`` and ``r`` is. The returned value
527                           is a ``float``.
528       :param atol: absolute tolerance
529       :type atol: non-negative ``float``
530       :param rtol: relative tolerance
531       :type rtol: non-negative ``float``
532       :param iter_max: maximum number of iteration steps
533       :type iter_max: ``int``
534       :return: the solution approximation and the corresponding residual
535       :rtype: ``tuple``
536       :warning: ``r`` and ``x`` are altered.
537       """
538       iter=0
539       rhat=Msolve(r)
540       d = rhat
541       rhat_dot_r = bilinearform(rhat, r)
542       if rhat_dot_r<0: raise NegativeNorm,"negative norm."
543       norm_r0=math.sqrt(rhat_dot_r)
544       atol2=atol+rtol*norm_r0
545       if atol2<=0:
546          raise ValueError,"Non-positive tolarance."
547       atol2=max(atol2, 100. * util.EPSILON * norm_r0)
548    
549       if verbose: print "PCG: initial residual norm = %e (absolute tolerance = %e)"%(norm_r0, atol2)
550    
551    
552       while not math.sqrt(rhat_dot_r) <= atol2:
553           iter+=1
554           if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
555    
556           q=Aprod(d)
557           alpha = rhat_dot_r / bilinearform(d, q)
558           x += alpha * d
559           if isinstance(q,ArithmeticTuple):
560           r += q * (-alpha)      # Doing it the other way calls the float64.__mul__ not AT.__rmul__
561           else:
562               r += (-alpha) * q
563           rhat=Msolve(r)
564           rhat_dot_r_new = bilinearform(rhat, r)
565           beta = rhat_dot_r_new / rhat_dot_r
566           rhat+=beta * d
567           d=rhat
568    
569           rhat_dot_r = rhat_dot_r_new
570           if rhat_dot_r<0: raise NegativeNorm,"negative norm."
571           if verbose: print "PCG: iteration step %s: residual norm = %e"%(iter, math.sqrt(rhat_dot_r))
572       if verbose: print "PCG: tolerance reached after %s steps."%iter
573       return x,r,math.sqrt(rhat_dot_r)
574    
575    class Defect(object):
576        """
577        Defines a non-linear defect F(x) of a variable x.
578        """
579        def __init__(self):
580            """
581            Initializes defect.
582            """
583            self.setDerivativeIncrementLength()
584    
585        def bilinearform(self, x0, x1):
586            """
587            Returns the inner product of x0 and x1
588    
589            :param x0: value for x0
590            :param x1: value for x1
591            :return: the inner product of x0 and x1
592            :rtype: ``float``
593            """
594            return 0
595    
596        def norm(self,x):
597            """
598            Returns the norm of argument ``x``.
599    
600            :param x: a value
601            :return: norm of argument x
602            :rtype: ``float``
603            :note: by default ``sqrt(self.bilinearform(x,x)`` is returned.
604            """
605            s=self.bilinearform(x,x)
606            if s<0: raise NegativeNorm,"negative norm."
607            return math.sqrt(s)
608    
609        def eval(self,x):
610            """
611            Returns the value F of a given ``x``.
612    
613            :param x: value for which the defect ``F`` is evaluated
614            :return: value of the defect at ``x``
615            """
616            return 0
617    
618        def __call__(self,x):
619            return self.eval(x)
620    
621        def setDerivativeIncrementLength(self,inc=1000.*math.sqrt(util.EPSILON)):
622            """
623            Sets the relative length of the increment used to approximate the
624            derivative of the defect. The increment is inc*norm(x)/norm(v)*v in the
625            direction of v with x as a starting point.
626    
627            :param inc: relative increment length
628            :type inc: positive ``float``
629            """
630            if inc<=0: raise ValueError,"positive increment required."
631            self.__inc=inc
632    
633        def getDerivativeIncrementLength(self):
634            """
635            Returns the relative increment length used to approximate the
636            derivative of the defect.
637            :return: value of the defect at ``x``
638            :rtype: positive ``float``
639            """
640            return self.__inc
641    
642        def derivative(self, F0, x0, v, v_is_normalised=True):
643            """
644            Returns the directional derivative at ``x0`` in the direction of ``v``.
645    
646            :param F0: value of this defect at x0
647            :param x0: value at which derivative is calculated
648            :param v: direction
649            :param v_is_normalised: True to indicate that ``v`` is nomalized
650                                    (self.norm(v)=0)
651            :return: derivative of this defect at x0 in the direction of ``v``
652            :note: by default numerical evaluation (self.eval(x0+eps*v)-F0)/eps is
653                   used but this method maybe overwritten to use exact evaluation.
654            """
655            normx=self.norm(x0)
656            if normx>0:
657                 epsnew = self.getDerivativeIncrementLength() * normx
658            else:
659                 epsnew = self.getDerivativeIncrementLength()
660            if not v_is_normalised:
661                normv=self.norm(v)
662                if normv<=0:
663                   return F0*0
664                else:
665                   epsnew /= normv
666            F1=self.eval(x0 + epsnew * v)
667            return (F1-F0)/epsnew
668    
669    ######################################
670    def NewtonGMRES(defect, x, iter_max=100, sub_iter_max=20, atol=0,rtol=1.e-4, subtol_max=0.5, gamma=0.9, verbose=False):
671       """
672       Solves a non-linear problem *F(x)=0* for unknown *x* using the stopping
673       criterion:
674    
675       *norm(F(x) <= atol + rtol * norm(F(x0)*
676    
677       where *x0* is the initial guess.
678    
679       :param defect: object defining the function *F*. ``defect.norm`` defines the
680                      *norm* used in the stopping criterion.
681       :type defect: `Defect`
682       :param x: initial guess for the solution, ``x`` is altered.
683       :type x: any object type allowing basic operations such as
684                ``numpy.ndarray``, `Data`
685       :param iter_max: maximum number of iteration steps
686       :type iter_max: positive ``int``
687       :param sub_iter_max: maximum number of inner iteration steps
688       :type sub_iter_max: positive ``int``
689       :param atol: absolute tolerance for the solution
690       :type atol: positive ``float``
691       :param rtol: relative tolerance for the solution
692       :type rtol: positive ``float``
693       :param gamma: tolerance safety factor for inner iteration
694       :type gamma: positive ``float``, less than 1
695       :param subtol_max: upper bound for inner tolerance
696       :type subtol_max: positive ``float``, less than 1
697       :return: an approximation of the solution with the desired accuracy
698       :rtype: same type as the initial guess
699       """
700       lmaxit=iter_max
701       if atol<0: raise ValueError,"atol needs to be non-negative."
702       if rtol<0: raise ValueError,"rtol needs to be non-negative."
703       if rtol+atol<=0: raise ValueError,"rtol or atol needs to be non-negative."
704       if gamma<=0 or gamma>=1: raise ValueError,"tolerance safety factor for inner iteration (gamma =%s) needs to be positive and less than 1."%gamma
705       if subtol_max<=0 or subtol_max>=1: raise ValueError,"upper bound for inner tolerance for inner iteration (subtol_max =%s) needs to be positive and less than 1."%subtol_max
706    
707       F=defect(x)
708       fnrm=defect.norm(F)
709       stop_tol=atol + rtol*fnrm
710       subtol=subtol_max
711       if verbose: print "NewtonGMRES: initial residual = %e."%fnrm
712       if verbose: print "             tolerance = %e."%subtol
713       iter=1
714       #
715       # main iteration loop
716       #
717       while not fnrm<=stop_tol:
718                if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
719                #
720            #   adjust subtol_
721            #
722                if iter > 1:
723               rat=fnrm/fnrmo
724                   subtol_old=subtol
725               subtol=gamma*rat**2
726               if gamma*subtol_old**2 > .1: subtol=max(subtol,gamma*subtol_old**2)
727               subtol=max(min(subtol,subtol_max), .5*stop_tol/fnrm)
728            #
729            # calculate newton increment xc
730                #     if iter_max in __FDGMRES is reached MaxIterReached is thrown
731                #     if iter_restart -1 is returned as sub_iter
732                #     if  atol is reached sub_iter returns the numer of steps performed to get there
733                #
734                #
735                if verbose: print "             subiteration (GMRES) is called with relative tolerance %e."%subtol
736                try:
737                   xc, sub_iter=__FDGMRES(F, defect, x, subtol*fnrm, iter_max=iter_max-iter, iter_restart=sub_iter_max)
738                except MaxIterReached:
739                   raise MaxIterReached,"maximum number of %s steps reached."%iter_max
740                if sub_iter<0:
741                   iter+=sub_iter_max
742                else:
743                   iter+=sub_iter
744                # ====
745            x+=xc
746                F=defect(x)
747            iter+=1
748                fnrmo, fnrm=fnrm, defect.norm(F)
749                if verbose: print "             step %s: residual %e."%(iter,fnrm)
750       if verbose: print "NewtonGMRES: completed after %s steps."%iter
751       return x
752    
753    def __givapp(c,s,vin):
754        """
755        Applies a sequence of Givens rotations (c,s) recursively to the vector
756        ``vin``
757    
758        :warning: ``vin`` is altered.
