/[escript]/trunk/escript/py_src/pdetools.py
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revision 1956 by gross, Mon Nov 3 05:08:42 2008 UTC revision 2264 by gross, Wed Feb 11 06:48:28 2009 UTC
# Line 20  http://www.opensource.org/licenses/osl-3 Line 20  http://www.opensource.org/licenses/osl-3
20  __url__="http://www.uq.edu.au/esscc/escript-finley"  __url__="http://www.uq.edu.au/esscc/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
# Line 52  import math Line 52  import math
52    
53  class TimeIntegrationManager:  class TimeIntegrationManager:
54    """    """
55    a simple mechanism to manage time dependend values.    A simple mechanism to manage time dependend values.
56    
57    typical usage is::    Typical usage is::
58    
59       dt=0.1 # time increment       dt=0.1 # time increment
60       tm=TimeIntegrationManager(inital_value,p=1)       tm=TimeIntegrationManager(inital_value,p=1)
# Line 68  class TimeIntegrationManager: Line 68  class TimeIntegrationManager:
68    """    """
69    def __init__(self,*inital_values,**kwargs):    def __init__(self,*inital_values,**kwargs):
70       """       """
71       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 C{inital_values} are the initial values
72         and p is the order used for extrapolation.
73       """       """
74       if kwargs.has_key("p"):       if kwargs.has_key("p"):
75              self.__p=kwargs["p"]              self.__p=kwargs["p"]
# Line 85  class TimeIntegrationManager: Line 86  class TimeIntegrationManager:
86    
87    def getTime(self):    def getTime(self):
88        return self.__t        return self.__t
89    
90    def getValue(self):    def getValue(self):
91        out=self.__v_mem[0]        out=self.__v_mem[0]
92        if len(out)==1:        if len(out)==1:
# Line 94  class TimeIntegrationManager: Line 96  class TimeIntegrationManager:
96    
97    def checkin(self,dt,*values):    def checkin(self,dt,*values):
98        """        """
99        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.
100        """        """
101        o=min(self.__order+1,self.__p)        o=min(self.__order+1,self.__p)
102        self.__order=min(self.__order+1,self.__p)        self.__order=min(self.__order+1,self.__p)
# Line 111  class TimeIntegrationManager: Line 113  class TimeIntegrationManager:
113    
114    def extrapolate(self,dt):    def extrapolate(self,dt):
115        """        """
116        extrapolates to dt forward in time.        Extrapolates to C{dt} forward in time.
117        """        """
118        if self.__order==0:        if self.__order==0:
119           out=self.__v_mem[0]           out=self.__v_mem[0]
# Line 126  class TimeIntegrationManager: Line 128  class TimeIntegrationManager:
128           return out[0]           return out[0]
129        else:        else:
130           return out           return out
131    
132    
133  class Projector:  class Projector:
134    """    """
135    The Projector is a factory which projects a discontiuous function onto a    The Projector is a factory which projects a discontinuous function onto a
136    continuous function on the a given domain.    continuous function on a given domain.
137    """    """
138    def __init__(self, domain, reduce = True, fast=True):    def __init__(self, domain, reduce=True, fast=True):
139      """      """
140      Create a continuous function space projector for a domain.      Creates a continuous function space projector for a domain.
141    
142      @param domain: Domain of the projection.      @param domain: Domain of the projection.
143      @param reduce: Flag to reduce projection order (default is True)      @param reduce: Flag to reduce projection order
144      @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
145      """      """
146      self.__pde = linearPDEs.LinearPDE(domain)      self.__pde = linearPDEs.LinearPDE(domain)
147      if fast:      if fast:
148        self.__pde.setSolverMethod(linearPDEs.LinearPDE.LUMPING)          self.__pde.setSolverMethod(linearPDEs.LinearPDE.LUMPING)
149      self.__pde.setSymmetryOn()      self.__pde.setSymmetryOn()
150      self.__pde.setReducedOrderTo(reduce)      self.__pde.setReducedOrderTo(reduce)
151      self.__pde.setValue(D = 1.)      self.__pde.setValue(D = 1.)
# Line 151  class Projector: Line 153  class Projector:
153    
154    def __call__(self, input_data):    def __call__(self, input_data):
155      """      """
156      Projects input_data onto a continuous function      Projects C{input_data} onto a continuous function.
157    
158      @param input_data: The input_data to be projected.      @param input_data: the data to be projected
159      """      """
160      out=escript.Data(0.,input_data.getShape(),self.__pde.getFunctionSpaceForSolution())      out=escript.Data(0.,input_data.getShape(),self.__pde.getFunctionSpaceForSolution())
161      self.__pde.setValue(Y = escript.Data(), Y_reduced = escript.Data())      self.__pde.setValue(Y = escript.Data(), Y_reduced = escript.Data())
# Line 186  class Projector: Line 188  class Projector:
188    
189  class NoPDE:  class NoPDE:
190       """       """
191       solves the following problem for u:       Solves the following problem for u:
192    
193       M{kronecker[i,j]*D[j]*u[j]=Y[i]}       M{kronecker[i,j]*D[j]*u[j]=Y[i]}
194    
195       with constraint       with constraint
196    
197       M{u[j]=r[j]}  where M{q[j]>0}       M{u[j]=r[j]}  where M{q[j]>0}
198    
199       where D, Y, r and q are given functions of rank 1.       where M{D}, M{Y}, M{r} and M{q} are given functions of rank 1.
200    
201       In the case of scalars this takes the form       In the case of scalars this takes the form
202    
203       M{D*u=Y}       M{D*u=Y}
204    
205       with constraint       with constraint
206    
207       M{u=r}  where M{q>0}       M{u=r} where M{q>0}
208    
209       where D, Y, r and q are given scalar functions.       where M{D}, M{Y}, M{r} and M{q} are given scalar functions.
210    
211       The constraint is overwriting any other condition.       The constraint overwrites any other condition.
212    
213       @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 L{linearPDEs.LinearPDE} class with
214              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
215              thing is a bit strange and I blame Robert.Woodcock@csiro.au for this.              in L{Solution} or L{ReducedSolution}.
216       """       """
217         # The whole thing is a bit strange and I blame Rob Woodcock (CSIRO) for
218         # this.
219       def __init__(self,domain,D=None,Y=None,q=None,r=None):       def __init__(self,domain,D=None,Y=None,q=None,r=None):
220           """           """
221           initialize the problem           Initializes the problem.
