/[escript]/trunk/escript/py_src/pdetools.py
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revision 351 by gross, Tue Dec 13 09:12:15 2005 UTC revision 4446 by caltinay, Tue Jun 11 04:00:15 2013 UTC
# Line 1  Line 1 
1  # $Id$  
2    ##############################################################################
3    #
4    # Copyright (c) 2003-2013 by University of Queensland
5    # http://www.uq.edu.au
6    #
7    # Primary Business: Queensland, Australia
8    # Licensed under the Open Software License version 3.0
9    # http://www.opensource.org/licenses/osl-3.0.php
10    #
11    # Development until 2012 by Earth Systems Science Computational Center (ESSCC)
12    # Development since 2012 by School of Earth Sciences
13    #
14    ##############################################################################
15    
16    __copyright__="""Copyright (c) 2003-2013 by University of Queensland
17    http://www.uq.edu.au
18    Primary Business: Queensland, Australia"""
19    __license__="""Licensed under the Open Software License version 3.0
20    http://www.opensource.org/licenses/osl-3.0.php"""
21    __url__="https://launchpad.net/escript-finley"
22    
23  """  """
24  Provides some tools related to PDEs.  Provides some tools related to PDEs.
25    
26  Currently includes:  Currently includes:
27      - Projector - to project a discontinuous      - Projector - to project a discontinuous function onto a continuous function
28      - Locator - to trace values in data objects at a certain location      - Locator - to trace values in data objects at a certain location
29      - TimeIntegrationManager - to handel extraplotion in time      - TimeIntegrationManager - to handle extrapolation in time
30        - SaddlePointProblem - solver for Saddle point problems using the inexact uszawa scheme
31    
32    :var __author__: name of author
33    :var __copyright__: copyrights
34    :var __license__: licence agreement
35    :var __url__: url entry point on documentation
36    :var __version__: version
37    :var __date__: date of the version
38  """  """
39    
40  import escript  __author__="Lutz Gross, l.gross@uq.edu.au"
41  import linearPDEs  
42  import numarray  
43  import util  from . import escriptcpp as escore
44    from . import linearPDEs
45  class TimeIntegrationManager:  from . import util
46    """  import math
47    a simple mechanism to manage time dependend values.  import numpy
48    
49    typical usage is:  class TimeIntegrationManager(object):
50      """
51    dt=0.1 # time increment    A simple mechanism to manage time dependend values.
52    tm=TimeIntegrationManager(inital_value,p=1)  
53    while t<1.    Typical usage is::
54        v_guess=tm.extrapolate(dt) # extrapolate to t+dt  
55        v=...       dt=0.1 # time increment
56        tm.checkin(dt,v)       tm=TimeIntegrationManager(inital_value,p=1)
57        t+=dt       while t<1.
58             v_guess=tm.extrapolate(dt) # extrapolate to t+dt
59             v=...
60             tm.checkin(dt,v)
61             t+=dt
62    
63    @remark: currently only p=1 is supported.    :note: currently only p=1 is supported.
64    """    """
65    def __init__(self,*inital_values,**kwargs):    def __init__(self,*inital_values,**kwargs):
66       """       """
67       sets up the value manager where 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
68         and p is the order used for extrapolation.
69       """       """
70       if kwargs.has_key("p"):       if "p" in kwargs:
71              self.__p=kwargs["p"]              self.__p=kwargs["p"]
72       else:       else:
73              self.__p=1              self.__p=1
74       if kwargs.has_key("time"):       if "time" in kwargs:
75              self.__t=kwargs["time"]              self.__t=kwargs["time"]
76       else:       else:
77              self.__t=0.              self.__t=0.
# Line 50  class TimeIntegrationManager: Line 83  class TimeIntegrationManager:
83    def getTime(self):    def getTime(self):
84        return self.__t        return self.__t
85    
86      def getValue(self):
87          out=self.__v_mem[0]
88          if len(out)==1:
89              return out[0]
90          else:
91              return out
92    
93    def checkin(self,dt,*values):    def checkin(self,dt,*values):
94        """        """
95        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.
96        """        """
97        o=min(self.__order+1,self.__p)        o=min(self.__order+1,self.__p)
98        self.__order=min(self.__order+1,self.__p)        self.__order=min(self.__order+1,self.__p)
# Line 69  class TimeIntegrationManager: Line 109  class TimeIntegrationManager:
109    
110    def extrapolate(self,dt):    def extrapolate(self,dt):
111        """        """
112        extrapolates to dt forward in time.        Extrapolates to ``dt`` forward in time.
113        """        """
114        if self.__order==0:        if self.__order==0:
115           out=self.__v_mem[0]           out=self.__v_mem[0]
# Line 85  class TimeIntegrationManager: Line 125  class TimeIntegrationManager:
125        else:        else:
126           return out           return out
127    
128  class Projector:  
129    class Projector(object):
130    """    """
131    The Projector is a factory which projects a discontiuous function onto a    The Projector is a factory which projects a discontinuous function onto a
132    continuous function on the a given domain.    continuous function on a given domain.
133    """    """
134    def __init__(self, domain, reduce = True, fast=True):    def __init__(self, domain, reduce=True, fast=True):
135      """      """
136      Create a continuous function space projector for a domain.      Creates a continuous function space projector for a domain.
137    
138      @param domain: Domain of the projection.      :param domain: Domain of the projection.
139      @param reduce: Flag to reduce projection order (default is True)      :param reduce: Flag to reduce projection order
140      @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
141      """      """
142      self.__pde = linearPDEs.LinearPDE(domain)      self.__pde = linearPDEs.LinearPDE(domain)
143      if fast:      if fast:
144        self.__pde.setSolverMethod(linearPDEs.LinearPDE.LUMPING)          self.__pde.getSolverOptions().setSolverMethod(linearPDEs.SolverOptions.LUMPING)
145      self.__pde.setSymmetryOn()      self.__pde.setSymmetryOn()
146      self.__pde.setReducedOrderTo(reduce)      self.__pde.setReducedOrderTo(reduce)
147      self.__pde.setValue(D = 1.)      self.__pde.setValue(D = 1.)
148      return      return
149      def getSolverOptions(self):
150        """
151        Returns the solver options of the PDE solver.
152        
153        :rtype: `linearPDEs.SolverOptions`
154        """
155        return self.__pde.getSolverOptions()
156    
157    def __del__(self):    def getValue(self, input_data):
158      return      """
159        Projects ``input_data`` onto a continuous function.
160    
161        :param input_data: the data to be projected
162        """
163        return self(input_data)
164    
165    def __call__(self, input_data):    def __call__(self, input_data):
166      """      """
167      Projects input_data onto a continuous function      Projects ``input_data`` onto a continuous function.
168    
169      @param input_data: The input_data to be projected.      :param input_data: the data to be projected
170      """      """
171      out=escript.Data(0.,input_data.getShape(),what=escript.ContinuousFunction(self.__pde.getDomain()))      out=escore.Data(0.,input_data.getShape(),self.__pde.getFunctionSpaceForSolution())
172        self.__pde.setValue(Y = escore.Data(), Y_reduced = escore.Data())
173      if input_data.getRank()==0:      if input_data.getRank()==0:
174          self.__pde.setValue(Y = input_data)          self.__pde.setValue(Y = input_data)
175          out=self.__pde.getSolution()          out=self.__pde.getSolution()
# Line 143  class Projector: Line 197  class Projector:
197                      out[i0,i1,i2,i3]=self.__pde.getSolution()                      out[i0,i1,i2,i3]=self.__pde.getSolution()
198      return out      return out
199    
200    class NoPDE(object):
201         """
202         Solves the following problem for u:
203    
204         *kronecker[i,j]*D[j]*u[j]=Y[i]*
205    
206         with constraint
207    
208         *u[j]=r[j]*  where *q[j]>0*
209    
210         where *D*, *Y*, *r* and *q* are given functions of rank 1.
211    
212         In the case of scalars this takes the form
213    
214  class Locator:       *D*u=Y*
215    
216         with constraint
217    
218         *u=r* where *q>0*
219    
220         where *D*, *Y*, *r* and *q* are given scalar functions.
221    
222         The constraint overwrites any other condition.
223    
224         :note: This class is similar to the `linearPDEs.LinearPDE` class with
225                A=B=C=X=0 but has the intention that all input parameters are given
226                in `Solution` or `ReducedSolution`.
227         """
228         # The whole thing is a bit strange and I blame Rob Woodcock (CSIRO) for
229         # this.