759        """
760        vrot=vin
761        if isinstance(c,float):
762            vrot=[c*vrot[0]-s*vrot[1],s*vrot[0]+c*vrot[1]]
763        else:
764            for i in range(len(c)):
765                w1=c[i]*vrot[i]-s[i]*vrot[i+1]
766            w2=s[i]*vrot[i]+c[i]*vrot[i+1]
767                vrot[i]=w1
768                vrot[i+1]=w2
769        return vrot
770    
771    def __FDGMRES(F0, defect, x0, atol, iter_max=100, iter_restart=20):
772       h=numpy.zeros((iter_restart,iter_restart),numpy.float64)
773       c=numpy.zeros(iter_restart,numpy.float64)
774       s=numpy.zeros(iter_restart,numpy.float64)
775       g=numpy.zeros(iter_restart,numpy.float64)
776       v=[]
777    
778       rho=defect.norm(F0)
779       if rho<=0.: return x0*0
780    
781       v.append(-F0/rho)
782       g[0]=rho
783       iter=0
784       while rho > atol and iter<iter_restart-1:
785            if iter  >= iter_max:
786                raise MaxIterReached,"maximum number of %s steps reached."%iter_max
787    
788            p=defect.derivative(F0,x0,v[iter], v_is_normalised=True)
789            v.append(p)
790    
791            v_norm1=defect.norm(v[iter+1])
792    
793            # Modified Gram-Schmidt
794            for j in range(iter+1):
795                h[j,iter]=defect.bilinearform(v[j],v[iter+1])
796                v[iter+1]-=h[j,iter]*v[j]
797    
798            h[iter+1,iter]=defect.norm(v[iter+1])
799            v_norm2=h[iter+1,iter]
800    
801            # Reorthogonalize if needed
802            if v_norm1 + 0.001*v_norm2 == v_norm1:   #Brown/Hindmarsh condition (default)
803                for j in range(iter+1):
804                    hr=defect.bilinearform(v[j],v[iter+1])
805                    h[j,iter]=h[j,iter]+hr
806                    v[iter+1] -= hr*v[j]
807    
808                v_norm2=defect.norm(v[iter+1])
809                h[iter+1,iter]=v_norm2
810            #   watch out for happy breakdown
811            if not v_norm2 == 0:
812                v[iter+1]=v[iter+1]/h[iter+1,iter]
813    
814            #   Form and store the information for the new Givens rotation
815            if iter > 0 :
816                hhat=numpy.zeros(iter+1,numpy.float64)
817                for i in range(iter+1) : hhat[i]=h[i,iter]
818                hhat=__givapp(c[0:iter],s[0:iter],hhat);
819                for i in range(iter+1) : h[i,iter]=hhat[i]
820    
821            mu=math.sqrt(h[iter,iter]*h[iter,iter]+h[iter+1,iter]*h[iter+1,iter])
822    
823            if mu!=0 :
824                c[iter]=h[iter,iter]/mu
825                s[iter]=-h[iter+1,iter]/mu
826                h[iter,iter]=c[iter]*h[iter,iter]-s[iter]*h[iter+1,iter]
827                h[iter+1,iter]=0.0
828                gg=__givapp(c[iter],s[iter],[g[iter],g[iter+1]])
829                g[iter]=gg[0]
830                g[iter+1]=gg[1]
831    
832            # Update the residual norm
833            rho=abs(g[iter+1])
834            iter+=1
835    
836       # At this point either iter > iter_max or rho < tol.
837       # It's time to compute x and leave.
838       if iter > 0 :
839         y=numpy.zeros(iter,numpy.float64)
840         y[iter-1] = g[iter-1] / h[iter-1,iter-1]
841         if iter > 1 :
842            i=iter-2
843            while i>=0 :
844              y[i] = ( g[i] - numpy.dot(h[i,i+1:iter], y[i+1:iter])) / h[i,i]
845              i=i-1
846         xhat=v[iter-1]*y[iter-1]
847         for i in range(iter-1):
848        xhat += v[i]*y[i]
849       else :
850          xhat=v[0] * 0
851    
852       if iter<iter_restart-1:
853          stopped=iter
854       else:
855          stopped=-1
856    
857       return xhat,stopped
858    
859    def GMRES(r, Aprod, x, bilinearform, atol=0, rtol=1.e-8, iter_max=100, iter_restart=20, verbose=False,P_R=None):
860       """
861       Solver for
862    
863       *Ax=b*
864    
865       with a general operator A (more details required!).
866       It uses the generalized minimum residual method (GMRES).
867    
868       The iteration is terminated if
869    
870       *|r| <= atol+rtol*|r0|*
871    
872       where *r0* is the initial residual and *|.|* is the energy norm. In fact
873    
874       *|r| = sqrt( bilinearform(r,r))*
875    
876       :param r: initial residual *r=b-Ax*. ``r`` is altered.
877       :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
878       :param x: an initial guess for the solution
879       :type x: same like ``r``
880       :param Aprod: returns the value Ax
881       :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
882                    argument ``x``. The returned object needs to be of the same
883                    type like argument ``r``.
884       :param bilinearform: inner product ``<x,r>``
885       :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
886                           type like argument ``x`` and ``r``. The returned value is
887                           a ``float``.
888       :param atol: absolute tolerance
889       :type atol: non-negative ``float``
890       :param rtol: relative tolerance
891       :type rtol: non-negative ``float``
892       :param iter_max: maximum number of iteration steps
893       :type iter_max: ``int``
894       :param iter_restart: in order to save memory the orthogonalization process
895                            is terminated after ``iter_restart`` steps and the
896                            iteration is restarted.
897       :type iter_restart: ``int``
898       :return: the solution approximation and the corresponding residual
899       :rtype: ``tuple``
900       :warning: ``r`` and ``x`` are altered.
901       """
902       m=iter_restart
903       restarted=False
904       iter=0
905       if rtol>0:
906          r_dot_r = bilinearform(r, r)
907          if r_dot_r<0: raise NegativeNorm,"negative norm."
908          atol2=atol+rtol*math.sqrt(r_dot_r)
909          if verbose: print "GMRES: norm of right hand side = %e (absolute tolerance = %e)"%(math.sqrt(r_dot_r), atol2)
910       else:
911          atol2=atol
912          if verbose: print "GMRES: absolute tolerance = %e"%atol2
913       if atol2<=0:
914          raise ValueError,"Non-positive tolarance."
915    
916       while True:
917          if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached"%iter_max
918          if restarted:
919             r2 = r-Aprod(x-x2)
920          else:
921             r2=1*r
922          x2=x*1.
923          x,stopped=_GMRESm(r2, Aprod, x, bilinearform, atol2, iter_max=iter_max-iter, iter_restart=m, verbose=verbose,P_R=P_R)
924          iter+=iter_restart
925          if stopped: break
926          if verbose: print "GMRES: restart."
927          restarted=True
928       if verbose: print "GMRES: tolerance has been reached."
929       return x
930    
931    def _GMRESm(r, Aprod, x, bilinearform, atol, iter_max=100, iter_restart=20, verbose=False, P_R=None):
932       iter=0
933    
934       h=numpy.zeros((iter_restart+1,iter_restart),numpy.float64)
935       c=numpy.zeros(iter_restart,numpy.float64)
936       s=numpy.zeros(iter_restart,numpy.float64)
937       g=numpy.zeros(iter_restart+1,numpy.float64)
938       v=[]
939    
940       r_dot_r = bilinearform(r, r)
941       if r_dot_r<0: raise NegativeNorm,"negative norm."