222    
223           @param domain: domain of the PDE.           @param domain: domain of the PDE
224           @type domain: L{Domain}           @type domain: L{Domain}
225           @param D: coefficient of the solution.           @param D: coefficient of the solution
226           @type D: C{float}, C{int}, L{numarray.NumArray}, L{Data}           @type D: C{float}, C{int}, C{numarray.NumArray}, L{Data}
227           @param Y: right hand side           @param Y: right hand side
228           @type Y: C{float}, C{int}, L{numarray.NumArray}, L{Data}           @type Y: C{float}, C{int}, C{numarray.NumArray}, L{Data}
229           @param q: location of constraints           @param q: location of constraints
230           @type q: C{float}, C{int}, L{numarray.NumArray}, L{Data}           @type q: C{float}, C{int}, C{numarray.NumArray}, L{Data}
231           @param r: value of solution at locations of constraints           @param r: value of solution at locations of constraints
232           @type r: C{float}, C{int}, L{numarray.NumArray}, L{Data}           @type r: C{float}, C{int}, C{numarray.NumArray}, L{Data}
233           """           """
234           self.__domain=domain           self.__domain=domain
235           self.__D=D           self.__D=D
# Line 234  class NoPDE: Line 238  class NoPDE:
238           self.__r=r           self.__r=r
239           self.__u=None           self.__u=None
240           self.__function_space=escript.Solution(self.__domain)           self.__function_space=escript.Solution(self.__domain)
241    
242       def setReducedOn(self):       def setReducedOn(self):
243           """           """
244           sets the L{FunctionSpace} of the solution to L{ReducedSolution}           Sets the L{FunctionSpace} of the solution to L{ReducedSolution}.
245           """           """
246           self.__function_space=escript.ReducedSolution(self.__domain)           self.__function_space=escript.ReducedSolution(self.__domain)
247           self.__u=None           self.__u=None
248    
249       def setReducedOff(self):       def setReducedOff(self):
250           """           """
251           sets the L{FunctionSpace} of the solution to L{Solution}           Sets the L{FunctionSpace} of the solution to L{Solution}.
252           """           """
253           self.__function_space=escript.Solution(self.__domain)           self.__function_space=escript.Solution(self.__domain)
254           self.__u=None           self.__u=None
255            
256       def setValue(self,D=None,Y=None,q=None,r=None):       def setValue(self,D=None,Y=None,q=None,r=None):
257           """           """
258           assigns values to the parameters.           Assigns values to the parameters.
259    
260           @param D: coefficient of the solution.           @param D: coefficient of the solution
261           @type D: C{float}, C{int}, L{numarray.NumArray}, L{Data}           @type D: C{float}, C{int}, C{numarray.NumArray}, L{Data}
262           @param Y: right hand side           @param Y: right hand side
263           @type Y: C{float}, C{int}, L{numarray.NumArray}, L{Data}           @type Y: C{float}, C{int}, C{numarray.NumArray}, L{Data}
264           @param q: location of constraints           @param q: location of constraints
265           @type q: C{float}, C{int}, L{numarray.NumArray}, L{Data}           @type q: C{float}, C{int}, C{numarray.NumArray}, L{Data}
266           @param r: value of solution at locations of constraints           @param r: value of solution at locations of constraints
267           @type r: C{float}, C{int}, L{numarray.NumArray}, L{Data}           @type r: C{float}, C{int}, C{numarray.NumArray}, L{Data}
268           """           """
269           if not D==None:           if not D==None:
270              self.__D=D              self.__D=D
# Line 276  class NoPDE: Line 281  class NoPDE:
281    
282       def getSolution(self):       def getSolution(self):
283           """           """
284           returns the solution           Returns the solution.
285            
286           @return: the solution of the problem           @return: the solution of the problem
287           @rtype: L{Data} object in the L{FunctionSpace} L{Solution} or L{ReducedSolution}.           @rtype: L{Data} object in the L{FunctionSpace} L{Solution} or
288                     L{ReducedSolution}
289           """           """
290           if self.__u==None:           if self.__u==None:
291              if self.__D==None:              if self.__D==None:
# Line 296  class NoPDE: Line 302  class NoPDE:
302                  self.__u*=(1.-q)                  self.__u*=(1.-q)
303                  if not self.__r==None: self.__u+=q*self.__r                  if not self.__r==None: self.__u+=q*self.__r
304           return self.__u           return self.__u
305                
306  class Locator:  class Locator:
307       """       """
308       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
309       spatial coordinate x.         coordinate x.
310        
311       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
312       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.  
313       """       """
314    
315       def __init__(self,where,x=numarray.zeros((3,))):       def __init__(self,where,x=numarray.zeros((3,))):
316         """         """
317         Initializes a Locator to access values in Data objects on the Doamin         Initializes a Locator to access values in Data objects on the Doamin
318         or FunctionSpace where for the sample point which         or FunctionSpace for the sample point which is closest to the given
319         closest to the given point x.         point x.
320    
321         @param where: function space         @param where: function space
322         @type where: L{escript.FunctionSpace}         @type where: L{escript.FunctionSpace}
323         @param x: coefficient of the solution.         @param x: coefficient of the solution
324         @type x: L{numarray.NumArray} or C{list} of L{numarray.NumArray}         @type x: C{numarray.NumArray} or C{list} of C{numarray.NumArray}
325         """         """
326         if isinstance(where,escript.FunctionSpace):         if isinstance(where,escript.FunctionSpace):
327            self.__function_space=where            self.__function_space=where
# Line 350  class Locator: Line 355  class Locator:
355    
356       def getX(self):       def getX(self):
357          """          """
358      Returns the exact coordinates of the Locator.          Returns the exact coordinates of the Locator.
359      """          """
360          return self(self.getFunctionSpace().getX())          return self(self.getFunctionSpace().getX())
361    
362       def getFunctionSpace(self):       def getFunctionSpace(self):
363          """          """
364      Returns the function space of the Locator.          Returns the function space of the Locator.
365      """          """
366          return self.__function_space          return self.__function_space
367    
368       def getId(self,item=None):       def getId(self,item=None):
369          """          """
370      Returns the identifier of the location.          Returns the identifier of the location.
371      """          """
372          if item == None:          if item == None:
373             return self.__id             return self.__id
374          else:          else:
# Line 375  class Locator: Line 380  class Locator:
380    
381       def __call__(self,data):       def __call__(self,data):
382          """          """
383      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.