230         def __init__(self,domain,D=None,Y=None,q=None,r=None):
231             """
232             Initializes the problem.
233    
234             :param domain: domain of the PDE
235             :type domain: `Domain`
236             :param D: coefficient of the solution
237             :type D: ``float``, ``int``, ``numpy.ndarray``, `Data`
238             :param Y: right hand side
239             :type Y: ``float``, ``int``, ``numpy.ndarray``, `Data`
240             :param q: location of constraints
241             :type q: ``float``, ``int``, ``numpy.ndarray``, `Data`
242             :param r: value of solution at locations of constraints
243             :type r: ``float``, ``int``, ``numpy.ndarray``, `Data`
244             """
245             self.__domain=domain
246             self.__D=D
247             self.__Y=Y
248             self.__q=q
249             self.__r=r
250             self.__u=None
251             self.__function_space=escore.Solution(self.__domain)
252    
253         def setReducedOn(self):
254             """
255             Sets the `FunctionSpace` of the solution to `ReducedSolution`.
256             """
257             self.__function_space=escore.ReducedSolution(self.__domain)
258             self.__u=None
259    
260         def setReducedOff(self):
261             """
262             Sets the `FunctionSpace` of the solution to `Solution`.
263             """
264             self.__function_space=escore.Solution(self.__domain)
265             self.__u=None
266    
267         def setValue(self,D=None,Y=None,q=None,r=None):
268             """
269             Assigns values to the parameters.
270    
271             :param D: coefficient of the solution
272             :type D: ``float``, ``int``, ``numpy.ndarray``, `Data`
273             :param Y: right hand side
274             :type Y: ``float``, ``int``, ``numpy.ndarray``, `Data`
275             :param q: location of constraints
276             :type q: ``float``, ``int``, ``numpy.ndarray``, `Data`
277             :param r: value of solution at locations of constraints
278             :type r: ``float``, ``int``, ``numpy.ndarray``, `Data`
279             """
280             if not D==None:
281                self.__D=D
282                self.__u=None
283             if not Y==None:
284                self.__Y=Y
285                self.__u=None
286             if not q==None:
287                self.__q=q
288                self.__u=None
289             if not r==None:
290                self.__r=r
291                self.__u=None
292    
293         def getSolution(self):
294             """
295             Returns the solution.
296    
297             :return: the solution of the problem
298             :rtype: `Data` object in the `FunctionSpace` `Solution` or
299                     `ReducedSolution`
300             """
301             if self.__u==None:
302                if self.__D==None:
303                   raise ValueError("coefficient D is undefined")
304                D=escore.Data(self.__D,self.__function_space)
305                if D.getRank()>1:
306                   raise ValueError("coefficient D must have rank 0 or 1")
307                if self.__Y==None:
308                   self.__u=escore.Data(0.,D.getShape(),self.__function_space)
309                else:
310                   self.__u=1./D*self.__Y
311                if not self.__q==None:
312                    q=util.wherePositive(escore.Data(self.__q,self.__function_space))
313                    self.__u*=(1.-q)
314                    if not self.__r==None: self.__u+=q*self.__r
315             return self.__u
316    
317    class Locator(object):
318       """       """
319       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
320       spatial coordinate x.         coordinate x.
321        
322       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
323       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.  
324       """       """
325    
326       def __init__(self,where,x=numarray.zeros((3,))):       def __init__(self,where,x=numpy.zeros((3,))):
327         """         """
328         Initializes a Locator to access values in Data objects on the Doamin         Initializes a Locator to access values in Data objects on the Doamin
329         or FunctionSpace where for the sample point which         or FunctionSpace for the sample point which is closest to the given
330         closest to the given point x.         point x.
331    
332           :param where: function space
333           :type where: `escript.FunctionSpace`
334           :param x: location(s) of the Locator
335           :type x: ``numpy.ndarray`` or ``list`` of ``numpy.ndarray``
336         """         """
337         if isinstance(where,escript.FunctionSpace):         if isinstance(where,escore.FunctionSpace):
338            self.__function_space=where            self.__function_space=where
339         else:         else:
340            self.__function_space=escript.ContinuousFunction(where)            self.__function_space=escore.ContinuousFunction(where)
341         self.__id=util.length(x[:self.__function_space.getDim()]-self.__function_space.getX()).mindp()         iterative=False
342           if isinstance(x, list):
343               if len(x)==0:
344                  raise ValueError("At least one point must be given.")
345               try:
346                 iter(x[0])
347                 iterative=True
348               except TypeError:
349                 iterative=False
350           xxx=self.__function_space.getX()
351           if iterative:
352               self.__id=[]
353               for p in x:
354                  self.__id.append(util.length(xxx-p[:self.__function_space.getDim()]).minGlobalDataPoint())
355           else:
356               self.__id=util.length(xxx-x[:self.__function_space.getDim()]).minGlobalDataPoint()
357    
358       def __str__(self):       def __str__(self):
359         """         """
360         Returns the coordinates of the Locator as a string.         Returns the coordinates of the Locator as a string.
361         """         """
362         return "<Locator %s>"%str(self.getX())         x=self.getX()
363           if isinstance(x,list):
364              out="["
365              first=True
366              for xx in x:
367                if not first:
368                    out+=","
369                else:
370                    first=False
371                out+=str(xx)
372              out+="]>"
373           else:
374              out=str(x)
375           return out
376    
377         def getX(self):
378            """
379            Returns the exact coordinates of the Locator.
380            """
381            return self(self.getFunctionSpace().getX())
382    
383       def getFunctionSpace(self):       def getFunctionSpace(self):
384          """          """
385      Returns the function space of the Locator.          Returns the function space of the Locator.
386      """          """
387          return self.__function_space          return self.__function_space
388    
389       def getId(self):       def getId(self,item=None):
390          """          """
391      Returns the identifier of the location.          Returns the identifier of the location.
     """  
         return self.__id  
   
      def getX(self):  
392          """          """
393      Returns the exact coordinates of the Locator.          if item == None:
394      """             return self.__id
395          return self(self.getFunctionSpace().getX())          else:
396               if isinstance(self.__id,list):
397                  return self.__id[item]
398               else:
399                  return self.__id
400    
401    
402       def __call__(self,data):       def __call__(self,data):
403          """          """
404      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.
405      the object is returned.          """
     """  
406          return self.getValue(data)          return self.getValue(data)
407    
408       def getValue(self,data):       def getValue(self,data):
409          """          """
410      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`
411      otherwise the object is returned.          object otherwise the object is returned.
412      """          """
413          if isinstance(data,escript.Data):          if isinstance(data,escore.Data):
414             if data.getFunctionSpace()==self.getFunctionSpace():             dat=util.interpolate(data,self.getFunctionSpace())
415               out=data.convertToNumArrayFromDPNo(self.getId()[0],self.getId()[1])             id=self.getId()
416             else:             r=data.getRank()
417               out=data.interpolate(self.getFunctionSpace()).convertToNumArrayFromDPNo(self.getId()[0],self.getId()[1])             if isinstance(id,list):
418             if data.getRank()==0:                 out=[]
419                return out[0]                 for i in id:
420                      o=numpy.array(dat.getTupleForGlobalDataPoint(*i))
421                      if data.getRank()==0:
422                         out.append(o[0])
423                      else:
424                         out.append(o)
425                   return out
426             else:             else:
427                return out               out=numpy.array(dat.getTupleForGlobalDataPoint(*id))
428                 if data.getRank()==0:
429                    return out[0]
430                 else:
431                    return out
432          else:          else:
433             return data             return data
434              
435         def setValue(self, data, v):
436          """
437          Sets the value of the ``data`` at the Locator.
438          """
439          if isinstance(data, escore.Data):
440             if data.getFunctionSpace()!=self.getFunctionSpace():
441               raise TypeError("setValue: FunctionSpace of Locator and Data object must match.")
442             data.expand()  
443             id=self.getId()
444             if isinstance(id, list):
445              for i in id:
446               data._setTupleForGlobalDataPoint(i[1], i[0], v)
447             else:
448               data._setTupleForGlobalDataPoint(id[1], id[0], v)
449          else:
450               raise TypeError("setValue: Invalid argument type.")
451    
452    
453    def getInfLocator(arg):
454        """
455        Return a Locator for a point with the inf value over all arg.
456        """
457        if not isinstance(arg, escore.Data):
458           raise TypeError("getInfLocator: Unknown argument type.")
459        a_inf=util.inf(arg)
460        loc=util.length(arg-a_inf).minGlobalDataPoint()     # This gives us the location but not coords
461        x=arg.getFunctionSpace().getX()
462        x_min=x.getTupleForGlobalDataPoint(*loc)
463        return Locator(arg.getFunctionSpace(),x_min)
464    
465    def getSupLocator(arg):
466        """
467        Return a Locator for a point with the sup value over all arg.
468        """
469        if not isinstance(arg, escore.Data):
470           raise TypeError("getInfLocator: Unknown argument type.")
471        a_inf=util.sup(arg)
472        loc=util.length(arg-a_inf).minGlobalDataPoint()     # This gives us the location but not coords
473        x=arg.getFunctionSpace().getX()
474        x_min=x.getTupleForGlobalDataPoint(*loc)
475        return Locator(arg.getFunctionSpace(),x_min)
476            
477    
478    class SolverSchemeException(Exception):
479       """
480       This is a generic exception thrown by solvers.
481       """
482       pass
483    
484    class IndefinitePreconditioner(SolverSchemeException):
485       """
486       Exception thrown if the preconditioner is not positive definite.
487       """
488       pass
489    
490    class MaxIterReached(SolverSchemeException):
491       """
492       Exception thrown if the maximum number of iteration steps is reached.
493       """
494       pass
495    
496    class CorrectionFailed(SolverSchemeException):
497       """
498       Exception thrown if no convergence has been achieved in the solution
499       correction scheme.
500       """
501       pass
502    
503    class IterationBreakDown(SolverSchemeException):
504       """
505       Exception thrown if the iteration scheme encountered an incurable breakdown.
506       """
507       pass
508    
509    class NegativeNorm(SolverSchemeException):
510       """
511       Exception thrown if a norm calculation returns a negative norm.
512       """
513       pass
514    
515    def PCG(r, Aprod, x, Msolve, bilinearform, atol=0, rtol=1.e-8, iter_max=100, initial_guess=True, verbose=False):
516       """
517       Solver for
518    
519       *Ax=b*
520    
521       with a symmetric and positive definite operator A (more details required!).
522       It uses the conjugate gradient method with preconditioner M providing an
523       approximation of A.