942       rho=math.sqrt(r_dot_r)
943    
944       v.append(r/rho)
945       g[0]=rho
946    
947       if verbose: print "GMRES: initial residual %e (absolute tolerance = %e)"%(rho,atol)
948       while not (rho<=atol or iter==iter_restart):
949    
950        if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
951    
952            if P_R!=None:
953                p=Aprod(P_R(v[iter]))
954            else:
955            p=Aprod(v[iter])
956        v.append(p)
957    
958        v_norm1=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
959    
960    # Modified Gram-Schmidt
961        for j in range(iter+1):
962          h[j,iter]=bilinearform(v[j],v[iter+1])
963          v[iter+1]-=h[j,iter]*v[j]
964    
965        h[iter+1,iter]=math.sqrt(bilinearform(v[iter+1],v[iter+1]))
966        v_norm2=h[iter+1,iter]
967    
968    # Reorthogonalize if needed
969        if v_norm1 + 0.001*v_norm2 == v_norm1:   #Brown/Hindmarsh condition (default)
970         for j in range(iter+1):
971            hr=bilinearform(v[j],v[iter+1])
972                h[j,iter]=h[j,iter]+hr
973                v[iter+1] -= hr*v[j]
974    
975         v_norm2=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
976         h[iter+1,iter]=v_norm2
977    
978    #   watch out for happy breakdown
979            if not v_norm2 == 0:
980             v[iter+1]=v[iter+1]/h[iter+1,iter]
981    
982    #   Form and store the information for the new Givens rotation
983        if iter > 0: h[:iter+1,iter]=__givapp(c[:iter],s[:iter],h[:iter+1,iter])
984        mu=math.sqrt(h[iter,iter]*h[iter,iter]+h[iter+1,iter]*h[iter+1,iter])
985    
986        if mu!=0 :
987            c[iter]=h[iter,iter]/mu
988            s[iter]=-h[iter+1,iter]/mu
989            h[iter,iter]=c[iter]*h[iter,iter]-s[iter]*h[iter+1,iter]
990            h[iter+1,iter]=0.0
991                    gg=__givapp(c[iter],s[iter],[g[iter],g[iter+1]])
992                    g[iter]=gg[0]
993                    g[iter+1]=gg[1]
994    # Update the residual norm
995    
996            rho=abs(g[iter+1])
997            if verbose: print "GMRES: iteration step %s: residual %e"%(iter,rho)
998        iter+=1
999    
1000    # At this point either iter > iter_max or rho < tol.
1001    # It's time to compute x and leave.
1002    
1003       if verbose: print "GMRES: iteration stopped after %s step."%iter
1004       if iter > 0 :
1005         y=numpy.zeros(iter,numpy.float64)
1006         y[iter-1] = g[iter-1] / h[iter-1,iter-1]
1007         if iter > 1 :
1008            i=iter-2
1009            while i>=0 :
1010              y[i] = ( g[i] - numpy.dot(h[i,i+1:iter], y[i+1:iter])) / h[i,i]
1011              i=i-1
1012         xhat=v[iter-1]*y[iter-1]
1013         for i in range(iter-1):
1014        xhat += v[i]*y[i]
1015       else:
1016         xhat=v[0] * 0
1017       if P_R!=None:
1018          x += P_R(xhat)
1019       else:
1020          x += xhat
1021       if iter<iter_restart-1:
1022          stopped=True
1023       else:
1024          stopped=False
1025    
1026       return x,stopped
1027    
1028    def MINRES(r, Aprod, x, Msolve, bilinearform, atol=0, rtol=1.e-8, iter_max=100):
1029        """
1030        Solver for
1031    
1032        *Ax=b*
1033    
1034        with a symmetric and positive definite operator A (more details required!).
1035        It uses the minimum residual method (MINRES) with preconditioner M
1036        providing an approximation of A.
1037    
1038        The iteration is terminated if
1039    
1040        *|r| <= atol+rtol*|r0|*
1041    
1042        where *r0* is the initial residual and *|.|* is the energy norm. In fact
1043    
1044        *|r| = sqrt( bilinearform(Msolve(r),r))*
1045    
1046        For details on the preconditioned conjugate gradient method see the book:
1047    
1048        I{Templates for the Solution of Linear Systems by R. Barrett, M. Berry,
1049        T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo,
1050        C. Romine, and H. van der Vorst}.
1051    
1052        :param r: initial residual *r=b-Ax*. ``r`` is altered.
1053        :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1054        :param x: an initial guess for the solution
1055        :type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1056        :param Aprod: returns the value Ax
1057        :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
1058                     argument ``x``. The returned object needs to be of the same
1059                     type like argument ``r``.
1060        :param Msolve: solves Mx=r
1061        :type Msolve: function ``Msolve(r)`` where ``r`` is of the same type like
1062                      argument ``r``. The returned object needs to be of the same
1063                      type like argument ``x``.
1064        :param bilinearform: inner product ``<x,r>``
1065        :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
1066                            type like argument ``x`` and ``r`` is. The returned value
1067                            is a ``float``.
1068        :param atol: absolute tolerance
1069        :type atol: non-negative ``float``
1070        :param rtol: relative tolerance
1071        :type rtol: non-negative ``float``
1072        :param iter_max: maximum number of iteration steps
1073        :type iter_max: ``int``
1074        :return: the solution approximation and the corresponding residual
1075        :rtype: ``tuple``
1076        :warning: ``r`` and ``x`` are altered.
1077        """
1078        #------------------------------------------------------------------
1079        # Set up y and v for the first Lanczos vector v1.
1080        # y  =  beta1 P' v1,  where  P = C**(-1).
1081        # v is really P' v1.
1082        #------------------------------------------------------------------
1083        r1    = r
1084        y = Msolve(r)
1085        beta1 = bilinearform(y,r)
1086    
1087        if beta1< 0: raise NegativeNorm,"negative norm."
1088    
1089        #  If r = 0 exactly, stop with x
1090        if beta1==0: return x
1091    
1092        if beta1> 0: beta1  = math.sqrt(beta1)
1093    
1094        #------------------------------------------------------------------
1095        # Initialize quantities.
1096        # ------------------------------------------------------------------
1097        iter   = 0
1098        Anorm = 0
1099        ynorm = 0
1100        oldb   = 0
1101        beta   = beta1
1102        dbar   = 0
1103        epsln  = 0
1104        phibar = beta1
1105        rhs1   = beta1
1106        rhs2   = 0
1107        rnorm  = phibar
1108        tnorm2 = 0
1109        ynorm2 = 0
1110        cs     = -1
1111        sn     = 0
1112        w      = r*0.
1113        w2     = r*0.
1114        r2     = r1
1115        eps    = 0.0001
1116    
1117        #---------------------------------------------------------------------
1118        # Main iteration loop.
1119        # --------------------------------------------------------------------
1120        while not rnorm<=atol+rtol*Anorm*ynorm:    #  checks ||r|| < (||A|| ||x||) * TOL
1121    
1122        if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
1123            iter    = iter  +  1
1124    
1125            #-----------------------------------------------------------------
1126            # Obtain quantities for the next Lanczos vector vk+1, k = 1, 2,...
1127            # The general iteration is similar to the case k = 1 with v0 = 0:
1128            #
1129            #   p1      = Operator * v1  -  beta1 * v0,
1130            #   alpha1  = v1'p1,
1131            #   q2      = p2  -  alpha1 * v1,
1132            #   beta2^2 = q2'q2,
1133            #   v2      = (1/beta2) q2.
1134            #
1135            # Again, y = betak P vk,  where  P = C**(-1).
1136            #-----------------------------------------------------------------
1137            s = 1/beta                 # Normalize previous vector (in y).