384      the object is returned.          """
     """  
385          return self.getValue(data)          return self.getValue(data)
386    
387       def getValue(self,data):       def getValue(self,data):
388          """          """
389      Returns the value of data at the Locator if data is a Data object          Returns the value of C{data} at the Locator if C{data} is a L{Data}
390      otherwise the object is returned.          object otherwise the object is returned.
391      """          """
392          if isinstance(data,escript.Data):          if isinstance(data,escript.Data):
393             if data.getFunctionSpace()==self.getFunctionSpace():             if data.getFunctionSpace()==self.getFunctionSpace():
394               dat=data               dat=data
# Line 412  class Locator: Line 416  class Locator:
416    
417  class SolverSchemeException(Exception):  class SolverSchemeException(Exception):
418     """     """
419     exceptions thrown by solvers     This is a generic exception thrown by solvers.
420     """     """
421     pass     pass
422    
423  class IndefinitePreconditioner(SolverSchemeException):  class IndefinitePreconditioner(SolverSchemeException):
424     """     """
425     the preconditioner is not positive definite.     Exception thrown if the preconditioner is not positive definite.
426     """     """
427     pass     pass
428    
429  class MaxIterReached(SolverSchemeException):  class MaxIterReached(SolverSchemeException):
430     """     """
431     maxium number of iteration steps is reached.     Exception thrown if the maximum number of iteration steps is reached.
432     """     """
433     pass     pass
434  class IterationBreakDown(SolverSchemeException):  
435    class CorrectionFailed(SolverSchemeException):
436     """     """
437     iteration scheme econouters an incurable breakdown.     Exception thrown if no convergence has been achieved in the solution
438       correction scheme.
439     """     """
440     pass     pass
441  class NegativeNorm(SolverSchemeException):  
442    class IterationBreakDown(SolverSchemeException):
443     """     """
444     a norm calculation returns a negative norm.     Exception thrown if the iteration scheme encountered an incurable breakdown.
445     """     """
446     pass     pass
447    
448  class IterationHistory(object):  class NegativeNorm(SolverSchemeException):
449     """     """
450     The IterationHistory class is used to define a stopping criterium. It keeps track of the     Exception thrown if a norm calculation returns a negative norm.
    residual norms. The stoppingcriterium indicates termination if the residual norm has been reduced by  
    a given tolerance.  
451     """     """
452     def __init__(self,tolerance=math.sqrt(util.EPSILON),verbose=False):     pass
       """  
       Initialization  
   
       @param tolerance: tolerance  
       @type tolerance: positive C{float}  
       @param verbose: switches on the printing out some information  
       @type verbose: C{bool}  
       """  
       if not tolerance>0.:  
           raise ValueError,"tolerance needs to be positive."  
       self.tolerance=tolerance  
       self.verbose=verbose  
       self.history=[]  
    def stoppingcriterium(self,norm_r,r,x):  
        """  
        returns True if the C{norm_r} is C{tolerance}*C{norm_r[0]} where C{norm_r[0]}  is the residual norm at the first call.  
   
         
        @param norm_r: current residual norm  
        @type norm_r: non-negative C{float}  
        @param r: current residual (not used)  
        @param x: current solution approximation (not used)  
        @return: C{True} is the stopping criterium is fullfilled. Otherwise C{False} is returned.  
        @rtype: C{bool}  
   
        """  
        self.history.append(norm_r)  
        if self.verbose: print "iter: %s:  inner(rhat,r) = %e"#(len(self.history)-1, self.history[-1])  
        return self.history[-1]<=self.tolerance * self.history[0]  
   
    def stoppingcriterium2(self,norm_r,norm_b,solver="GMRES",TOL=None):  
        """  
        returns True if the C{norm_r} is C{tolerance}*C{norm_b}  
   
         
        @param norm_r: current residual norm  
        @type norm_r: non-negative C{float}  
        @param norm_b: norm of right hand side  
        @type norm_b: non-negative C{float}  
        @return: C{True} is the stopping criterium is fullfilled. Otherwise C{False} is returned.  
        @rtype: C{bool}  
   
        """  
        if TOL==None:  
           TOL=self.tolerance  
        self.history.append(norm_r)  
        if self.verbose: print "iter: %s:  norm(r) = %e"#(len(self.history)-1, self.history[-1])  
        return self.history[-1]<=TOL * norm_b  
453    
454  def PCG(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100):  def PCG(r, Aprod, x, Msolve, bilinearform, atol=0, rtol=1.e-8, iter_max=100, initial_guess=True, verbose=False):
455     """     """
456     Solver for     Solver for
457    
458     M{Ax=b}     M{Ax=b}
459    
460     with a symmetric and positive definite operator A (more details required!).     with a symmetric and positive definite operator A (more details required!).
461     It uses the conjugate gradient method with preconditioner M providing an approximation of A.     It uses the conjugate gradient method with preconditioner M providing an
462       approximation of A.
463    
464     The iteration is terminated if the C{stoppingcriterium} function return C{True}.     The iteration is terminated if
465    
466     For details on the preconditioned conjugate gradient method see the book:     M{|r| <= atol+rtol*|r0|}
467    
468     Templates for the Solution of Linear Systems by R. Barrett, M. Berry,     where M{r0} is the initial residual and M{|.|} is the energy norm. In fact
    T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo,  
    C. Romine, and H. van der Vorst.  
469    
470     @param b: the right hand side of the liner system. C{b} is altered.     M{|r| = sqrt( bilinearform(Msolve(r),r))}
471     @type b: any object supporting inplace add (x+=y) and scaling (x=scalar*y)  
472     @param Aprod: returns the value Ax     For details on the preconditioned conjugate gradient method see the book:
473     @type Aprod: function C{Aprod(x)} where C{x} is of the same object like argument C{x}. The returned object needs to be of the same type like argument C{b}.  
474     @param Msolve: solves Mx=r     I{Templates for the Solution of Linear Systems by R. Barrett, M. Berry,
475     @type Msolve: function C{Msolve(r)} where C{r} is of the same type like argument C{b}. The returned object needs to be of the same     T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo,
476  type like argument C{x}.     C. Romine, and H. van der Vorst}.
477     @param bilinearform: inner product C{<x,r>}  
478     @type bilinearform: function C{bilinearform(x,r)} where C{x} is of the same type like argument C{x} and C{r} is . The returned value is a C{float}.     @param r: initial residual M{r=b-Ax}. C{r} is altered.