524    
525       The iteration is terminated if
526    
527       *|r| <= atol+rtol*|r0|*
528    
529       where *r0* is the initial residual and *|.|* is the energy norm. In fact
530    
531       *|r| = sqrt( bilinearform(Msolve(r),r))*
532    
533       For details on the preconditioned conjugate gradient method see the book:
534    
535       I{Templates for the Solution of Linear Systems by R. Barrett, M. Berry,
536       T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo,
537       C. Romine, and H. van der Vorst}.
538    
539       :param r: initial residual *r=b-Ax*. ``r`` is altered.
540       :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
541       :param x: an initial guess for the solution
542       :type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
543       :param Aprod: returns the value Ax
544       :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
545                    argument ``x``. The returned object needs to be of the same type
546                    like argument ``r``.
547       :param Msolve: solves Mx=r
548       :type Msolve: function ``Msolve(r)`` where ``r`` is of the same type like
549                     argument ``r``. The returned object needs to be of the same
550                     type like argument ``x``.
551       :param bilinearform: inner product ``<x,r>``
552       :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
553                           type like argument ``x`` and ``r`` is. The returned value
554                           is a ``float``.
555       :param atol: absolute tolerance
556       :type atol: non-negative ``float``
557       :param rtol: relative tolerance
558       :type rtol: non-negative ``float``
559       :param iter_max: maximum number of iteration steps
560       :type iter_max: ``int``
561       :return: the solution approximation and the corresponding residual
562       :rtype: ``tuple``
563       :warning: ``r`` and ``x`` are altered.
564       """
565       iter=0
566       rhat=Msolve(r)
567       d = rhat
568       rhat_dot_r = bilinearform(rhat, r)
569       if rhat_dot_r<0: raise NegativeNorm("negative norm.")
570       norm_r0=math.sqrt(rhat_dot_r)
571       atol2=atol+rtol*norm_r0
572       if atol2<=0:
573          raise ValueError("Non-positive tolarance.")
574       atol2=max(atol2, 100. * util.EPSILON * norm_r0)
575    
576       if verbose: print(("PCG: initial residual norm = %e (absolute tolerance = %e)"%(norm_r0, atol2)))
577    
578    
579       while not math.sqrt(rhat_dot_r) <= atol2:
580           iter+=1
581           if iter  >= iter_max: raise MaxIterReached("maximum number of %s steps reached."%iter_max)
582    
583           q=Aprod(d)
584           alpha = rhat_dot_r / bilinearform(d, q)
585           x += alpha * d
586           if isinstance(q,ArithmeticTuple):
587              r += q * (-alpha)      # Doing it the other way calls the float64.__mul__ not AT.__rmul__
588           else:
589               r += (-alpha) * q
590           rhat=Msolve(r)
591           rhat_dot_r_new = bilinearform(rhat, r)
592           beta = rhat_dot_r_new / rhat_dot_r
593           rhat+=beta * d
594           d=rhat
595    
596           rhat_dot_r = rhat_dot_r_new
597           if rhat_dot_r<0: raise NegativeNorm("negative norm.")
598           if verbose: print(("PCG: iteration step %s: residual norm = %e"%(iter, math.sqrt(rhat_dot_r))))
599       if verbose: print(("PCG: tolerance reached after %s steps."%iter))
600       return x,r,math.sqrt(rhat_dot_r)
601    
602    class Defect(object):
603        """
604        Defines a non-linear defect F(x) of a variable x.
605        """
606        def __init__(self):
607            """
608            Initializes defect.
609            """
610            self.setDerivativeIncrementLength()
611    
612        def bilinearform(self, x0, x1):
613            """
614            Returns the inner product of x0 and x1
615    
616            :param x0: value for x0
617            :param x1: value for x1
618            :return: the inner product of x0 and x1
619            :rtype: ``float``
620            """
621            return 0
622    
623        def norm(self,x):
624            """
625            Returns the norm of argument ``x``.
626    
627            :param x: a value
628            :return: norm of argument x
629            :rtype: ``float``
630            :note: by default ``sqrt(self.bilinearform(x,x)`` is returned.
631            """
632            s=self.bilinearform(x,x)
633            if s<0: raise NegativeNorm("negative norm.")
634            return math.sqrt(s)
635    
636        def eval(self,x):
637            """
638            Returns the value F of a given ``x``.
639    
640            :param x: value for which the defect ``F`` is evaluated
641            :return: value of the defect at ``x``
642            """
643            return 0
644    
645        def __call__(self,x):
646            return self.eval(x)
647    
648        def setDerivativeIncrementLength(self,inc=1000.*math.sqrt(util.EPSILON)):
649            """
650            Sets the relative length of the increment used to approximate the
651            derivative of the defect. The increment is inc*norm(x)/norm(v)*v in the
652            direction of v with x as a starting point.
653    
654            :param inc: relative increment length
655            :type inc: positive ``float``
656            """
657            if inc<=0: raise ValueError("positive increment required.")
658            self.__inc=inc
659    
660        def getDerivativeIncrementLength(self):
661            """
662            Returns the relative increment length used to approximate the
663            derivative of the defect.
664            :return: value of the defect at ``x``
665            :rtype: positive ``float``
666            """
667            return self.__inc
668    
669        def derivative(self, F0, x0, v, v_is_normalised=True):
670            """
671            Returns the directional derivative at ``x0`` in the direction of ``v``.
672    
673            :param F0: value of this defect at x0
674            :param x0: value at which derivative is calculated
675            :param v: direction
676            :param v_is_normalised: True to indicate that ``v`` is nomalized
677                                    (self.norm(v)=0)
678            :return: derivative of this defect at x0 in the direction of ``v``
679            :note: by default numerical evaluation (self.eval(x0+eps*v)-F0)/eps is
680                   used but this method maybe overwritten to use exact evaluation.
681            """
682            normx=self.norm(x0)
683            if normx>0:
684                 epsnew = self.getDerivativeIncrementLength() * normx
685            else:
686                 epsnew = self.getDerivativeIncrementLength()
687            if not v_is_normalised:
688                normv=self.norm(v)
689                if normv<=0:
690                   return F0*0
691                else:
692                   epsnew /= normv
693            F1=self.eval(x0 + epsnew * v)
694            return (F1-F0)/epsnew
695    
696    ######################################
697    def NewtonGMRES(defect, x, iter_max=100, sub_iter_max=20, atol=0,rtol=1.e-4, subtol_max=0.5, gamma=0.9, verbose=False):
698       """
699       Solves a non-linear problem *F(x)=0* for unknown *x* using the stopping
700       criterion:
701    
702       *norm(F(x) <= atol + rtol * norm(F(x0)*
703    
704       where *x0* is the initial guess.
705    
706       :param defect: object defining the function *F*. ``defect.norm`` defines the
707                      *norm* used in the stopping criterion.
708       :type defect: `Defect`
709       :param x: initial guess for the solution, ``x`` is altered.
710       :type x: any object type allowing basic operations such as
711                ``numpy.ndarray``, `Data`
712       :param iter_max: maximum number of iteration steps
713       :type iter_max: positive ``int``
714       :param sub_iter_max: maximum number of inner iteration steps
715       :type sub_iter_max: positive ``int``
716       :param atol: absolute tolerance for the solution
717       :type atol: positive ``float``
718       :param rtol: relative tolerance for the solution
719       :type rtol: positive ``float``
720       :param gamma: tolerance safety factor for inner iteration
721       :type gamma: positive ``float``, less than 1
722       :param subtol_max: upper bound for inner tolerance
723       :type subtol_max: positive ``float``, less than 1
724       :return: an approximation of the solution with the desired accuracy
725       :rtype: same type as the initial guess
726       """
727       lmaxit=iter_max
728       if atol<0: raise ValueError("atol needs to be non-negative.")
729       if rtol<0: raise ValueError("rtol needs to be non-negative.")
730       if rtol+atol<=0: raise ValueError("rtol or atol needs to be non-negative.")
731       if gamma<=0 or gamma>=1: raise ValueError("tolerance safety factor for inner iteration (gamma =%s) needs to be positive and less than 1."%gamma)
732       if subtol_max<=0 or subtol_max>=1: raise ValueError("upper bound for inner tolerance for inner iteration (subtol_max =%s) needs to be positive and less than 1."%subtol_max)
733    
734       F=defect(x)
735       fnrm=defect.norm(F)
736       stop_tol=atol + rtol*fnrm
737       subtol=subtol_max
738       if verbose: print(("NewtonGMRES: initial residual = %e."%fnrm))
739       if verbose: print(("             tolerance = %e."%subtol))
740       iter=1
741       #
742       # main iteration loop
743       #
744       while not fnrm<=stop_tol:
745                if iter  >= iter_max: raise MaxIterReached("maximum number of %s steps reached."%iter_max)
746                #
747                #   adjust subtol_
748                #
749                if iter > 1:
750                   rat=fnrm/fnrmo
751                   subtol_old=subtol
752                   subtol=gamma*rat**2
753                   if gamma*subtol_old**2 > .1: subtol=max(subtol,gamma*subtol_old**2)
754                   subtol=max(min(subtol,subtol_max), .5*stop_tol/fnrm)
755                #
756                # calculate newton increment xc
757                #     if iter_max in __FDGMRES is reached MaxIterReached is thrown
758                #     if iter_restart -1 is returned as sub_iter
759                #     if  atol is reached sub_iter returns the numer of steps performed to get there
760                #
761                #
762                if verbose: print(("             subiteration (GMRES) is called with relative tolerance %e."%subtol))
763                try:
764                   xc, sub_iter=__FDGMRES(F, defect, x, subtol*fnrm, iter_max=iter_max-iter, iter_restart=sub_iter_max)
765                except MaxIterReached:
766                   raise MaxIterReached("maximum number of %s steps reached."%iter_max)
767                if sub_iter<0:
768                   iter+=sub_iter_max
769                else:
770                   iter+=sub_iter
771                # ====
772                x+=xc
773                F=defect(x)
774                iter+=1
775                fnrmo, fnrm=fnrm, defect.norm(F)
776                if verbose: print(("             step %s: residual %e."%(iter,fnrm)))
777       if verbose: print(("NewtonGMRES: completed after %s steps."%iter))
778       return x
779    
780    def __givapp(c,s,vin):
781        """
782        Applies a sequence of Givens rotations (c,s) recursively to the vector
783        ``vin``
784    
785        :warning: ``vin`` is altered.