1138            v = s*y                    # v = vk if P = I
1139    
1140            y      = Aprod(v)
1141    
1142            if iter >= 2:
1143              y = y - (beta/oldb)*r1
1144    
1145            alfa   = bilinearform(v,y)              # alphak
1146            y      += (- alfa/beta)*r2
1147            r1     = r2
1148            r2     = y
1149            y = Msolve(r2)
1150            oldb   = beta                           # oldb = betak
1151            beta   = bilinearform(y,r2)             # beta = betak+1^2
1152            if beta < 0: raise NegativeNorm,"negative norm."
1153    
1154            beta   = math.sqrt( beta )
1155            tnorm2 = tnorm2 + alfa*alfa + oldb*oldb + beta*beta
1156    
1157            if iter==1:                 # Initialize a few things.
1158              gmax   = abs( alfa )      # alpha1
1159              gmin   = gmax             # alpha1
1160    
1161            # Apply previous rotation Qk-1 to get
1162            #   [deltak epslnk+1] = [cs  sn][dbark    0   ]
1163            #   [gbar k dbar k+1]   [sn -cs][alfak betak+1].
1164    
1165            oldeps = epsln
1166            delta  = cs * dbar  +  sn * alfa  # delta1 = 0         deltak
1167            gbar   = sn * dbar  -  cs * alfa  # gbar 1 = alfa1     gbar k
1168            epsln  =               sn * beta  # epsln2 = 0         epslnk+1
1169            dbar   =            -  cs * beta  # dbar 2 = beta2     dbar k+1
1170    
1171            # Compute the next plane rotation Qk
1172    
1173            gamma  = math.sqrt(gbar*gbar+beta*beta)  # gammak
1174            gamma  = max(gamma,eps)
1175            cs     = gbar / gamma             # ck
1176            sn     = beta / gamma             # sk
1177            phi    = cs * phibar              # phik
1178            phibar = sn * phibar              # phibark+1
1179    
1180            # Update  x.
1181    
1182            denom = 1/gamma
1183            w1    = w2
1184            w2    = w
1185            w     = (v - oldeps*w1 - delta*w2) * denom
1186            x     +=  phi*w
1187    
1188            # Go round again.
1189    
1190            gmax   = max(gmax,gamma)
1191            gmin   = min(gmin,gamma)
1192            z      = rhs1 / gamma
1193            ynorm2 = z*z  +  ynorm2
1194            rhs1   = rhs2 -  delta*z
1195            rhs2   =      -  epsln*z
1196    
1197            # Estimate various norms and test for convergence.
1198    
1199            Anorm  = math.sqrt( tnorm2 )
1200            ynorm  = math.sqrt( ynorm2 )
1201    
1202            rnorm  = phibar
1203    
1204        return x
1205    
1206    def TFQMR(r, Aprod, x, bilinearform, atol=0, rtol=1.e-8, iter_max=100):
1207      """
1208      Solver for
1209    
1210     M{f(u,p)=0}    *Ax=b*
    M{g(u)=0}  
1211    
1212     for u and p. The problem is solved with an inexact Uszawa scheme for p:    with a general operator A (more details required!).
1213      It uses the Transpose-Free Quasi-Minimal Residual method (TFQMR).
1214    
1215     M{Q_f (u^{k+1}-u^{k}) = - f(u^{k},p^{k})}    The iteration is terminated if
1216     M{Q_g (p^{k+1}-p^{k}) =   g(u^{k+1})}  
1217      *|r| <= atol+rtol*|r0|*
1218    
1219      where *r0* is the initial residual and *|.|* is the energy norm. In fact
1220    
1221      *|r| = sqrt( bilinearform(r,r))*
1222    
1223      :param r: initial residual *r=b-Ax*. ``r`` is altered.
1224      :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1225      :param x: an initial guess for the solution
1226      :type x: same like ``r``
1227      :param Aprod: returns the value Ax
1228      :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
1229                   argument ``x``. The returned object needs to be of the same type
1230                   like argument ``r``.
1231      :param bilinearform: inner product ``<x,r>``
1232      :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
1233                          type like argument ``x`` and ``r``. The returned value is
1234                          a ``float``.
1235      :param atol: absolute tolerance
1236      :type atol: non-negative ``float``
1237      :param rtol: relative tolerance
1238      :type rtol: non-negative ``float``
1239      :param iter_max: maximum number of iteration steps
1240      :type iter_max: ``int``
1241      :rtype: ``tuple``
1242      :warning: ``r`` and ``x`` are altered.
1243      """
1244      u1=0
1245      u2=0
1246      y1=0
1247      y2=0
1248    
1249      w = r
1250      y1 = r
1251      iter = 0
1252      d = 0
1253      v = Aprod(y1)
1254      u1 = v
1255    
1256      theta = 0.0;
1257      eta = 0.0;
1258      rho=bilinearform(r,r)
1259      if rho < 0: raise NegativeNorm,"negative norm."
1260      tau = math.sqrt(rho)
1261      norm_r0=tau
1262      while tau>atol+rtol*norm_r0:
1263        if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
1264    
1265        sigma = bilinearform(r,v)
1266        if sigma == 0.0: raise IterationBreakDown,'TFQMR breakdown, sigma=0'
1267    
1268        alpha = rho / sigma
1269    
1270        for j in range(2):
1271    #
1272    #   Compute y2 and u2 only if you have to
1273    #
1274          if ( j == 1 ):
1275            y2 = y1 - alpha * v
1276            u2 = Aprod(y2)
1277    
1278          m = 2 * (iter+1) - 2 + (j+1)
1279          if j==0:
1280             w = w - alpha * u1
1281             d = y1 + ( theta * theta * eta / alpha ) * d
1282          if j==1:
1283             w = w - alpha * u2
1284             d = y2 + ( theta * theta * eta / alpha ) * d
1285    
1286          theta = math.sqrt(bilinearform(w,w))/ tau
1287          c = 1.0 / math.sqrt ( 1.0 + theta * theta )
1288          tau = tau * theta * c
1289          eta = c * c * alpha
1290          x = x + eta * d
1291    #
1292    #  Try to terminate the iteration at each pass through the loop
1293    #
1294        if rho == 0.0: raise IterationBreakDown,'TFQMR breakdown, rho=0'
1295    
1296        rhon = bilinearform(r,w)
1297        beta = rhon / rho;
1298        rho = rhon;
1299        y1 = w + beta * y2;
1300        u1 = Aprod(y1)
1301        v = u1 + beta * ( u2 + beta * v )
1302    
1303        iter += 1
1304    
1305      return x
1306    
1307    
1308    #############################################
1309    
1310    class ArithmeticTuple(object):
1311       """
1312       Tuple supporting inplace update x+=y and scaling x=a*y where ``x,y`` is an
1313       ArithmeticTuple and ``a`` is a float.
1314    
1315       Example of usage::
1316    
1317           from esys.escript import Data
1318           from numpy import array
1319           a=Data(...)
1320           b=array([1.,4.])
1321           x=ArithmeticTuple(a,b)
1322           y=5.*x
1323    
    where Q_f is an approximation of the Jacobiean A_f of f with respect to u  and Q_f is an approximation of  
    A_g A_f^{-1} A_g with A_g is the jacobiean of g with respect to p. As a the construction of a 'proper'  
    Q_g can be difficult, non-linear conjugate gradient method is applied to solve for p, so Q_g plays  
    in fact the role of a preconditioner.  
1324     """     """
1325     def __init__(self,verbose=False,*args):     def __init__(self,*args):
1326         """         """
1327         initializes the problem         Initializes object with elements ``args``.
1328    
1329         @param verbose: switches on the printing out some information         :param args: tuple of objects that support inplace add (x+=y) and
1330         @type verbose: C{bool}                      scaling (x=a*y)
        @note: this method may be overwritten by a particular saddle point problem  
1331         """         """
1332         if verbose:         self.__items=list(args)
           self.__verbose=True  
        else:  
           self.__verbose=False  
        self.relaxation=1.  
1333    
1334     def trace(self,text):     def __len__(self):
1335         """         """
1336         prints text if verbose has been set         Returns the number of items.
1337    
1338         @param text: a text message         :return: number of items
1339         @type text: C{str}         :rtype: ``int``
1340         """         """
1341         if self.__verbose: print "%s: %s"%(str(self),text)         return len(self.__items)
1342    
1343     def solve_f(self,u,p,tol=1.e-8):     def __getitem__(self,index):
1344         """         """
1345         solves         Returns item at specified position.