479     @param stoppingcriterium: function which returns True if a stopping criterium is meet. C{stoppingcriterium} has the arguments C{norm_r}, C{r} and C{x} giving the current norm of the residual (=C{sqrt(bilinearform(Msolve(r),r)}), the current residual and the current solution approximation. C{stoppingcriterium} is called in each iteration step.     @type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
480     @type stoppingcriterium: function that returns C{True} or C{False}     @param x: an initial guess for the solution
    @param x: an initial guess for the solution. If no C{x} is given 0*b is used.  
481     @type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)     @type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
482     @param iter_max: maximum number of iteration steps.     @param Aprod: returns the value Ax
483       @type Aprod: function C{Aprod(x)} where C{x} is of the same object like
484                    argument C{x}. The returned object needs to be of the same type
485                    like argument C{r}.
486       @param Msolve: solves Mx=r
487       @type Msolve: function C{Msolve(r)} where C{r} is of the same type like
488                     argument C{r}. The returned object needs to be of the same
489                     type like argument C{x}.
490       @param bilinearform: inner product C{<x,r>}
491       @type bilinearform: function C{bilinearform(x,r)} where C{x} is of the same
492                           type like argument C{x} and C{r} is. The returned value
493                           is a C{float}.
494       @param atol: absolute tolerance
495       @type atol: non-negative C{float}
496       @param rtol: relative tolerance
497       @type rtol: non-negative C{float}
498       @param iter_max: maximum number of iteration steps
499     @type iter_max: C{int}     @type iter_max: C{int}
500     @return: the solution approximation and the corresponding residual     @return: the solution approximation and the corresponding residual
501     @rtype: C{tuple}     @rtype: C{tuple}
502     @warning: C{b} and C{x} are altered.     @warning: C{r} and C{x} are altered.
503     """     """
504     iter=0     iter=0
    if x==None:  
       x=0*b  
    else:  
       b += (-1)*Aprod(x)  
    r=b  
505     rhat=Msolve(r)     rhat=Msolve(r)
506     d = rhat     d = rhat
507     rhat_dot_r = bilinearform(rhat, r)     rhat_dot_r = bilinearform(rhat, r)
508     if rhat_dot_r<0: raise NegativeNorm,"negative norm."     if rhat_dot_r<0: raise NegativeNorm,"negative norm."
509       norm_r0=math.sqrt(rhat_dot_r)
510       atol2=atol+rtol*norm_r0
511       if atol2<=0:
512          raise ValueError,"Non-positive tolarance."
513       atol2=max(atol2, 100. * util.EPSILON * norm_r0)
514    
515       if verbose: print "PCG: initial residual norm = %e (absolute tolerance = %e)"%(norm_r0, atol2)
516    
517    
518     while not stoppingcriterium(math.sqrt(rhat_dot_r),r,x):     while not math.sqrt(rhat_dot_r) <= atol2:
519         iter+=1         iter+=1
520         if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max         if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
521    
# Line 557  type like argument C{x}. Line 532  type like argument C{x}.
532    
533         rhat_dot_r = rhat_dot_r_new         rhat_dot_r = rhat_dot_r_new
534         if rhat_dot_r<0: raise NegativeNorm,"negative norm."         if rhat_dot_r<0: raise NegativeNorm,"negative norm."
535           if verbose: print "PCG: iteration step %s: residual norm = %e"%(iter, math.sqrt(rhat_dot_r))
536     return x,r     if verbose: print "PCG: tolerance reached after %s steps."%iter
537       return x,r,math.sqrt(rhat_dot_r)
538    
539  class Defect(object):  class Defect(object):
540      """      """
541      defines a non-linear defect F(x) of a variable x      Defines a non-linear defect F(x) of a variable x.
542      """      """
543      def __init__(self):      def __init__(self):
544          """          """
545          initialize defect          Initializes defect.
546          """          """
547          self.setDerivativeIncrementLength()          self.setDerivativeIncrementLength()
548    
549      def bilinearform(self, x0, x1):      def bilinearform(self, x0, x1):
550          """          """
551          returns the inner product of x0 and x1          Returns the inner product of x0 and x1
552          @param x0: a value for x  
553          @param x1: a value for x          @param x0: value for x0
554            @param x1: value for x1
555          @return: the inner product of x0 and x1          @return: the inner product of x0 and x1
556          @rtype: C{float}          @rtype: C{float}
557          """          """
558          return 0          return 0
559          
560      def norm(self,x):      def norm(self,x):
561          """          """
562          the norm of argument C{x}          Returns the norm of argument C{x}.
563    
564          @param x: a value for x          @param x: a value
565          @return: norm of argument x          @return: norm of argument x
566          @rtype: C{float}          @rtype: C{float}
567          @note: by default C{sqrt(self.bilinearform(x,x)} is retrurned.          @note: by default C{sqrt(self.bilinearform(x,x)} is returned.
568          """          """
569          s=self.bilinearform(x,x)          s=self.bilinearform(x,x)
570          if s<0: raise NegativeNorm,"negative norm."          if s<0: raise NegativeNorm,"negative norm."
571          return math.sqrt(s)          return math.sqrt(s)
572    
   
573      def eval(self,x):      def eval(self,x):
574          """          """
575          returns the value F of a given x          Returns the value F of a given C{x}.
576    
577          @param x: value for which the defect C{F} is evalulated.          @param x: value for which the defect C{F} is evaluated
578          @return: value of the defect at C{x}          @return: value of the defect at C{x}
579          """          """
580          return 0          return 0
# Line 608  class Defect(object): Line 584  class Defect(object):
584    
585      def setDerivativeIncrementLength(self,inc=math.sqrt(util.EPSILON)):      def setDerivativeIncrementLength(self,inc=math.sqrt(util.EPSILON)):
586          """          """
587          sets the relative length of the increment used to approximate the derivative of the defect          Sets the relative length of the increment used to approximate the
588          the increment is inc*norm(x)/norm(v)*v in the direction of v with x as a staring point.          derivative of the defect. The increment is inc*norm(x)/norm(v)*v in the
589            direction of v with x as a starting point.
590    
591          @param inc: relative increment length          @param inc: relative increment length
592          @type inc: positive C{float}          @type inc: positive C{float}
# Line 619  class Defect(object): Line 596  class Defect(object):
596    
597      def getDerivativeIncrementLength(self):      def getDerivativeIncrementLength(self):
598          """          """
599          returns the relative increment length used to approximate the derivative of the defect          Returns the relative increment length used to approximate the
600            derivative of the defect.