786        """
787        vrot=vin
788        if isinstance(c,float):
789            vrot=[c*vrot[0]-s*vrot[1],s*vrot[0]+c*vrot[1]]
790        else:
791            for i in range(len(c)):
792                w1=c[i]*vrot[i]-s[i]*vrot[i+1]
793                w2=s[i]*vrot[i]+c[i]*vrot[i+1]
794                vrot[i]=w1
795                vrot[i+1]=w2
796        return vrot
797    
798    def __FDGMRES(F0, defect, x0, atol, iter_max=100, iter_restart=20):
799       h=numpy.zeros((iter_restart,iter_restart),numpy.float64)
800       c=numpy.zeros(iter_restart,numpy.float64)
801       s=numpy.zeros(iter_restart,numpy.float64)
802       g=numpy.zeros(iter_restart,numpy.float64)
803       v=[]
804    
805       rho=defect.norm(F0)
806       if rho<=0.: return x0*0
807    
808       v.append(-F0/rho)
809       g[0]=rho
810       iter=0
811       while rho > atol and iter<iter_restart-1:
812            if iter  >= iter_max:
813                raise MaxIterReached("maximum number of %s steps reached."%iter_max)
814    
815            p=defect.derivative(F0,x0,v[iter], v_is_normalised=True)
816            v.append(p)
817    
818            v_norm1=defect.norm(v[iter+1])
819    
820            # Modified Gram-Schmidt
821            for j in range(iter+1):
822                h[j,iter]=defect.bilinearform(v[j],v[iter+1])
823                v[iter+1]-=h[j,iter]*v[j]
824    
825            h[iter+1,iter]=defect.norm(v[iter+1])
826            v_norm2=h[iter+1,iter]
827    
828            # Reorthogonalize if needed
829            if v_norm1 + 0.001*v_norm2 == v_norm1:   #Brown/Hindmarsh condition (default)
830                for j in range(iter+1):
831                    hr=defect.bilinearform(v[j],v[iter+1])
832                    h[j,iter]=h[j,iter]+hr
833                    v[iter+1] -= hr*v[j]
834    
835                v_norm2=defect.norm(v[iter+1])
836                h[iter+1,iter]=v_norm2
837            #   watch out for happy breakdown
838            if not v_norm2 == 0:
839                v[iter+1]=v[iter+1]/h[iter+1,iter]
840    
841            #   Form and store the information for the new Givens rotation
842            if iter > 0 :
843                hhat=numpy.zeros(iter+1,numpy.float64)
844                for i in range(iter+1) : hhat[i]=h[i,iter]
845                hhat=__givapp(c[0:iter],s[0:iter],hhat);
846                for i in range(iter+1) : h[i,iter]=hhat[i]
847    
848            mu=math.sqrt(h[iter,iter]*h[iter,iter]+h[iter+1,iter]*h[iter+1,iter])
849    
850            if mu!=0 :
851                c[iter]=h[iter,iter]/mu
852                s[iter]=-h[iter+1,iter]/mu
853                h[iter,iter]=c[iter]*h[iter,iter]-s[iter]*h[iter+1,iter]
854                h[iter+1,iter]=0.0
855                gg=__givapp(c[iter],s[iter],[g[iter],g[iter+1]])
856                g[iter]=gg[0]
857                g[iter+1]=gg[1]
858    
859            # Update the residual norm
860            rho=abs(g[iter+1])
861            iter+=1
862    
863       # At this point either iter > iter_max or rho < tol.
864       # It's time to compute x and leave.
865       if iter > 0 :
866         y=numpy.zeros(iter,numpy.float64)
867         y[iter-1] = g[iter-1] / h[iter-1,iter-1]
868         if iter > 1 :
869            i=iter-2
870            while i>=0 :
871              y[i] = ( g[i] - numpy.dot(h[i,i+1:iter], y[i+1:iter])) / h[i,i]
872              i=i-1
873         xhat=v[iter-1]*y[iter-1]
874         for i in range(iter-1):
875           xhat += v[i]*y[i]
876       else :
877          xhat=v[0] * 0
878    
879       if iter<iter_restart-1:
880          stopped=iter
881       else:
882          stopped=-1
883    
884       return xhat,stopped
885    
886    def GMRES(r, Aprod, x, bilinearform, atol=0, rtol=1.e-8, iter_max=100, iter_restart=20, verbose=False,P_R=None):
887       """
888       Solver for
889    
890       *Ax=b*
891    
892       with a general operator A (more details required!).
893       It uses the generalized minimum residual method (GMRES).
894    
895       The iteration is terminated if
896    
897       *|r| <= atol+rtol*|r0|*
898    
899       where *r0* is the initial residual and *|.|* is the energy norm. In fact
900    
901       *|r| = sqrt( bilinearform(r,r))*
902    
903       :param r: initial residual *r=b-Ax*. ``r`` is altered.
904       :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
905       :param x: an initial guess for the solution
906       :type x: same like ``r``
907       :param Aprod: returns the value Ax
908       :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
909                    argument ``x``. The returned object needs to be of the same
910                    type like argument ``r``.
911       :param bilinearform: inner product ``<x,r>``
912       :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
913                           type like argument ``x`` and ``r``. The returned value is
914                           a ``float``.
915       :param atol: absolute tolerance
916       :type atol: non-negative ``float``
917       :param rtol: relative tolerance
918       :type rtol: non-negative ``float``
919       :param iter_max: maximum number of iteration steps
920       :type iter_max: ``int``
921       :param iter_restart: in order to save memory the orthogonalization process
922                            is terminated after ``iter_restart`` steps and the
923                            iteration is restarted.
924       :type iter_restart: ``int``
925       :return: the solution approximation and the corresponding residual
926       :rtype: ``tuple``
927       :warning: ``r`` and ``x`` are altered.
928       """
929       m=iter_restart
930       restarted=False
931       iter=0
932       if rtol>0:
933          r_dot_r = bilinearform(r, r)
934          if r_dot_r<0: raise NegativeNorm("negative norm.")
935          atol2=atol+rtol*math.sqrt(r_dot_r)
936          if verbose: print(("GMRES: norm of right hand side = %e (absolute tolerance = %e)"%(math.sqrt(r_dot_r), atol2)))
937       else:
938          atol2=atol
939          if verbose: print(("GMRES: absolute tolerance = %e"%atol2))
940       if atol2<=0:
941          raise ValueError("Non-positive tolarance.")
942    
943       while True:
944          if iter  >= iter_max: raise MaxIterReached("maximum number of %s steps reached"%iter_max)
945          if restarted:
946             r2 = r-Aprod(x-x2)
947          else:
948             r2=1*r
949          x2=x*1.
950          x,stopped=_GMRESm(r2, Aprod, x, bilinearform, atol2, iter_max=iter_max-iter, iter_restart=m, verbose=verbose,P_R=P_R)
951          iter+=iter_restart
952          if stopped: break
953          if verbose: print("GMRES: restart.")
954          restarted=True
955       if verbose: print("GMRES: tolerance has been reached.")
956       return x
957    
958    def _GMRESm(r, Aprod, x, bilinearform, atol, iter_max=100, iter_restart=20, verbose=False, P_R=None):
959       iter=0
960    
961       h=numpy.zeros((iter_restart+1,iter_restart),numpy.float64)
962       c=numpy.zeros(iter_restart,numpy.float64)
963       s=numpy.zeros(iter_restart,numpy.float64)
964       g=numpy.zeros(iter_restart+1,numpy.float64)
965       v=[]
966    
967       r_dot_r = bilinearform(r, r)
968       if r_dot_r<0: raise NegativeNorm("negative norm.")