1346    
1347         A_f du = f(u,p)         :param index: index of item to be returned
1348           :type index: ``int``
1349           :return: item with index ``index``
1350           """
1351           return self.__items.__getitem__(index)
1352    
1353         with tolerance C{tol} and return du. A_f is Jacobiean of f with respect to u.     def __mul__(self,other):
1354           """
1355           Scales by ``other`` from the right.
1356    
1357         @param u: current approximation of u         :param other: scaling factor
1358         @type u: L{escript.Data}         :type other: ``float``
1359         @param p: current approximation of p         :return: itemwise self*other
1360         @type p: L{escript.Data}         :rtype: `ArithmeticTuple`
        @param tol: tolerance expected for du  
        @type tol: C{float}  
        @return: increment du  
        @rtype: L{escript.Data}  
        @note: this method has to be overwritten by a particular saddle point problem  
1361         """         """
1362         pass         out=[]
1363           try:
1364               l=len(other)
1365               if l!=len(self):
1366                   raise ValueError,"length of arguments don't match."
1367               for i in range(l): out.append(self[i]*other[i])
1368           except TypeError:
1369               for i in range(len(self)): out.append(self[i]*other)
1370           return ArithmeticTuple(*tuple(out))
1371    
1372     def solve_g(self,u,tol=1.e-8):     def __rmul__(self,other):
1373         """         """
1374         solves         Scales by ``other`` from the left.
1375    
1376         Q_g dp = g(u)         :param other: scaling factor
1377           :type other: ``float``
1378           :return: itemwise other*self
1379           :rtype: `ArithmeticTuple`
1380           """
1381           out=[]
1382           try:
1383               l=len(other)
1384               if l!=len(self):
1385                   raise ValueError,"length of arguments don't match."
1386               for i in range(l): out.append(other[i]*self[i])
1387           except TypeError:
1388               for i in range(len(self)): out.append(other*self[i])
1389           return ArithmeticTuple(*tuple(out))
1390    
1391         with Q_g is a preconditioner for A_g A_f^{-1} A_g with  A_g is the jacobiean of g with respect to p.     def __div__(self,other):
1392           """
1393           Scales by (1/``other``) from the right.
1394    
1395           :param other: scaling factor
1396           :type other: ``float``
1397           :return: itemwise self/other
1398           :rtype: `ArithmeticTuple`
1399           """
1400           return self*(1/other)
1401    
1402         @param u: current approximation of u     def __rdiv__(self,other):
        @type u: L{escript.Data}  
        @param tol: tolerance expected for dp  
        @type tol: C{float}  
        @return: increment dp  
        @rtype: L{escript.Data}  
        @note: this method has to be overwritten by a particular saddle point problem  
1403         """         """
1404         pass         Scales by (1/``other``) from the left.
1405    
1406     def inner(self,p0,p1):         :param other: scaling factor
1407           :type other: ``float``
1408           :return: itemwise other/self
1409           :rtype: `ArithmeticTuple`
1410         """         """
1411         inner product of p0 and p1 approximating p. Typically this returns integrate(p0*p1)         out=[]
1412         @return: inner product of p0 and p1         try:
1413         @rtype: C{float}             l=len(other)
1414               if l!=len(self):
1415                   raise ValueError,"length of arguments don't match."
1416               for i in range(l): out.append(other[i]/self[i])
1417           except TypeError:
1418               for i in range(len(self)): out.append(other/self[i])
1419           return ArithmeticTuple(*tuple(out))
1420    
1421       def __iadd__(self,other):
1422           """
1423           Inplace addition of ``other`` to self.
1424    
1425           :param other: increment
1426           :type other: ``ArithmeticTuple``
1427           """
1428           if len(self) != len(other):
1429               raise ValueError,"tuple lengths must match."
1430           for i in range(len(self)):
1431               self.__items[i]+=other[i]
1432           return self
1433    
1434       def __add__(self,other):
1435           """
1436           Adds ``other`` to self.
1437    
1438           :param other: increment
1439           :type other: ``ArithmeticTuple``
1440           """
1441           out=[]
1442           try:
1443               l=len(other)
1444               if l!=len(self):
1445                   raise ValueError,"length of arguments don't match."
1446               for i in range(l): out.append(self[i]+other[i])
1447           except TypeError:
1448               for i in range(len(self)): out.append(self[i]+other)
1449           return ArithmeticTuple(*tuple(out))
1450    
1451       def __sub__(self,other):
1452         """         """
1453         pass         Subtracts ``other`` from self.
1454    
1455           :param other: decrement
1456           :type other: ``ArithmeticTuple``
1457           """
1458           out=[]
1459           try:
1460               l=len(other)
1461               if l!=len(self):
1462                   raise ValueError,"length of arguments don't match."
1463               for i in range(l): out.append(self[i]-other[i])
1464           except TypeError:
1465               for i in range(len(self)): out.append(self[i]-other)
1466           return ArithmeticTuple(*tuple(out))
1467    
1468       def __isub__(self,other):
1469           """
1470           Inplace subtraction of ``other`` from self.
1471    
1472           :param other: decrement
1473           :type other: ``ArithmeticTuple``
1474           """
1475           if len(self) != len(other):
1476               raise ValueError,"tuple length must match."
1477           for i in range(len(self)):
1478               self.__items[i]-=other[i]
1479           return self
1480    
1481       def __neg__(self):
1482           """
1483           Negates values.
1484           """
1485           out=[]
1486           for i in range(len(self)):
1487               out.append(-self[i])
1488           return ArithmeticTuple(*tuple(out))
1489    
1490    
1491    class HomogeneousSaddlePointProblem(object):
1492          """
1493          This class provides a framework for solving linear homogeneous saddle
1494          point problems of the form::
1495    
1496              *Av+B^*p=f*
1497              *Bv     =0*
1498    
1499          for the unknowns *v* and *p* and given operators *A* and *B* and
1500          given right hand side *f*. *B^** is the adjoint operator of *B*.
1501          *A* may depend weakly on *v* and *p*.
1502          """
1503          def __init__(self, **kwargs):
1504        """
1505        initializes the saddle point problem
1506        """
1507            self.resetControlParameters()
1508            self.setTolerance()
1509            self.setAbsoluteTolerance()
1510          def resetControlParameters(self,gamma=0.85, gamma_min=1.e-2,chi_max=0.1, omega_div=0.2, omega_conv=1.1, rtol_min=1.e-7, rtol_max=0.9, chi=1., C_p=1., C_v=1., safety_factor=0.3):
1511             """
1512             sets a control parameter
1513    
1514             :param gamma: ``1/(1-gamma)`` controls the perturbation of the converegence rate due to termination errors in the subproblems.
1515             :type gamma: ``float``
1516             :param gamma_min: minimum value for ``gamma``.
1517             :type gamma_min: ``float``
1518             :param chi_max: maximum tolerable converegence rate.
1519             :type chi_max: ``float``
1520             :param omega_div: reduction fact for ``gamma`` if no convergence is detected.
1521             :type omega_div: ``float``
1522             :param omega_conv: raise fact for ``gamma`` if convergence is detected.
1523             :type omega_conv: ``float``
1524             :param rtol_min: minimum relative tolerance used to calculate presssure and velocity increment.
1525             :type rtol_min: ``float``
1526             :param rtol_max: maximuim relative tolerance used to calculate presssure and velocity increment.
1527             :type rtol_max: ``float``
1528             :param chi: initial convergence measure.
1529             :type chi: ``float``
1530             :param C_p: initial value for constant to adjust pressure tolerance
1531             :type C_p: ``float``
1532             :param C_v: initial value for constant to adjust velocity tolerance
1533             :type C_v: ``float``
1534             :param safety_factor: safety factor for addjustment of pressure and velocity tolerance from stopping criteria
1535             :type safety_factor: ``float``
1536             """
1537             self.setControlParameter(gamma, gamma_min ,chi_max , omega_div , omega_conv, rtol_min , rtol_max, chi,C_p, C_v,safety_factor)
1538    
1539          def setControlParameter(self,gamma=None, gamma_min=None ,chi_max=None, omega_div=None, omega_conv=None, rtol_min=None, rtol_max=None, chi=None, C_p=None, C_v=None, safety_factor=None):
1540             """
1541             sets a control parameter
1542    
1543             :param gamma: ``1/(1-gamma)`` controls the perturbation of the converegence rate due to termination errors in the subproblems.