601          @return: value of the defect at C{x}          @return: value of the defect at C{x}
602          @rtype: positive C{float}          @rtype: positive C{float}
603          """          """
# Line 627  class Defect(object): Line 605  class Defect(object):
605    
606      def derivative(self, F0, x0, v, v_is_normalised=True):      def derivative(self, F0, x0, v, v_is_normalised=True):
607          """          """
608          returns the directional derivative at x0 in the direction of v          Returns the directional derivative at C{x0} in the direction of C{v}.
609    
610          @param F0: value of this defect at x0          @param F0: value of this defect at x0
611          @param x0: value at which derivative is calculated.          @param x0: value at which derivative is calculated
612          @param v: direction          @param v: direction
613          @param v_is_normalised: is true to indicate that C{v} is nomalized (self.norm(v)=0)          @param v_is_normalised: True to indicate that C{v} is nomalized
614                                    (self.norm(v)=0)
615          @return: derivative of this defect at x0 in the direction of C{v}          @return: derivative of this defect at x0 in the direction of C{v}
616          @note: by default numerical evaluation (self.eval(x0+eps*v)-F0)/eps is used but this method          @note: by default numerical evaluation (self.eval(x0+eps*v)-F0)/eps is
617          maybe oepsnew verwritten to use exact evalution.                 used but this method maybe overwritten to use exact evaluation.
618          """          """
619          normx=self.norm(x0)          normx=self.norm(x0)
620          if normx>0:          if normx>0:
# Line 651  class Defect(object): Line 630  class Defect(object):
630          F1=self.eval(x0 + epsnew * v)          F1=self.eval(x0 + epsnew * v)
631          return (F1-F0)/epsnew          return (F1-F0)/epsnew
632    
633  ######################################      ######################################
634  def NewtonGMRES(defect, x, iter_max=100, sub_iter_max=20, atol=0,rtol=1.e-4, sub_tol_max=0.5, gamma=0.9, verbose=False):  def NewtonGMRES(defect, x, iter_max=100, sub_iter_max=20, atol=0,rtol=1.e-4, sub_tol_max=0.5, gamma=0.9, verbose=False):
635     """     """
636     solves a non-linear problem M{F(x)=0} for unknown M{x} using the stopping criterion:     Solves a non-linear problem M{F(x)=0} for unknown M{x} using the stopping
637       criterion:
638    
639     M{norm(F(x) <= atol + rtol * norm(F(x0)}     M{norm(F(x) <= atol + rtol * norm(F(x0)}
640      
641     where M{x0} is the initial guess.     where M{x0} is the initial guess.
642    
643     @param defect: object defining the the function M{F}, C{defect.norm} defines the M{norm} used in the stopping criterion.     @param defect: object defining the function M{F}. C{defect.norm} defines the
644                      M{norm} used in the stopping criterion.
645     @type defect: L{Defect}     @type defect: L{Defect}
646     @param x: initial guess for the solution, C{x} is altered.     @param x: initial guess for the solution, C{x} is altered.
647     @type x: any object type allowing basic operations such as  L{numarray.NumArray}, L{Data}     @type x: any object type allowing basic operations such as
648     @param iter_max: maximum number of iteration steps              C{numarray.NumArray}, L{Data}
649       @param iter_max: maximum number of iteration steps
650     @type iter_max: positive C{int}     @type iter_max: positive C{int}
651     @param sub_iter_max:     @param sub_iter_max: maximum number of inner iteration steps
652     @type sub_iter_max:     @type sub_iter_max: positive C{int}
653     @param atol: absolute tolerance for the solution     @param atol: absolute tolerance for the solution
654     @type atol: positive C{float}     @type atol: positive C{float}
655     @param rtol: relative tolerance for the solution     @param rtol: relative tolerance for the solution
656     @type rtol: positive C{float}     @type rtol: positive C{float}
657     @param gamma: tolerance safety factor for inner iteration     @param gamma: tolerance safety factor for inner iteration
658     @type gamma: positive C{float}, less than 1     @type gamma: positive C{float}, less than 1
659     @param sub_tol_max: upper bound for inner tolerance.     @param sub_tol_max: upper bound for inner tolerance
660     @type sub_tol_max: positive C{float}, less than 1     @type sub_tol_max: positive C{float}, less than 1
661     @return: an approximation of the solution with the desired accuracy     @return: an approximation of the solution with the desired accuracy
662     @rtype: same type as the initial guess.     @rtype: same type as the initial guess
663     """     """
664     lmaxit=iter_max     lmaxit=iter_max
665     if atol<0: raise ValueError,"atol needs to be non-negative."     if atol<0: raise ValueError,"atol needs to be non-negative."
# Line 697  def NewtonGMRES(defect, x, iter_max=100, Line 679  def NewtonGMRES(defect, x, iter_max=100,
679     # main iteration loop     # main iteration loop
680     #     #
681     while not fnrm<=stop_tol:     while not fnrm<=stop_tol:
682              if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max              if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
683              #              #
684          #   adjust sub_tol_          #   adjust sub_tol_
685          #          #
686              if iter > 1:              if iter > 1:
687             rat=fnrm/fnrmo             rat=fnrm/fnrmo
# Line 712  def NewtonGMRES(defect, x, iter_max=100, Line 694  def NewtonGMRES(defect, x, iter_max=100,
694              #     if iter_max in __FDGMRES is reached MaxIterReached is thrown              #     if iter_max in __FDGMRES is reached MaxIterReached is thrown
695              #     if iter_restart -1 is returned as sub_iter              #     if iter_restart -1 is returned as sub_iter
696              #     if  atol is reached sub_iter returns the numer of steps performed to get there              #     if  atol is reached sub_iter returns the numer of steps performed to get there
697              #              #
698              #                #
699              if verbose: print "             subiteration (GMRES) is called with relative tolerance %e."%sub_tol              if verbose: print "             subiteration (GMRES) is called with relative tolerance %e."%sub_tol
700              try:              try:
701                 xc, sub_iter=__FDGMRES(F, defect, x, sub_tol*fnrm, iter_max=iter_max-iter, iter_restart=sub_iter_max)                 xc, sub_iter=__FDGMRES(F, defect, x, sub_tol*fnrm, iter_max=iter_max-iter, iter_restart=sub_iter_max)
# Line 734  def NewtonGMRES(defect, x, iter_max=100, Line 716  def NewtonGMRES(defect, x, iter_max=100,
716    
717  def __givapp(c,s,vin):  def __givapp(c,s,vin):
718      """      """
719      apply a sequence of Givens rotations (c,s) to the recuirsively to the vector vin      Applies a sequence of Givens rotations (c,s) recursively to the vector
720        C{vin}
721    
722      @warning: C{vin} is altered.      @warning: C{vin} is altered.