969       rho=math.sqrt(r_dot_r)
970    
971       v.append(r/rho)
972       g[0]=rho
973    
974       if verbose: print(("GMRES: initial residual %e (absolute tolerance = %e)"%(rho,atol)))
975       while not (rho<=atol or iter==iter_restart):
976    
977            if iter  >= iter_max: raise MaxIterReached("maximum number of %s steps reached."%iter_max)
978    
979            if P_R!=None:
980                p=Aprod(P_R(v[iter]))
981            else:
982                p=Aprod(v[iter])
983            v.append(p)
984    
985            v_norm1=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
986    
987    # Modified Gram-Schmidt
988            for j in range(iter+1):
989              h[j,iter]=bilinearform(v[j],v[iter+1])
990              v[iter+1]-=h[j,iter]*v[j]
991    
992            h[iter+1,iter]=math.sqrt(bilinearform(v[iter+1],v[iter+1]))
993            v_norm2=h[iter+1,iter]
994    
995    # Reorthogonalize if needed
996            if v_norm1 + 0.001*v_norm2 == v_norm1:   #Brown/Hindmarsh condition (default)
997             for j in range(iter+1):
998                hr=bilinearform(v[j],v[iter+1])
999                h[j,iter]=h[j,iter]+hr
1000                v[iter+1] -= hr*v[j]
1001    
1002             v_norm2=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
1003             h[iter+1,iter]=v_norm2
1004    
1005    #   watch out for happy breakdown
1006            if not v_norm2 == 0:
1007             v[iter+1]=v[iter+1]/h[iter+1,iter]
1008    
1009    #   Form and store the information for the new Givens rotation
1010            if iter > 0: h[:iter+1,iter]=__givapp(c[:iter],s[:iter],h[:iter+1,iter])
1011            mu=math.sqrt(h[iter,iter]*h[iter,iter]+h[iter+1,iter]*h[iter+1,iter])
1012    
1013            if mu!=0 :
1014                    c[iter]=h[iter,iter]/mu
1015                    s[iter]=-h[iter+1,iter]/mu
1016                    h[iter,iter]=c[iter]*h[iter,iter]-s[iter]*h[iter+1,iter]
1017                    h[iter+1,iter]=0.0
1018                    gg=__givapp(c[iter],s[iter],[g[iter],g[iter+1]])
1019                    g[iter]=gg[0]
1020                    g[iter+1]=gg[1]
1021    # Update the residual norm
1022    
1023            rho=abs(g[iter+1])
1024            if verbose: print(("GMRES: iteration step %s: residual %e"%(iter,rho)))
1025            iter+=1
1026    
1027    # At this point either iter > iter_max or rho < tol.
1028    # It's time to compute x and leave.
1029    
1030       if verbose: print(("GMRES: iteration stopped after %s step."%iter))
1031       if iter > 0 :
1032         y=numpy.zeros(iter,numpy.float64)
1033         y[iter-1] = g[iter-1] / h[iter-1,iter-1]
1034         if iter > 1 :
1035            i=iter-2
1036            while i>=0 :
1037              y[i] = ( g[i] - numpy.dot(h[i,i+1:iter], y[i+1:iter])) / h[i,i]
1038              i=i-1
1039         xhat=v[iter-1]*y[iter-1]
1040         for i in range(iter-1):
1041           xhat += v[i]*y[i]
1042       else:
1043         xhat=v[0] * 0
1044       if P_R!=None:
1045          x += P_R(xhat)
1046       else:
1047          x += xhat
1048       if iter<iter_restart-1:
1049          stopped=True
1050       else:
1051          stopped=False
1052    
1053       return x,stopped
1054    
1055    def MINRES(r, Aprod, x, Msolve, bilinearform, atol=0, rtol=1.e-8, iter_max=100):
1056        """
1057        Solver for
1058    
1059        *Ax=b*
1060    
1061        with a symmetric and positive definite operator A (more details required!).
1062        It uses the minimum residual method (MINRES) with preconditioner M
1063        providing an approximation of A.
1064    
1065        The iteration is terminated if
1066    
1067        *|r| <= atol+rtol*|r0|*
1068    
1069        where *r0* is the initial residual and *|.|* is the energy norm. In fact
1070    
1071        *|r| = sqrt( bilinearform(Msolve(r),r))*
1072    
1073        For details on the preconditioned conjugate gradient method see the book:
1074    
1075        I{Templates for the Solution of Linear Systems by R. Barrett, M. Berry,
1076        T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo,
1077        C. Romine, and H. van der Vorst}.
1078    
1079        :param r: initial residual *r=b-Ax*. ``r`` is altered.
1080        :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1081        :param x: an initial guess for the solution
1082        :type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1083        :param Aprod: returns the value Ax
1084        :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
1085                     argument ``x``. The returned object needs to be of the same
1086                     type like argument ``r``.
1087        :param Msolve: solves Mx=r
1088        :type Msolve: function ``Msolve(r)`` where ``r`` is of the same type like
1089                      argument ``r``. The returned object needs to be of the same
1090                      type like argument ``x``.
1091        :param bilinearform: inner product ``<x,r>``
1092        :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
1093                            type like argument ``x`` and ``r`` is. The returned value
1094                            is a ``float``.
1095        :param atol: absolute tolerance
1096        :type atol: non-negative ``float``
1097        :param rtol: relative tolerance
1098        :type rtol: non-negative ``float``
1099        :param iter_max: maximum number of iteration steps
1100        :type iter_max: ``int``
1101        :return: the solution approximation and the corresponding residual
1102        :rtype: ``tuple``
1103        :warning: ``r`` and ``x`` are altered.
1104        """
1105        #------------------------------------------------------------------
1106        # Set up y and v for the first Lanczos vector v1.
1107        # y  =  beta1 P' v1,  where  P = C**(-1).
1108        # v is really P' v1.
1109        #------------------------------------------------------------------
1110        r1    = r
1111        y = Msolve(r)
1112        beta1 = bilinearform(y,r)
1113    
1114        if beta1< 0: raise NegativeNorm("negative norm.")
1115    
1116        #  If r = 0 exactly, stop with x
1117        if beta1==0: return x
1118    
1119        if beta1> 0: beta1  = math.sqrt(beta1)
1120    
1121        #------------------------------------------------------------------
1122        # Initialize quantities.
1123        # ------------------------------------------------------------------
1124        iter   = 0
1125        Anorm = 0
1126        ynorm = 0
1127        oldb   = 0
1128        beta   = beta1
1129        dbar   = 0
1130        epsln  = 0
1131        phibar = beta1
1132        rhs1   = beta1
1133        rhs2   = 0
1134        rnorm  = phibar
1135        tnorm2 = 0
1136        ynorm2 = 0
1137        cs     = -1
1138        sn     = 0
1139        w      = r*0.
1140        w2     = r*0.
1141        r2     = r1
1142        eps    = 0.0001
1143    
1144        #---------------------------------------------------------------------
1145        # Main iteration loop.
1146        # --------------------------------------------------------------------
1147        while not rnorm<=atol+rtol*Anorm*ynorm:    #  checks ||r|| < (||A|| ||x||) * TOL
1148    
1149            if iter  >= iter_max: raise MaxIterReached("maximum number of %s steps reached."%iter_max)
1150            iter    = iter  +  1
1151    
1152            #-----------------------------------------------------------------
1153            # Obtain quantities for the next Lanczos vector vk+1, k = 1, 2,...
1154            # The general iteration is similar to the case k = 1 with v0 = 0:
1155            #
1156            #   p1      = Operator * v1  -  beta1 * v0,
1157            #   alpha1  = v1'p1,
1158            #   q2      = p2  -  alpha1 * v1,
1159            #   beta2^2 = q2'q2,
1160            #   v2      = (1/beta2) q2.
1161            #
1162            # Again, y = betak P vk,  where  P = C**(-1).
1163            #-----------------------------------------------------------------
1164            s = 1/beta                 # Normalize previous vector (in y).
1165            v = s*y                    # v = vk if P = I
1166    
1167            y      = Aprod(v)
1168    
1169            if iter >= 2:
1170              y = y - (beta/oldb)*r1
1171    
1172            alfa   = bilinearform(v,y)              # alphak
1173            y      += (- alfa/beta)*r2
1174            r1     = r2
1175            r2     = y
1176            y = Msolve(r2)
1177            oldb   = beta                           # oldb = betak
1178            beta   = bilinearform(y,r2)             # beta = betak+1^2
1179            if beta < 0: raise NegativeNorm("negative norm.")
1180    
1181            beta   = math.sqrt( beta )
1182            tnorm2 = tnorm2 + alfa*alfa + oldb*oldb + beta*beta
1183    
1184            if iter==1:                 # Initialize a few things.
1185              gmax   = abs( alfa )      # alpha1
1186              gmin   = gmax             # alpha1
1187    
1188            # Apply previous rotation Qk-1 to get
1189            #   [deltak epslnk+1] = [cs  sn][dbark    0   ]
1190            #   [gbar k dbar k+1]   [sn -cs][alfak betak+1].
1191    
1192            oldeps = epsln
1193            delta  = cs * dbar  +  sn * alfa  # delta1 = 0         deltak
1194            gbar   = sn * dbar  -  cs * alfa  # gbar 1 = alfa1     gbar k
1195            epsln  =               sn * beta  # epsln2 = 0         epslnk+1
1196            dbar   =            -  cs * beta  # dbar 2 = beta2     dbar k+1
1197    
1198            # Compute the next plane rotation Qk
1199    
1200            gamma  = math.sqrt(gbar*gbar+beta*beta)  # gammak
1201            gamma  = max(gamma,eps)
1202            cs     = gbar / gamma             # ck
1203            sn     = beta / gamma             # sk
1204            phi    = cs * phibar              # phik
1205            phibar = sn * phibar              # phibark+1
1206    
1207            # Update  x.
1208    
1209            denom = 1/gamma
1210            w1    = w2
1211            w2    = w
1212            w     = (v - oldeps*w1 - delta*w2) * denom
1213            x     +=  phi*w
1214    
1215            # Go round again.
1216    
1217            gmax   = max(gmax,gamma)
1218            gmin   = min(gmin,gamma)
1219            z      = rhs1 / gamma
1220            ynorm2 = z*z  +  ynorm2
1221            rhs1   = rhs2 -  delta*z
1222            rhs2   =      -  epsln*z
1223    
1224            # Estimate various norms and test for convergence.