1544             :type gamma: ``float``
1545             :param gamma_min: minimum value for ``gamma``.
1546             :type gamma_min: ``float``
1547             :param chi_max: maximum tolerable converegence rate.
1548             :type chi_max: ``float``
1549             :param omega_div: reduction fact for ``gamma`` if no convergence is detected.
1550             :type omega_div: ``float``
1551             :param omega_conv: raise fact for ``gamma`` if convergence is detected.
1552             :type omega_conv: ``float``
1553             :param rtol_min: minimum relative tolerance used to calculate presssure and velocity increment.
1554             :type rtol_min: ``float``
1555             :param rtol_max: maximuim relative tolerance used to calculate presssure and velocity increment.
1556             :type rtol_max: ``float``
1557             :param chi: initial convergence measure.
1558             :type chi: ``float``
1559             :param C_p: initial value for constant to adjust pressure tolerance
1560             :type C_p: ``float``
1561             :param C_v: initial value for constant to adjust velocity tolerance
1562             :type C_v: ``float``
1563             :param safety_factor: safety factor for addjustment of pressure and velocity tolerance from stopping criteria
1564             :type safety_factor: ``float``
1565             """
1566             if not gamma == None:
1567                if gamma<=0 or gamma>=1:
1568                   raise ValueError,"Initial gamma needs to be positive and less than 1."
1569             else:
1570                gamma=self.__gamma
1571    
1572             if not gamma_min == None:
1573                if gamma_min<=0 or gamma_min>=1:
1574                   raise ValueError,"gamma_min needs to be positive and less than 1."
1575             else:
1576                gamma_min = self.__gamma_min
1577    
1578             if not chi_max == None:
1579                if chi_max<=0 or chi_max>=1:
1580                   raise ValueError,"chi_max needs to be positive and less than 1."
1581             else:
1582                chi_max = self.__chi_max
1583    
1584             if not omega_div == None:
1585                if omega_div<=0 or omega_div >=1:
1586                   raise ValueError,"omega_div needs to be positive and less than 1."
1587             else:
1588                omega_div=self.__omega_div
1589    
1590             if not omega_conv == None:
1591                if omega_conv<1:
1592                   raise ValueError,"omega_conv needs to be greater or equal to 1."
1593             else:
1594                omega_conv=self.__omega_conv
1595    
1596             if not rtol_min == None:
1597                if rtol_min<=0 or rtol_min>=1:
1598                   raise ValueError,"rtol_min needs to be positive and less than 1."
1599             else:
1600                rtol_min=self.__rtol_min
1601    
1602             if not rtol_max == None:
1603                if rtol_max<=0 or rtol_max>=1:
1604                   raise ValueError,"rtol_max needs to be positive and less than 1."
1605             else:
1606                rtol_max=self.__rtol_max
1607    
1608             if not chi == None:
1609                if chi<=0 or chi>1:
1610                   raise ValueError,"chi needs to be positive and less than 1."
1611             else:
1612                chi=self.__chi
1613    
1614             if not C_p == None:
1615                if C_p<1:
1616                   raise ValueError,"C_p need to be greater or equal to 1."
1617             else:
1618                C_p=self.__C_p
1619    
1620             if not C_v == None:
1621                if C_v<1:
1622                   raise ValueError,"C_v need to be greater or equal to 1."
1623             else:
1624                C_v=self.__C_v
1625    
1626             if not safety_factor == None:
1627                if safety_factor<=0 or safety_factor>1:
1628                   raise ValueError,"safety_factor need to be between zero and one."
1629             else:
1630                safety_factor=self.__safety_factor
1631    
1632             if gamma<gamma_min:
1633                   raise ValueError,"gamma = %e needs to be greater or equal gamma_min = %e."%(gamma,gamma_min)
1634             if rtol_max<=rtol_min:
1635                   raise ValueError,"rtol_max = %e needs to be greater rtol_min = %e."%(rtol_max,rtol_min)
1636                
1637             self.__gamma = gamma
1638             self.__gamma_min = gamma_min
1639             self.__chi_max = chi_max
1640             self.__omega_div = omega_div
1641             self.__omega_conv = omega_conv
1642             self.__rtol_min = rtol_min
1643             self.__rtol_max = rtol_max
1644             self.__chi = chi
1645             self.__C_p = C_p
1646             self.__C_v = C_v
1647             self.__safety_factor = safety_factor
1648    
1649          #=============================================================
1650          def inner_pBv(self,p,Bv):
1651             """
1652             Returns inner product of element p and Bv (overwrite).
1653    
1654             :param p: a pressure increment
1655             :param Bv: a residual
1656             :return: inner product of element p and Bv
1657             :rtype: ``float``
1658             :note: used if PCG is applied.
1659             """
1660             raise NotImplementedError,"no inner product for p and Bv implemented."
1661    
1662          def inner_p(self,p0,p1):
1663             """
1664             Returns inner product of p0 and p1 (overwrite).
1665    
1666             :param p0: a pressure
1667             :param p1: a pressure
1668             :return: inner product of p0 and p1
1669             :rtype: ``float``
1670             """
1671             raise NotImplementedError,"no inner product for p implemented."
1672      
1673          def norm_v(self,v):
1674             """
1675             Returns the norm of v (overwrite).
1676    
1677     subiter_max=3           :param v: a velovity
1678     def solve(self,u0,p0,tolerance=1.e-6,tolerance_u=None,iter_max=100,accepted_reduction=0.995,relaxation=None):           :return: norm of v
1679             :rtype: non-negative ``float``
1680             """
1681             raise NotImplementedError,"no norm of v implemented."
1682          def getDV(self, p, v, tol):
1683             """
1684             return a correction to the value for a given v and a given p with accuracy `tol` (overwrite)
1685    
1686             :param p: pressure
1687             :param v: pressure
1688             :return: dv given as *dv= A^{-1} (f-A v-B^*p)*
1689             :note: Only *A* may depend on *v* and *p*
1690             """
1691             raise NotImplementedError,"no dv calculation implemented."
1692    
1693            
1694          def Bv(self,v, tol):
1695          """          """
1696          runs the solver          Returns Bv with accuracy `tol` (overwrite)
1697    
1698          @param u0: initial guess for C{u}          :rtype: equal to the type of p
1699          @type u0: L{esys.escript.Data}          :note: boundary conditions on p should be zero!
         @param p0: initial guess for C{p}  
         @type p0: L{esys.escript.Data}  
         @param tolerance: tolerance for relative error in C{u} and C{p}  
         @type tolerance: positive C{float}  
         @param tolerance_u: tolerance for relative error in C{u} if different from C{tolerance}  
         @type tolerance_u: positive C{float}  
         @param iter_max: maximum number of iteration steps.  
         @type iter_max: C{int}  
         @param accepted_reduction: if the norm  g cannot be reduced by C{accepted_reduction} backtracking to adapt the  
                                    relaxation factor. If C{accepted_reduction=None} no backtracking is used.  
         @type accepted_reduction: positive C{float} or C{None}  
         @param relaxation: initial relaxation factor. If C{relaxation==None}, the last relaxation factor is used.  
         @type relaxation: C{float} or C{None}  
1700          """          """
1701          tol=1.e-2          raise NotImplementedError, "no operator B implemented."
1702          if tolerance_u==None: tolerance_u=tolerance  
1703          if not relaxation==None: self.relaxation=relaxation        def norm_Bv(self,Bv):
1704          if accepted_reduction ==None:          """
1705                angle_limit=0.          Returns the norm of Bv (overwrite).
1706          elif accepted_reduction>=1.:  
1707                angle_limit=0.          :rtype: equal to the type of p
1708          else:          :note: boundary conditions on p should be zero!
1709                angle_limit=util.sqrt(1-accepted_reduction**2)          """
1710          self.iter=0          raise NotImplementedError, "no norm of Bv implemented."