723      """      """
724      vrot=vin      vrot=vin
725      if isinstance(c,float):      if isinstance(c,float):
726          vrot=[c*vrot[0]-s*vrot[1],s*vrot[0]+c*vrot[1]]          vrot=[c*vrot[0]-s*vrot[1],s*vrot[0]+c*vrot[1]]
727      else:      else:
# Line 756  def __FDGMRES(F0, defect, x0, atol, iter Line 740  def __FDGMRES(F0, defect, x0, atol, iter
740    
741     rho=defect.norm(F0)     rho=defect.norm(F0)
742     if rho<=0.: return x0*0     if rho<=0.: return x0*0
743      
744     v.append(-F0/rho)     v.append(-F0/rho)
745     g[0]=rho     g[0]=rho
746     iter=0     iter=0
747     while rho > atol and iter<iter_restart-1:     while rho > atol and iter<iter_restart-1:
748            if iter  >= iter_max:
749      if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max              raise MaxIterReached,"maximum number of %s steps reached."%iter_max
750    
751          p=defect.derivative(F0,x0,v[iter], v_is_normalised=True)          p=defect.derivative(F0,x0,v[iter], v_is_normalised=True)
752      v.append(p)          v.append(p)
753    
754      v_norm1=defect.norm(v[iter+1])          v_norm1=defect.norm(v[iter+1])
755    
756          # Modified Gram-Schmidt          # Modified Gram-Schmidt
757      for j in range(iter+1):          for j in range(iter+1):
758           h[j][iter]=defect.bilinearform(v[j],v[iter+1])                h[j,iter]=defect.bilinearform(v[j],v[iter+1])
759           v[iter+1]-=h[j][iter]*v[j]              v[iter+1]-=h[j,iter]*v[j]
760          
761      h[iter+1][iter]=defect.norm(v[iter+1])          h[iter+1,iter]=defect.norm(v[iter+1])
762      v_norm2=h[iter+1][iter]          v_norm2=h[iter+1,iter]
763    
764          # Reorthogonalize if needed          # Reorthogonalize if needed
765      if v_norm1 + 0.001*v_norm2 == v_norm1:   #Brown/Hindmarsh condition (default)          if v_norm1 + 0.001*v_norm2 == v_norm1:   #Brown/Hindmarsh condition (default)
766          for j in range(iter+1):                for j in range(iter+1):
767             hr=defect.bilinearform(v[j],v[iter+1])                  hr=defect.bilinearform(v[j],v[iter+1])
768                 h[j][iter]=h[j][iter]+hr                  h[j,iter]=h[j,iter]+hr
769                 v[iter+1] -= hr*v[j]                  v[iter+1] -= hr*v[j]
770    
771          v_norm2=defect.norm(v[iter+1])              v_norm2=defect.norm(v[iter+1])
772          h[iter+1][iter]=v_norm2              h[iter+1,iter]=v_norm2
773          #   watch out for happy breakdown          #   watch out for happy breakdown
774          if not v_norm2 == 0:          if not v_norm2 == 0:
775                  v[iter+1]=v[iter+1]/h[iter+1][iter]              v[iter+1]=v[iter+1]/h[iter+1,iter]
776    
777          #   Form and store the information for the new Givens rotation          #   Form and store the information for the new Givens rotation
778      if iter > 0 :          if iter > 0 :
779          hhat=numarray.zeros(iter+1,numarray.Float64)              hhat=numarray.zeros(iter+1,numarray.Float64)
780          for i in range(iter+1) : hhat[i]=h[i][iter]              for i in range(iter+1) : hhat[i]=h[i,iter]
781          hhat=__givapp(c[0:iter],s[0:iter],hhat);              hhat=__givapp(c[0:iter],s[0:iter],hhat);
782              for i in range(iter+1) : h[i][iter]=hhat[i]              for i in range(iter+1) : h[i,iter]=hhat[i]
783    
784      mu=math.sqrt(h[iter][iter]*h[iter][iter]+h[iter+1][iter]*h[iter+1][iter])          mu=math.sqrt(h[iter,iter]*h[iter,iter]+h[iter+1,iter]*h[iter+1,iter])
785    
786      if mu!=0 :          if mu!=0 :
787          c[iter]=h[iter][iter]/mu              c[iter]=h[iter,iter]/mu
788          s[iter]=-h[iter+1][iter]/mu              s[iter]=-h[iter+1,iter]/mu
789          h[iter][iter]=c[iter]*h[iter][iter]-s[iter]*h[iter+1][iter]              h[iter,iter]=c[iter]*h[iter,iter]-s[iter]*h[iter+1,iter]
790          h[iter+1][iter]=0.0              h[iter+1,iter]=0.0
791          g[iter:iter+2]=__givapp(c[iter],s[iter],g[iter:iter+2])              g[iter:iter+2]=__givapp(c[iter],s[iter],g[iter:iter+2])
792    
793          # Update the residual norm          # Update the residual norm
794          rho=abs(g[iter+1])          rho=abs(g[iter+1])
795      iter+=1          iter+=1
796    
797     # At this point either iter > iter_max or rho < tol.     # At this point either iter > iter_max or rho < tol.
798     # It's time to compute x and leave.             # It's time to compute x and leave.