1225    
1226            Anorm  = math.sqrt( tnorm2 )
1227            ynorm  = math.sqrt( ynorm2 )
1228    
1229            rnorm  = phibar
1230    
1231        return x
1232    
1233    def TFQMR(r, Aprod, x, bilinearform, atol=0, rtol=1.e-8, iter_max=100):
1234      """
1235      Solver for
1236    
1237      *Ax=b*
1238    
1239      with a general operator A (more details required!).
1240      It uses the Transpose-Free Quasi-Minimal Residual method (TFQMR).
1241    
1242      The iteration is terminated if
1243    
1244      *|r| <= atol+rtol*|r0|*
1245    
1246      where *r0* is the initial residual and *|.|* is the energy norm. In fact
1247    
1248      *|r| = sqrt( bilinearform(r,r))*
1249    
1250      :param r: initial residual *r=b-Ax*. ``r`` is altered.
1251      :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1252      :param x: an initial guess for the solution
1253      :type x: same like ``r``
1254      :param Aprod: returns the value Ax
1255      :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
1256                   argument ``x``. The returned object needs to be of the same type
1257                   like argument ``r``.
1258      :param bilinearform: inner product ``<x,r>``
1259      :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
1260                          type like argument ``x`` and ``r``. The returned value is
1261                          a ``float``.
1262      :param atol: absolute tolerance
1263      :type atol: non-negative ``float``
1264      :param rtol: relative tolerance
1265      :type rtol: non-negative ``float``
1266      :param iter_max: maximum number of iteration steps
1267      :type iter_max: ``int``
1268      :rtype: ``tuple``
1269      :warning: ``r`` and ``x`` are altered.
1270      """
1271      u1=0
1272      u2=0
1273      y1=0
1274      y2=0
1275    
1276      w = r
1277      y1 = r
1278      iter = 0
1279      d = 0
1280      v = Aprod(y1)
1281      u1 = v
1282    
1283      theta = 0.0;
1284      eta = 0.0;
1285      rho=bilinearform(r,r)
1286      if rho < 0: raise NegativeNorm("negative norm.")
1287      tau = math.sqrt(rho)
1288      norm_r0=tau
1289      while tau>atol+rtol*norm_r0:
1290        if iter  >= iter_max: raise MaxIterReached("maximum number of %s steps reached."%iter_max)
1291    
1292        sigma = bilinearform(r,v)
1293        if sigma == 0.0: raise IterationBreakDown('TFQMR breakdown, sigma=0')
1294    
1295        alpha = rho / sigma
1296    
1297        for j in range(2):
1298    #
1299    #   Compute y2 and u2 only if you have to
1300    #
1301          if ( j == 1 ):
1302            y2 = y1 - alpha * v
1303            u2 = Aprod(y2)
1304    
1305          m = 2 * (iter+1) - 2 + (j+1)
1306          if j==0:
1307             w = w - alpha * u1
1308             d = y1 + ( theta * theta * eta / alpha ) * d
1309          if j==1:
1310             w = w - alpha * u2
1311             d = y2 + ( theta * theta * eta / alpha ) * d
1312    
1313          theta = math.sqrt(bilinearform(w,w))/ tau
1314          c = 1.0 / math.sqrt ( 1.0 + theta * theta )
1315          tau = tau * theta * c
1316          eta = c * c * alpha
1317          x = x + eta * d
1318    #
1319    #  Try to terminate the iteration at each pass through the loop
1320    #
1321        if rho == 0.0: raise IterationBreakDown('TFQMR breakdown, rho=0')
1322    
1323        rhon = bilinearform(r,w)
1324        beta = rhon / rho;
1325        rho = rhon;
1326        y1 = w + beta * y2;
1327        u1 = Aprod(y1)
1328        v = u1 + beta * ( u2 + beta * v )
1329    
1330        iter += 1
1331    
1332      return x
1333    
1334    
1335    #############################################
1336    
1337    class ArithmeticTuple(object):
1338       """
1339       Tuple supporting inplace update x+=y and scaling x=a*y where ``x,y`` is an
1340       ArithmeticTuple and ``a`` is a float.
1341    
1342       Example of usage::
1343    
1344           from esys.escript import Data
1345           from numpy import array
1346           a=eData(...)
1347           b=array([1.,4.])
1348           x=ArithmeticTuple(a,b)
1349           y=5.*x
1350    
1351       """
1352       def __init__(self,*args):
1353           """
1354           Initializes object with elements ``args``.
1355    
1356           :param args: tuple of objects that support inplace add (x+=y) and
1357                        scaling (x=a*y)
1358           """
1359           self.__items=list(args)
1360    
1361       def __len__(self):
1362           """
1363           Returns the number of items.
1364    
1365           :return: number of items
1366           :rtype: ``int``
1367           """
1368           return len(self.__items)
1369    
1370       def __getitem__(self,index):
1371           """
1372           Returns item at specified position.
1373    
1374           :param index: index of item to be returned
1375           :type index: ``int``
1376           :return: item with index ``index``
1377           """
1378           return self.__items.__getitem__(index)
1379    
1380       def __mul__(self,other):
1381           """
1382           Scales by ``other`` from the right.
1383    
1384           :param other: scaling factor
1385           :type other: ``float``
1386           :return: itemwise self*other
1387           :rtype: `ArithmeticTuple`
1388           """
1389           out=[]
1390           try:
1391               l=len(other)
1392               if l!=len(self):
1393                   raise ValueError("length of arguments don't match.")
1394               for i in range(l):
1395                    if self.__isEmpty(self[i]) or self.__isEmpty(other[i]):
1396                        out.append(escore.Data())
1397                    else:
1398                        out.append(self[i]*other[i])
1399           except TypeError:
1400               for i in range(len(self)):  
1401                    if self.__isEmpty(self[i]) or self.__isEmpty(other):
1402                        out.append(escore.Data())
1403                    else:
1404                        out.append(self[i]*other)
1405           return ArithmeticTuple(*tuple(out))
1406    
1407       def __rmul__(self,other):
1408          """
1409          Scales by ``other`` from the left.
1410    
1411          :param other: scaling factor
1412          :type other: ``float``
1413          :return: itemwise other*self
1414          :rtype: `ArithmeticTuple`
1415          """
1416          out=[]
1417          try:
1418              l=len(other)
1419              if l!=len(self):
1420                  raise ValueError("length of arguments don't match.")
1421              for i in range(l):
1422                    if self.__isEmpty(self[i]) or self.__isEmpty(other[i]):
1423                        out.append(escore.Data())
1424                    else:
1425                        out.append(other[i]*self[i])
1426          except TypeError:
1427              for i in range(len(self)):  
1428                    if self.__isEmpty(self[i]) or self.__isEmpty(other):
1429                        out.append(escore.Data())
1430                    else:
1431                        out.append(other*self[i])
1432          return ArithmeticTuple(*tuple(out))
1433    
1434       def __div__(self,other):
1435           """
1436           Scales by (1/``other``) from the right.
1437    
1438           :param other: scaling factor
1439           :type other: ``float``
1440           :return: itemwise self/other
1441           :rtype: `ArithmeticTuple`
1442           """
1443           return self*(1/other)
1444    
1445       def __rdiv__(self,other):
1446          """
1447          Scales by (1/``other``) from the left.
1448    
1449          :param other: scaling factor
1450          :type other: ``float``
1451          :return: itemwise other/self
1452          :rtype: `ArithmeticTuple`
1453          """
1454          out=[]
1455          try:
1456              l=len(other)
1457              if l!=len(self):
1458                  raise ValueError("length of arguments don't match.")
1459              
1460              for i in range(l):
1461                    if self.__isEmpty(self[i]):
1462                        raise ZeroDivisionError("in component %s"%i)
1463                    else:
1464                        if self.__isEmpty(other[i]):
1465                            out.append(escore.Data())
1466                        else:
1467                            out.append(other[i]/self[i])
1468          except TypeError:
1469              for i in range(len(self)):
1470                    if self.__isEmpty(self[i]):
1471                        raise ZeroDivisionError("in component %s"%i)
1472                    else:
1473                        if self.__isEmpty(other):
1474                            out.append(escore.Data())
1475                        else:
1476                            out.append(other/self[i])
1477          return ArithmeticTuple(*tuple(out))
1478    
1479       def __iadd__(self,other):
1480          """
1481          Inplace addition of ``other`` to self.
1482    
1483          :param other: increment
1484          :type other: ``ArithmeticTuple``
1485          """
1486          if len(self) != len(other):
1487              raise ValueError("tuple lengths must match.")
1488          for i in range(len(self)):
1489              if self.__isEmpty(self.__items[i]):
1490                  self.__items[i]=other[i]
1491              else:
1492                  self.__items[i]+=other[i]
1493                  
1494          return self
1495    
1496       def __add__(self,other):
1497          """
1498          Adds ``other`` to self.
1499    
1500          :param other: increment
1501          :type other: ``ArithmeticTuple``
1502          """
1503          out=[]
1504          try:
1505              l=len(other)
1506              if l!=len(self):
1507                  raise ValueError("length of arguments don't match.")
1508              for i in range(l):
1509                    if self.__isEmpty(self[i]):
1510                        out.append(other[i])
1511                    elif self.__isEmpty(other[i]):
1512                        out.append(self[i])
1513                    else:
1514                        out.append(self[i]+other[i])
1515          except TypeError:
1516                for i in range(len(self)):    
1517                    if self.__isEmpty(self[i]):
1518                        out.append(other)
1519                    elif self.__isEmpty(other):
1520                        out.append(self[i])
1521                    else:
1522                        out.append(self[i]+other)
1523          return ArithmeticTuple(*tuple(out))
1524    
1525       def __sub__(self,other):
1526          """
1527          Subtracts ``other`` from self.
1528    
1529          :param other: decrement
1530          :type other: ``ArithmeticTuple``
1531          """
1532          out=[]
1533          try:
1534              l=len(other)
1535              if l!=len(self):
1536                  raise ValueError("length of arguments don't match.")