1711          u=u0  
1712          p=p0        def solve_AinvBt(self,dp, tol):
1713          #           """
1714          #   initialize things:           Solves *A dv=B^*dp* with accuracy `tol`
1715          #  
1716          converged=False           :param dp: a pressure increment
1717          #           :return: the solution of *A dv=B^*dp*
1718          #  start loop:           :note: boundary conditions on dv should be zero! *A* is the operator used in ``getDV`` and must not be altered.
1719          #           """
1720          #  initial search direction is g           raise NotImplementedError,"no operator A implemented."
1721          #  
1722          while not converged :        def solve_prec(self,Bv, tol):
1723              if self.iter>iter_max:           """
1724                  raise ArithmeticError("no convergence after %s steps."%self.iter)           Provides a preconditioner for *(BA^{-1}B^ * )* applied to Bv with accuracy `tol`
1725              f_new=self.solve_f(u,p,tol)  
1726              norm_f_new = util.Lsup(f_new)           :rtype: equal to the type of p
1727              u_new=u-f_new           :note: boundary conditions on p should be zero!
1728              g_new=self.solve_g(u_new,tol)           """
1729              self.iter+=1           raise NotImplementedError,"no preconditioner for Schur complement implemented."
1730              norm_g_new = util.sqrt(self.inner(g_new,g_new))        #=============================================================
1731              if norm_f_new==0. and norm_g_new==0.: return u, p        def __Aprod_PCG(self,dp):
1732              if self.iter>1 and not accepted_reduction==None:            dv=self.solve_AinvBt(dp, self.__subtol)
1733                 #            return ArithmeticTuple(dv,self.Bv(dv, self.__subtol))
1734                 #   did we manage to reduce the norm of G? I  
1735                 #   if not we start a backtracking procedure        def __inner_PCG(self,p,r):
1736                 #           return self.inner_pBv(p,r[1])
1737                 # print "new/old norm = ",norm_g_new, norm_g, norm_g_new/norm_g  
1738                 if norm_g_new > accepted_reduction * norm_g:        def __Msolve_PCG(self,r):
1739                    sub_iter=0            return self.solve_prec(r[1], self.__subtol)
1740                    s=self.relaxation        #=============================================================
1741                    d=g        def __Aprod_GMRES(self,p):
1742                    g_last=g            return self.solve_prec(self.Bv(self.solve_AinvBt(p, self.__subtol), self.__subtol), self.__subtol)
1743                    self.trace("    start substepping: f = %s, g = %s, relaxation = %s."%(norm_f_new, norm_g_new, s))        def __inner_GMRES(self,p0,p1):
1744                    while sub_iter < self.subiter_max and  norm_g_new > accepted_reduction * norm_g:           return self.inner_p(p0,p1)
1745                       dg= g_new-g_last  
1746                       norm_dg=abs(util.sqrt(self.inner(dg,dg))/self.relaxation)        #=============================================================
1747                       rad=self.inner(g_new,dg)/self.relaxation        def norm_p(self,p):
1748                       # print "   ",sub_iter,": rad, norm_dg:",abs(rad), norm_dg*norm_g_new * angle_limit            """
1749                       # print "   ",sub_iter,": rad, norm_dg:",rad, norm_dg, norm_g_new, norm_g            calculates the norm of ``p``
                      if abs(rad) < norm_dg*norm_g_new * angle_limit:  
                          if sub_iter>0: self.trace("    no further improvements expected from backtracking.")  
                          break  
                      r=self.relaxation  
                      self.relaxation= - rad/norm_dg**2  
                      s+=self.relaxation  
                      #####  
                      # a=g_new+self.relaxation*dg/r  
                      # print "predicted new norm = ",util.sqrt(self.inner(a,a)),util.sqrt(self.inner(g_new,g_new)), self.relaxation  
                      #####  
                      g_last=g_new  
                      p+=self.relaxation*d  
                      f_new=self.solve_f(u,p,tol)  
                      u_new=u-f_new  
                      g_new=self.solve_g(u_new,tol)  
                      self.iter+=1  
                      norm_f_new = util.Lsup(f_new)  
                      norm_g_new = util.sqrt(self.inner(g_new,g_new))  
                      # print "   ",sub_iter," new g norm",norm_g_new  
                      self.trace("    %s th sub-step: f = %s, g = %s, relaxation = %s."%(sub_iter, norm_f_new, norm_g_new, s))  
                      #  
                      #   can we expect reduction of g?  
                      #  
                      # u_last=u_new  
                      sub_iter+=1  
                   self.relaxation=s  
             #  
             #  check for convergence:  
             #  
             norm_u_new = util.Lsup(u_new)  
             p_new=p+self.relaxation*g_new  
             norm_p_new = util.sqrt(self.inner(p_new,p_new))  
             self.trace("%s th step: f/u = %s, g/p = %s, relaxation = %s."%(self.iter,norm_f_new/norm_u_new, norm_g_new/norm_p_new, self.relaxation))  
   
             if self.iter>1:  
                dg2=g_new-g  
                df2=f_new-f  
                norm_dg2=util.sqrt(self.inner(dg2,dg2))  
                norm_df2=util.Lsup(df2)  
                # print norm_g_new, norm_g, norm_dg, norm_p, tolerance  
                tol_eq_g=tolerance*norm_dg2/(norm_g*abs(self.relaxation))*norm_p_new  
                tol_eq_f=tolerance_u*norm_df2/norm_f*norm_u_new  
                if norm_g_new <= tol_eq_g and norm_f_new <= tol_eq_f:  
                    converged=True  
             f, norm_f, u, norm_u, g, norm_g, p, norm_p = f_new, norm_f_new, u_new, norm_u_new, g_new, norm_g_new, p_new, norm_p_new  
         self.trace("convergence after %s steps."%self.iter)  
         return u,p  
 #   def solve(self,u0,p0,tolerance=1.e-6,iter_max=10,self.relaxation=1.):  
 #      tol=1.e-2  
 #      iter=0  
 #      converged=False  
 #      u=u0*1.  
 #      p=p0*1.  
 #      while not converged and iter<iter_max:  
 #          du=self.solve_f(u,p,tol)  
 #          u-=du  
 #          norm_du=util.Lsup(du)  
 #          norm_u=util.Lsup(u)  
 #          
 #          dp=self.relaxation*self.solve_g(u,tol)  
 #          p+=dp  
 #          norm_dp=util.sqrt(self.inner(dp,dp))  
 #          norm_p=util.sqrt(self.inner(p,p))  
 #          print iter,"-th step rel. errror u,p= (%s,%s) (%s,%s)(%s,%s)"%(norm_du,norm_dp,norm_du/norm_u,norm_dp/norm_p,norm_u,norm_p)  
 #          iter+=1  
 #  
 #          converged = (norm_du <= tolerance*norm_u) and  (norm_dp <= tolerance*norm_p)  
 #      if converged:  
 #          print "convergence after %s steps."%iter  
 #      else:  
 #          raise ArithmeticError("no convergence after %s steps."%iter)  
 #  
 #      return u,p  
1750                        
1751  # vim: expandtab shiftwidth=4:            :param p: a pressure
1752              :return: the norm of ``p`` using the inner product for pressure
1753              :rtype: ``float``
1754              """
1755              f=self.inner_p(p,p)
1756              if f<0: raise ValueError,"negative pressure norm."
1757              return math.sqrt(f)
1758          
1759          def solve(self,v,p,max_iter=20, verbose=False, usePCG=True, iter_restart=20, max_correction_steps=10):
1760             """
1761             Solves the saddle point problem using initial guesses v and p.
1762    
1763             :param v: initial guess for velocity
1764             :param p: initial guess for pressure
1765             :type v: `Data`
1766             :type p: `Data`
1767             :param usePCG: indicates the usage of the PCG rather than GMRES scheme.
1768             :param max_iter: maximum number of iteration steps per correction
1769                              attempt
1770             :param verbose: if True, shows information on the progress of the
1771                             saddlepoint problem solver.
1772             :param iter_restart: restart the iteration after ``iter_restart`` steps
1773                                  (only used if useUzaw=False)
1774             :type usePCG: ``bool``
1775             :type max_iter: ``int``
1776             :type verbose: ``bool``
1777             :type iter_restart: ``int``
1778             :rtype: ``tuple`` of `Data` objects
1779             :note: typically this method is overwritten by a subclass. It provides a wrapper for the ``_solve`` method.