799     if iter > 0 :     if iter > 0 :
800       y=numarray.zeros(iter,numarray.Float64)           y=numarray.zeros(iter,numarray.Float64)
801       y[iter-1] = g[iter-1] / h[iter-1][iter-1]       y[iter-1] = g[iter-1] / h[iter-1,iter-1]
802       if iter > 1 :         if iter > 1 :
803          i=iter-2            i=iter-2
804          while i>=0 :          while i>=0 :
805            y[i] = ( g[i] - numarray.dot(h[i][i+1:iter], y[i+1:iter])) / h[i][i]            y[i] = ( g[i] - numarray.dot(h[i,i+1:iter], y[i+1:iter])) / h[i,i]
806            i=i-1            i=i-1
807       xhat=v[iter-1]*y[iter-1]       xhat=v[iter-1]*y[iter-1]
808       for i in range(iter-1):       for i in range(iter-1):
809      xhat += v[i]*y[i]      xhat += v[i]*y[i]
810     else :     else :
811        xhat=v[0] * 0        xhat=v[0] * 0
812    
813     if iter<iter_restart-1:     if iter<iter_restart-1:
814        stopped=iter        stopped=iter
815     else:     else:
816        stopped=-1        stopped=-1
817    
818     return xhat,stopped     return xhat,stopped
819    
820  ##############################################  def GMRES(r, Aprod, x, bilinearform, atol=0, rtol=1.e-8, iter_max=100, iter_restart=20, verbose=False,P_R=None):
821  def GMRES(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100, iter_restart=20):     """
822  ################################################     Solver for
823    
824       M{Ax=b}
825    
826       with a general operator A (more details required!).
827       It uses the generalized minimum residual method (GMRES).
828    
829       The iteration is terminated if
830    
831       M{|r| <= atol+rtol*|r0|}
832    
833       where M{r0} is the initial residual and M{|.|} is the energy norm. In fact
834    
835       M{|r| = sqrt( bilinearform(r,r))}
836    
837       @param r: initial residual M{r=b-Ax}. C{r} is altered.
838       @type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
839       @param x: an initial guess for the solution
840       @type x: same like C{r}
841       @param Aprod: returns the value Ax
842       @type Aprod: function C{Aprod(x)} where C{x} is of the same object like
843                    argument C{x}. The returned object needs to be of the same
844                    type like argument C{r}.
845       @param bilinearform: inner product C{<x,r>}
846       @type bilinearform: function C{bilinearform(x,r)} where C{x} is of the same
847                           type like argument C{x} and C{r}. The returned value is
848                           a C{float}.
849       @param atol: absolute tolerance
850       @type atol: non-negative C{float}
851       @param rtol: relative tolerance
852       @type rtol: non-negative C{float}
853       @param iter_max: maximum number of iteration steps
854       @type iter_max: C{int}
855       @param iter_restart: in order to save memory the orthogonalization process
856                            is terminated after C{iter_restart} steps and the
857                            iteration is restarted.
858       @type iter_restart: C{int}
859       @return: the solution approximation and the corresponding residual
860       @rtype: C{tuple}
861       @warning: C{r} and C{x} are altered.
862       """
863     m=iter_restart     m=iter_restart
864       restarted=False
865     iter=0     iter=0
866     xc=x     if rtol>0:
867          r_dot_r = bilinearform(r, r)
868          if r_dot_r<0: raise NegativeNorm,"negative norm."
869          atol2=atol+rtol*math.sqrt(r_dot_r)
870          if verbose: print "GMRES: norm of right hand side = %e (absolute tolerance = %e)"%(math.sqrt(r_dot_r), atol2)
871       else:
872          atol2=atol
873          if verbose: print "GMRES: absolute tolerance = %e"%atol2
874       if atol2<=0:
875          raise ValueError,"Non-positive tolarance."
876    
877     while True:     while True:
878        if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached"%iter_max        if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached"%iter_max
879        xc,stopped=__GMRESm(b*1, Aprod, Msolve, bilinearform, stoppingcriterium, x=xc*1, iter_max=iter_max-iter, iter_restart=m)        if restarted:
880             r2 = r-Aprod(x-x2)
881          else:
882             r2=1*r
883          x2=x*1.
884          x,stopped=_GMRESm(r2, Aprod, x, bilinearform, atol2, iter_max=iter_max-iter, iter_restart=m, verbose=verbose,P_R=P_R)
885          iter+=iter_restart
886        if stopped: break        if stopped: break
887        iter+=iter_restart            if verbose: print "GMRES: restart."
888     return xc        restarted=True
889       if verbose: print "GMRES: tolerance has been reached."
890       return x
891    
892  def __GMRESm(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100, iter_restart=20):  def _GMRESm(r, Aprod, x, bilinearform, atol, iter_max=100, iter_restart=20, verbose=False, P_R=None):
893     iter=0     iter=0
    r=Msolve(b)  
    r_dot_r = bilinearform(r, r)  
    if r_dot_r<0: raise NegativeNorm,"negative norm."  
    norm_b=math.sqrt(r_dot_r)  
894    
895     if x==None:     h=numarray.zeros((iter_restart+1,iter_restart),numarray.Float64)
       x=0*b  
    else:  
       r=Msolve(b-Aprod(x))  
       r_dot_r = bilinearform(r, r)  
       if r_dot_r<0: raise NegativeNorm,"negative norm."  
     
    h=numarray.zeros((iter_restart,iter_restart),numarray.Float64)  
896     c=numarray.zeros(iter_restart,numarray.Float64)     c=numarray.zeros(iter_restart,numarray.Float64)
897     s=numarray.zeros(iter_restart,numarray.Float64)     s=numarray.zeros(iter_restart,numarray.Float64)
898     g=numarray.zeros(iter_restart,numarray.Float64)     g=numarray.zeros(iter_restart+1,numarray.Float64)
899     v=[]     v=[]
900    
901       r_dot_r = bilinearform(r, r)
902       if r_dot_r<0: raise NegativeNorm,"negative norm."