1537              for i in range(l):
1538                    if self.__isEmpty(other[i]):
1539                        out.append(self[i])
1540                    elif self.__isEmpty(self[i]):
1541                        out.append(-other[i])
1542                    else:
1543                        out.append(self[i]-other[i])
1544          except TypeError:
1545                for i in range(len(self)):    
1546                    if  self.__isEmpty(other):
1547                        out.append(self[i])
1548                    elif self.__isEmpty(self[i]):
1549                        out.append(-other)
1550                    else:
1551                        out.append(self[i]-other)
1552                        
1553          return ArithmeticTuple(*tuple(out))
1554    
1555       def __isub__(self,other):
1556          """
1557          Inplace subtraction of ``other`` from self.
1558    
1559          :param other: decrement
1560          :type other: ``ArithmeticTuple``
1561          """
1562          if len(self) != len(other):
1563              raise ValueError("tuple length must match.")
1564          for i in range(len(self)):
1565              if not self.__isEmpty(other[i]):
1566                  if self.__isEmpty(self.__items[i]):
1567                      self.__items[i]=-other[i]
1568                  else:
1569                      self.__items[i]=other[i]
1570          return self
1571    
1572       def __neg__(self):
1573          """
1574          Negates values.
1575          """
1576          out=[]
1577          for i in range(len(self)):
1578              if self.__isEmpty(self[i]):
1579                  out.append(escore.Data())
1580              else:
1581                  out.append(-self[i])
1582              
1583          return ArithmeticTuple(*tuple(out))
1584       def __isEmpty(self, d):
1585        if isinstance(d, escore.Data):
1586            return d.isEmpty()
1587        else:
1588            return False
1589                    
1590       def __str__(self):
1591        s="("
1592        for i in self:
1593          s=s+str(i)+", "
1594        s=s+")"
1595        return s
1596    
1597    class HomogeneousSaddlePointProblem(object):
1598          """
1599          This class provides a framework for solving linear homogeneous saddle
1600          point problems of the form::
1601    
1602              *Av+B^*p=f*
1603              *Bv     =0*
1604    
1605          for the unknowns *v* and *p* and given operators *A* and *B* and
1606          given right hand side *f*. *B^** is the adjoint operator of *B*.
1607          *A* may depend weakly on *v* and *p*.
1608          """
1609          def __init__(self, **kwargs):
1610            """
1611            initializes the saddle point problem
1612            """
1613            self.resetControlParameters()
1614            self.setTolerance()
1615            self.setAbsoluteTolerance()
1616          def resetControlParameters(self, K_p=1., K_v=1., rtol_max=0.01, rtol_min = 1.e-7, chi_max=0.5, reduction_factor=0.3, theta = 0.1):
1617             """
1618             sets a control parameter
1619    
1620             :param K_p: initial value for constant to adjust pressure tolerance
1621             :type K_p: ``float``
1622             :param K_v: initial value for constant to adjust velocity tolerance
1623             :type K_v: ``float``
1624             :param rtol_max: maximuim relative tolerance used to calculate presssure and velocity increment.
1625             :type rtol_max: ``float``
1626             :param chi_max: maximum tolerable converegence rate.
1627             :type chi_max: ``float``
1628             :param reduction_factor: reduction factor for adjustment factors.
1629             :type reduction_factor: ``float``
1630             """
1631             self.setControlParameter(K_p, K_v, rtol_max, rtol_min, chi_max, reduction_factor, theta)
1632    
1633          def setControlParameter(self,K_p=None, K_v=None, rtol_max=None, rtol_min=None, chi_max=None, reduction_factor=None, theta=None):
1634             """
1635             sets a control parameter
1636    
1637    
1638             :param K_p: initial value for constant to adjust pressure tolerance
1639             :type K_p: ``float``
1640             :param K_v: initial value for constant to adjust velocity tolerance
1641             :type K_v: ``float``
1642             :param rtol_max: maximuim relative tolerance used to calculate presssure and velocity increment.
1643             :type rtol_max: ``float``
1644             :param chi_max: maximum tolerable converegence rate.
1645             :type chi_max: ``float``
1646             :type reduction_factor: ``float``
1647             """
1648             if not K_p == None:
1649                if K_p<1:
1650                   raise ValueError("K_p need to be greater or equal to 1.")
1651             else:
1652                K_p=self.__K_p
1653    
1654             if not K_v == None:
1655                if K_v<1:
1656                   raise ValueError("K_v need to be greater or equal to 1.")
1657             else:
1658                K_v=self.__K_v
1659    
1660             if not rtol_max == None:
1661                if rtol_max<=0 or rtol_max>=1:
1662                   raise ValueError("rtol_max needs to be positive and less than 1.")
1663             else:
1664                rtol_max=self.__rtol_max
1665    
1666             if not rtol_min == None:
1667                if rtol_min<=0 or rtol_min>=1:
1668                   raise ValueError("rtol_min needs to be positive and less than 1.")
1669             else:
1670                rtol_min=self.__rtol_min
1671    
1672             if not chi_max == None:
1673                if chi_max<=0 or chi_max>=1:
1674                   raise ValueError("chi_max needs to be positive and less than 1.")
1675             else:
1676                chi_max = self.__chi_max
1677    
1678             if not reduction_factor == None:
1679                if reduction_factor<=0 or reduction_factor>1:
1680                   raise ValueError("reduction_factor need to be between zero and one.")
1681             else:
1682                reduction_factor=self.__reduction_factor
1683    
1684             if not theta == None:
1685                if theta<=0 or theta>1:
1686                   raise ValueError("theta need to be between zero and one.")
1687             else:
1688                theta=self.__theta
1689    
1690             if rtol_min>=rtol_max:
1691                 raise ValueError("rtol_max = %e needs to be greater than rtol_min = %e"%(rtol_max,rtol_min))
1692             self.__chi_max = chi_max
1693             self.__rtol_max = rtol_max
1694             self.__K_p = K_p
1695             self.__K_v = K_v
1696             self.__reduction_factor = reduction_factor
1697             self.__theta = theta
1698             self.__rtol_min=rtol_min
1699    
1700          #=============================================================
1701          def inner_pBv(self,p,Bv):
1702             """
1703             Returns inner product of element p and Bv (overwrite).
1704    
1705             :param p: a pressure increment
1706             :param Bv: a residual
1707             :return: inner product of element p and Bv
1708             :rtype: ``float``
1709             :note: used if PCG is applied.
1710             """
1711             raise NotImplementedError("no inner product for p and Bv implemented.")
1712    
1713          def inner_p(self,p0,p1):
1714             """
1715             Returns inner product of p0 and p1 (overwrite).
1716    
1717             :param p0: a pressure
1718             :param p1: a pressure
1719             :return: inner product of p0 and p1
1720             :rtype: ``float``
1721             """
1722             raise NotImplementedError("no inner product for p implemented.")
1723      
1724          def norm_v(self,v):
1725             """
1726             Returns the norm of v (overwrite).
1727    
1728             :param v: a velovity
1729             :return: norm of v
1730             :rtype: non-negative ``float``
1731             """
1732             raise NotImplementedError("no norm of v implemented.")
1733          def getDV(self, p, v, tol):
1734             """
1735             return a correction to the value for a given v and a given p with accuracy `tol` (overwrite)
1736    
1737             :param p: pressure
1738             :param v: pressure
1739             :return: dv given as *dv= A^{-1} (f-A v-B^*p)*
1740             :note: Only *A* may depend on *v* and *p*
1741             """
1742             raise NotImplementedError("no dv calculation implemented.")
1743    
1744            
1745          def Bv(self,v, tol):
1746            """
1747            Returns Bv with accuracy `tol` (overwrite)
1748    
1749            :rtype: equal to the type of p
1750            :note: boundary conditions on p should be zero!
1751            """
1752            raise NotImplementedError("no operator B implemented.")
1753    
1754          def norm_Bv(self,Bv):
1755            """
1756            Returns the norm of Bv (overwrite).
1757    
1758            :rtype: equal to the type of p
1759            :note: boundary conditions on p should be zero!
1760            """
1761            raise NotImplementedError("no norm of Bv implemented.")
1762    
1763          def solve_AinvBt(self,dp, tol):
1764             """
1765             Solves *A dv=B^*dp* with accuracy `tol`
1766    
1767             :param dp: a pressure increment
1768             :return: the solution of *A dv=B^*dp*
1769             :note: boundary conditions on dv should be zero! *A* is the operator used in ``getDV`` and must not be altered.
1770             """
1771             raise NotImplementedError("no operator A implemented.")
1772    
1773          def solve_prec(self,Bv, tol):
1774             """
1775             Provides a preconditioner for *(BA^{-1}B^ * )* applied to Bv with accuracy `tol`
1776    
1777             :rtype: equal to the type of p
1778             :note: boundary conditions on p should be zero!
1779             """
1780             raise NotImplementedError("no preconditioner for Schur complement implemented.")