1780             """
1781             return self._solve(v=v,p=p,max_iter=max_iter,verbose=verbose, usePCG=usePCG, iter_restart=iter_restart, max_correction_steps=max_correction_steps)
1782    
1783          def _solve(self,v,p,max_iter=20, verbose=False, usePCG=True, iter_restart=20, max_correction_steps=10):
1784             """
1785             see `_solve` method.
1786             """
1787             self.verbose=verbose
1788             rtol=self.getTolerance()
1789             atol=self.getAbsoluteTolerance()
1790             correction_step=0
1791             converged=False
1792             error=None
1793             chi=None
1794             gamma=self.__gamma
1795             C_p=self.__C_p
1796             C_v=self.__C_v
1797             while not converged:
1798                  if error== None or chi == None:
1799                      gamma_new=gamma/self.__omega_conv
1800                  else:
1801                     if chi < self.__chi_max:
1802                        gamma_new=min(max(gamma*self.__omega_conv,1-chi*error/(self.__safety_factor*ATOL)), 1-chi/self.__chi_max)
1803                     else:
1804                        gamma_new=gamma*self.__omega_div
1805                  gamma=max(gamma_new, self.__gamma_min)
1806                  # calculate velocity for current pressure:
1807                  rtol_v=min(max(gamma/(1.+gamma)/C_v,self.__rtol_min), self.__rtol_max)
1808                  rtol_p=min(max(gamma/C_p, self.__rtol_min), self.__rtol_max)
1809                  self.__subtol=rtol_p**2
1810                  if self.verbose: print "HomogeneousSaddlePointProblem: step %s: gamma = %e, rtol_v= %e, rtol_p=%e"%(correction_step,gamma,rtol_v,rtol_p)
1811                  if self.verbose: print "HomogeneousSaddlePointProblem: subtolerance: %e"%self.__subtol
1812                  # calculate velocity for current pressure: A*dv= F-A*v-B*p
1813                  dv1=self.getDV(p,v,rtol_v)
1814                  v1=v+dv1
1815                  Bv1=self.Bv(v1, self.__subtol)
1816                  norm_Bv1=self.norm_Bv(Bv1)
1817                  norm_dv1=self.norm_v(dv1)
1818                  norm_v1=self.norm_v(v1)
1819                  ATOL=norm_v1*rtol+atol
1820                  if self.verbose: print "HomogeneousSaddlePointProblem: step %s: Bv = %e, dv = %e, v=%e"%(correction_step,norm_Bv1, norm_dv1, norm_v1)
1821                  if not ATOL>0: raise ValueError,"overall absolute tolerance needs to be positive."
1822                  if max(norm_Bv1, norm_dv1) <= ATOL:
1823                      converged=True
1824                      v=v1
1825                  else:
1826                      # now we solve for the pressure increment dp from B*A^{-1}B^* dp = Bv1
1827                      if usePCG:
1828                        dp,r,a_norm=PCG(ArithmeticTuple(v1,Bv1),self.__Aprod_PCG,0*p,self.__Msolve_PCG,self.__inner_PCG,atol=0, rtol=rtol_p,iter_max=max_iter, verbose=self.verbose)
1829                        v2=r[0]
1830                        Bv2=r[1]
1831                      else:
1832                        dp=GMRES(self.solve_prec(Bv1,self.__subtol),self.__Aprod_GMRES, 0*p, self.__inner_GMRES,atol=0, rtol=rtol_p,iter_max=max_iter, iter_restart=iter_restart, verbose=self.verbose)
1833                        dv2=self.solve_AinvBt(dp, self.__subtol)
1834                        v2=v1-dv2
1835                        Bv2=self.Bv(v2, self.__subtol)
1836                      #
1837                      # convergence indicators:
1838                      #
1839                      norm_v2=self.norm_v(v2)
1840                      norm_dv2=self.norm_v(v2-v)
1841                      error_new=max(norm_dv2, norm_Bv1)
1842                      norm_Bv2=self.norm_Bv(Bv2)
1843                      if self.verbose: print "HomogeneousSaddlePointProblem: step %s (part 2): Bv = %e, dv = %e, v=%e"%(correction_step,norm_Bv2, norm_dv2, norm_v2)
1844                      if error !=None:
1845                          chi_new=error_new/error
1846                          if self.verbose: print "HomogeneousSaddlePointProblem: step %s: convergence rate = %e, est. error = %e"%(correction_step,chi_new, error_new)
1847                          if chi != None:
1848                              gamma0=max(gamma, 1-chi/chi_new)
1849                              C_p*=gamma0/gamma
1850                              C_v*=gamma0/gamma*(1+gamma)/(1+gamma0)
1851                          chi=chi_new
1852                      else:
1853                          if self.verbose: print "HomogeneousSaddlePointProblem: step %s: est. error = %e"%(correction_step, error_new)
1854    
1855                      error = error_new
1856                      correction_step+=1
1857                      if correction_step>max_correction_steps:
1858                            raise CorrectionFailed,"Given up after %d correction steps."%correction_step
1859                      v,p=v2,p+dp
1860             if self.verbose: print "HomogeneousSaddlePointProblem: tolerance reached after %s steps."%correction_step
1861         return v,p
1862    
1863          #========================================================================
1864          def setTolerance(self,tolerance=1.e-4):
1865             """
1866             Sets the relative tolerance for (v,p).
1867    
1868             :param tolerance: tolerance to be used
1869             :type tolerance: non-negative ``float``
1870             """
1871             if tolerance<0:
1872                 raise ValueError,"tolerance must be positive."
1873             self.__rtol=tolerance
1874    
1875          def getTolerance(self):
1876             """
1877             Returns the relative tolerance.
1878    
1879             :return: relative tolerance
1880             :rtype: ``float``
1881             """
1882             return self.__rtol
1883    
1884          def setAbsoluteTolerance(self,tolerance=0.):
1885             """
1886             Sets the absolute tolerance.
1887    
1888             :param tolerance: tolerance to be used
1889             :type tolerance: non-negative ``float``
1890             """
1891             if tolerance<0:
1892                 raise ValueError,"tolerance must be non-negative."
1893             self.__atol=tolerance
1894    
1895          def getAbsoluteTolerance(self):
1896             """
1897             Returns the absolute tolerance.
1898    
1899             :return: absolute tolerance
1900             :rtype: ``float``
1901             """
1902             return self.__atol
1903    
1904    
1905    def MaskFromBoundaryTag(domain,*tags):
1906       """
1907       Creates a mask on the Solution(domain) function space where the value is
1908       one for samples that touch the boundary tagged by tags.
1909    
1910       Usage: m=MaskFromBoundaryTag(domain, "left", "right")
1911    
1912       :param domain: domain to be used
1913       :type domain: `escript.Domain`
1914       :param tags: boundary tags
1915       :type tags: ``str``
1916       :return: a mask which marks samples that are touching the boundary tagged
1917                by any of the given tags
1918       :rtype: `escript.Data` of rank 0
1919       """
1920       pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)
1921       d=escript.Scalar(0.,escript.FunctionOnBoundary(domain))
1922       for t in tags: d.setTaggedValue(t,1.)
1923       pde.setValue(y=d)
1924       return util.whereNonZero(pde.getRightHandSide())
1925    
1926    def MaskFromTag(domain,*tags):
1927       """
1928       Creates a mask on the Solution(domain) function space where the value is
1929       one for samples that touch regions tagged by tags.
1930    
1931       Usage: m=MaskFromTag(domain, "ham")
1932    
1933       :param domain: domain to be used
1934       :type domain: `escript.Domain`
1935       :param tags: boundary tags
1936       :type tags: ``str``
1937       :return: a mask which marks samples that are touching the boundary tagged
1938                by any of the given tags
1939       :rtype: `escript.Data` of rank 0
1940       """
1941       pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)
1942       d=escript.Scalar(0.,escript.Function(domain))
1943       for t in tags: d.setTaggedValue(t,1.)
1944       pde.setValue(Y=d)
1945       return util.whereNonZero(pde.getRightHandSide())
1946    
1947    

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