1781          #=============================================================
1782          def __Aprod_PCG(self,dp):
1783              dv=self.solve_AinvBt(dp, self.__subtol)
1784              return ArithmeticTuple(dv,self.Bv(dv, self.__subtol))
1785    
1786          def __inner_PCG(self,p,r):
1787             return self.inner_pBv(p,r[1])
1788    
1789          def __Msolve_PCG(self,r):
1790              return self.solve_prec(r[1], self.__subtol)
1791          #=============================================================
1792          def __Aprod_GMRES(self,p):
1793              return self.solve_prec(self.Bv(self.solve_AinvBt(p, self.__subtol), self.__subtol), self.__subtol)
1794          def __inner_GMRES(self,p0,p1):
1795             return self.inner_p(p0,p1)
1796    
1797          #=============================================================
1798          def norm_p(self,p):
1799              """
1800              calculates the norm of ``p``
1801              
1802              :param p: a pressure
1803              :return: the norm of ``p`` using the inner product for pressure
1804              :rtype: ``float``
1805              """
1806              f=self.inner_p(p,p)
1807              if f<0: raise ValueError("negative pressure norm.")
1808              return math.sqrt(f)
1809              
1810          def solve(self,v,p,max_iter=20, verbose=False, usePCG=True, iter_restart=20, max_correction_steps=10):
1811             """
1812             Solves the saddle point problem using initial guesses v and p.
1813    
1814             :param v: initial guess for velocity
1815             :param p: initial guess for pressure
1816             :type v: `Data`
1817             :type p: `Data`
1818             :param usePCG: indicates the usage of the PCG rather than GMRES scheme.
1819             :param max_iter: maximum number of iteration steps per correction
1820                              attempt
1821             :param verbose: if True, shows information on the progress of the
1822                             saddlepoint problem solver.
1823             :param iter_restart: restart the iteration after ``iter_restart`` steps
1824                                  (only used if useUzaw=False)
1825             :type usePCG: ``bool``
1826             :type max_iter: ``int``
1827             :type verbose: ``bool``
1828             :type iter_restart: ``int``
1829             :rtype: ``tuple`` of `Data` objects
1830             :note: typically this method is overwritten by a subclass. It provides a wrapper for the ``_solve`` method.
1831             """
1832             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)
1833    
1834          def _solve(self,v,p,max_iter=20, verbose=False, usePCG=True, iter_restart=20, max_correction_steps=10):
1835             """
1836             see `_solve` method.
1837             """
1838             self.verbose=verbose
1839             rtol=self.getTolerance()
1840             atol=self.getAbsoluteTolerance()
1841    
1842             K_p=self.__K_p
1843             K_v=self.__K_v
1844             correction_step=0
1845             converged=False
1846             chi=None
1847             eps=None
1848    
1849             if self.verbose: print(("HomogeneousSaddlePointProblem: start iteration: rtol= %e, atol=%e"%(rtol, atol)))
1850             while not converged:
1851    
1852                 # get tolerance for velecity increment:
1853                 if chi == None:
1854                    rtol_v=self.__rtol_max
1855                 else:
1856                    rtol_v=min(chi/K_v,self.__rtol_max)
1857                 rtol_v=max(rtol_v, self.__rtol_min)
1858                 if self.verbose: print(("HomogeneousSaddlePointProblem: step %s: rtol_v= %e"%(correction_step,rtol_v)))
1859                 # get velocity increment:
1860                 dv1=self.getDV(p,v,rtol_v)
1861                 v1=v+dv1
1862                 Bv1=self.Bv(v1, rtol_v)
1863                 norm_Bv1=self.norm_Bv(Bv1)
1864                 norm_dv1=self.norm_v(dv1)
1865                 if self.verbose: print(("HomogeneousSaddlePointProblem: step %s: norm_Bv1 = %e, norm_dv1 = %e"%(correction_step, norm_Bv1, norm_dv1)))
1866                 if norm_dv1*self.__theta < norm_Bv1:
1867                    # get tolerance for pressure increment:
1868                    large_Bv1=True
1869                    if chi == None or eps == None:
1870                       rtol_p=self.__rtol_max
1871                    else:
1872                       rtol_p=min(chi**2*eps/K_p/norm_Bv1, self.__rtol_max)
1873                    self.__subtol=max(rtol_p**2, self.__rtol_min)
1874                    if self.verbose: print(("HomogeneousSaddlePointProblem: step %s: rtol_p= %e"%(correction_step,rtol_p)))
1875                    # now we solve for the pressure increment dp from B*A^{-1}B^* dp = Bv1
1876                    if usePCG:
1877                        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)
1878                        v2=r[0]
1879                        Bv2=r[1]
1880                    else:
1881                        # don't use!!!!
1882                        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)
1883                        dv2=self.solve_AinvBt(dp, self.__subtol)
1884                        v2=v1-dv2
1885                        Bv2=self.Bv(v2, self.__subtol)
1886                    p2=p+dp
1887                 else:
1888                    large_Bv1=False
1889                    v2=v1
1890                    p2=p
1891                 # update business:
1892                 norm_dv2=self.norm_v(v2-v)
1893                 norm_v2=self.norm_v(v2)
1894                 if self.verbose: print(("HomogeneousSaddlePointProblem: step %s: v2 = %e, norm_dv2 = %e"%(correction_step, norm_v2, self.norm_v(v2-v))))
1895                 eps, eps_old = max(norm_Bv1, norm_dv2), eps
1896                 if eps_old == None:
1897                      chi, chi_old = None, chi
1898                 else:
1899                      chi, chi_old = min(eps/ eps_old, self.__chi_max), chi
1900                 if eps != None:
1901                     if chi !=None:
1902                        if self.verbose: print(("HomogeneousSaddlePointProblem: step %s: convergence rate = %e, correction = %e"%(correction_step,chi, eps)))
1903                     else:
1904                        if self.verbose: print(("HomogeneousSaddlePointProblem: step %s: correction = %e"%(correction_step, eps)))
1905                 if eps <= rtol*norm_v2+atol :
1906                     converged = True
1907                 else:
1908                     if correction_step>=max_correction_steps:
1909                          raise CorrectionFailed("Given up after %d correction steps."%correction_step)
1910                     if chi_old!=None:
1911                        K_p=max(1,self.__reduction_factor*K_p,(chi-chi_old)/chi_old**2*K_p)
1912                        K_v=max(1,self.__reduction_factor*K_v,(chi-chi_old)/chi_old**2*K_p)
1913                        if self.verbose: print(("HomogeneousSaddlePointProblem: step %s: new adjustment factor K = %e"%(correction_step,K_p)))
1914                 correction_step+=1
1915                 v,p =v2, p2
1916             if self.verbose: print(("HomogeneousSaddlePointProblem: tolerance reached after %s steps."%correction_step))
1917             return v,p
1918          #========================================================================
1919          def setTolerance(self,tolerance=1.e-4):
1920             """
1921             Sets the relative tolerance for (v,p).
1922    
1923             :param tolerance: tolerance to be used
1924             :type tolerance: non-negative ``float``
1925             """
1926             if tolerance<0:
1927                 raise ValueError("tolerance must be positive.")
1928             self.__rtol=tolerance
1929    
1930          def getTolerance(self):
1931             """
1932             Returns the relative tolerance.
1933    
1934             :return: relative tolerance
1935             :rtype: ``float``
1936             """
1937             return self.__rtol
1938    
1939          def setAbsoluteTolerance(self,tolerance=0.):
1940             """
1941             Sets the absolute tolerance.
1942    
1943             :param tolerance: tolerance to be used
1944             :type tolerance: non-negative ``float``
1945             """
1946             if tolerance<0:
1947                 raise ValueError("tolerance must be non-negative.")
1948             self.__atol=tolerance
1949    
1950          def getAbsoluteTolerance(self):
1951             """
1952             Returns the absolute tolerance.
1953    
1954             :return: absolute tolerance
1955             :rtype: ``float``
1956             """
1957             return self.__atol
1958    
1959    
1960    def MaskFromBoundaryTag(domain,*tags):
1961       """
1962       Creates a mask on the Solution(domain) function space where the value is
1963       one for samples that touch the boundary tagged by tags.
1964    
1965       Usage: m=MaskFromBoundaryTag(domain, "left", "right")
1966    
1967       :param domain: domain to be used
1968       :type domain: `escript.Domain`
1969       :param tags: boundary tags
1970       :type tags: ``str``
1971       :return: a mask which marks samples that are touching the boundary tagged
1972                by any of the given tags
1973       :rtype: `escript.Data` of rank 0
1974       """
1975       pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)
1976       d=escore.Scalar(0.,escore.FunctionOnBoundary(domain))
1977       for t in tags: d.setTaggedValue(t,1.)
1978       pde.setValue(y=d)
1979       return util.whereNonZero(pde.getRightHandSide())
1980    
1981    def MaskFromTag(domain,*tags):
1982       """
1983       Creates a mask on the Solution(domain) function space where the value is
1984       one for samples that touch regions tagged by tags.
1985    
1986       Usage: m=MaskFromTag(domain, "ham")
1987    
1988       :param domain: domain to be used
1989       :type domain: `escript.Domain`
1990       :param tags: boundary tags
1991       :type tags: ``str``
1992       :return: a mask which marks samples that are touching the boundary tagged
1993                by any of the given tags
1994       :rtype: `escript.Data` of rank 0
1995       """
1996       pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)
1997       d=escore.Scalar(0.,escore.Function(domain))
1998       for t in tags: d.setTaggedValue(t,1.)
1999       pde.setValue(Y=d)
2000       return util.whereNonZero(pde.getRightHandSide())
2001    
2002    
 # vim: expandtab shiftwidth=4:  

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