903     rho=math.sqrt(r_dot_r)     rho=math.sqrt(r_dot_r)
904      
905     v.append(r/rho)     v.append(r/rho)
906     g[0]=rho     g[0]=rho
907    
908     while not (stoppingcriterium(rho,norm_b) or iter==iter_restart-1):     if verbose: print "GMRES: initial residual %e (absolute tolerance = %e)"%(rho,atol)
909       while not (rho<=atol or iter==iter_restart):
910    
911      if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max      if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
912    
913      p=Msolve(Aprod(v[iter]))          if P_R!=None:
914                p=Aprod(P_R(v[iter]))
915            else:
916            p=Aprod(v[iter])
917      v.append(p)      v.append(p)
918    
919      v_norm1=math.sqrt(bilinearform(v[iter+1], v[iter+1]))        v_norm1=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
920    
921    # Modified Gram-Schmidt
922        for j in range(iter+1):
923          h[j,iter]=bilinearform(v[j],v[iter+1])
924          v[iter+1]-=h[j,iter]*v[j]
925    
926  # Modified Gram-Schmidt      h[iter+1,iter]=math.sqrt(bilinearform(v[iter+1],v[iter+1]))
927      for j in range(iter+1):      v_norm2=h[iter+1,iter]
       h[j][iter]=bilinearform(v[j],v[iter+1])    
       v[iter+1]-=h[j][iter]*v[j]  
         
     h[iter+1][iter]=math.sqrt(bilinearform(v[iter+1],v[iter+1]))  
     v_norm2=h[iter+1][iter]  
928    
929  # Reorthogonalize if needed  # Reorthogonalize if needed
930      if v_norm1 + 0.001*v_norm2 == v_norm1:   #Brown/Hindmarsh condition (default)      if v_norm1 + 0.001*v_norm2 == v_norm1:   #Brown/Hindmarsh condition (default)
931       for j in range(iter+1):         for j in range(iter+1):
932          hr=bilinearform(v[j],v[iter+1])          hr=bilinearform(v[j],v[iter+1])
933              h[j][iter]=h[j][iter]+hr              h[j,iter]=h[j,iter]+hr
934              v[iter+1] -= hr*v[j]              v[iter+1] -= hr*v[j]
935    
936       v_norm2=math.sqrt(bilinearform(v[iter+1], v[iter+1]))         v_norm2=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
937       h[iter+1][iter]=v_norm2       h[iter+1,iter]=v_norm2
938    
939  #   watch out for happy breakdown  #   watch out for happy breakdown
940          if not v_norm2 == 0:          if not v_norm2 == 0:
941           v[iter+1]=v[iter+1]/h[iter+1][iter]           v[iter+1]=v[iter+1]/h[iter+1,iter]
942    
943  #   Form and store the information for the new Givens rotation  #   Form and store the information for the new Givens rotation
944      if iter > 0 :      if iter > 0: h[:iter+1,iter]=__givapp(c[:iter],s[:iter],h[:iter+1,iter])
945          hhat=numarray.zeros(iter+1,numarray.Float64)      mu=math.sqrt(h[iter,iter]*h[iter,iter]+h[iter+1,iter]*h[iter+1,iter])
         for i in range(iter+1) : hhat[i]=h[i][iter]  
         hhat=__givapp(c[0:iter],s[0:iter],hhat);  
             for i in range(iter+1) : h[i][iter]=hhat[i]  
   
     mu=math.sqrt(h[iter][iter]*h[iter][iter]+h[iter+1][iter]*h[iter+1][iter])  
946    
947      if mu!=0 :      if mu!=0 :
948          c[iter]=h[iter][iter]/mu          c[iter]=h[iter,iter]/mu
949          s[iter]=-h[iter+1][iter]/mu          s[iter]=-h[iter+1,iter]/mu
950          h[iter][iter]=c[iter]*h[iter][iter]-s[iter]*h[iter+1][iter]          h[iter,iter]=c[iter]*h[iter,iter]-s[iter]*h[iter+1,iter]
951          h[iter+1][iter]=0.0          h[iter+1,iter]=0.0
952          g[iter:iter+2]=__givapp(c[iter],s[iter],g[iter:iter+2])          g[iter:iter+2]=__givapp(c[iter],s[iter],g[iter:iter+2])
   
953  # Update the residual norm  # Update the residual norm
954                  
955          rho=abs(g[iter+1])          rho=abs(g[iter+1])
956            if verbose: print "GMRES: iteration step %s: residual %e"%(iter,rho)
957      iter+=1      iter+=1
958    
959  # At this point either iter > iter_max or rho < tol.  # At this point either iter > iter_max or rho < tol.
960  # It's time to compute x and leave.          # It's time to compute x and leave.
961    
962     if iter > 0 :     if verbose: print "GMRES: iteration stopped after %s step."%iter
963       y=numarray.zeros(iter,numarray.Float64)         if iter > 0 :
964       y[iter-1] = g[iter-1] / h[iter-1][iter-1]       y=numarray.zeros(iter,numarray.Float64)
965       if iter > 1 :         y[iter-1] = g[iter-1] / h[iter-1,iter-1]
966          i=iter-2         if iter > 1 :
967            i=iter-2
968          while i>=0 :          while i>=0 :
969            y[i] = ( g[i] - numarray.dot(h[i][i+1:iter], y[i+1:iter])) / h[i][i]            y[i] = ( g[i] - numarray.dot(h[i,i+1:iter], y[i+1:iter])) / h[i,i]
970            i=i-1            i=i-1
971       xhat=v[iter-1]*y[iter-1]       xhat=v[iter-1]*y[iter-1]
972       for i in range(iter-1):       for i in range(iter-1):
973      xhat += v[i]*y[i]      xhat += v[i]*y[i]
974     else : xhat=v[0]     else:
975         xhat=v[0] * 0
976     x += xhat     if P_R!=None:
977     if iter<iter_restart-1:        x += P_R(xhat)
978        stopped=True     else:
979     else:        x += xhat
980       if iter<iter_restart-1:
981          stopped=True
982       else:
983        stopped=False        stopped=False
984    
985     return x,stopped     return x,stopped
986    
987  #################################################  def MINRES(r, Aprod, x, Msolve, bilinearform, atol=0, rtol=1.e-8, iter_max=100):
988  def MINRES(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100):      """
989  #################################################      Solver for
990      #  
991      #  minres solves the system of linear equations Ax = b      M{Ax=b}
992      #  where A is a symmetric matrix (possibly indefinite or singular)  
993      #  and b is a given vector.      with a symmetric and positive definite operator A (more details required!).
994      #        It uses the minimum residual method (MINRES) with preconditioner M
995      #  "A" may be a dense or sparse matrix (preferably sparse!)      providing an approximation of A.
996      #  or the name of a function such that  
997      #               y = A(x)      The iteration is terminated if
998      #  returns the product y = Ax for any given vector x.  
999      #      M{|r| <= atol+rtol*|r0|}
1000      #  "M" defines a positive-definite preconditioner M = C C'.  
1001      #  "M" may be a dense or sparse matrix (preferably sparse!)      where M{r0} is the initial residual and M{|.|} is the energy norm. In fact
1002      #  or the name of a function such that  
1003      #  solves the system My = x for any given vector x.      M{|r| = sqrt( bilinearform(Msolve(r),r))}
1004      #  
1005      #      For details on the preconditioned conjugate gradient method see the book:
1006        
1007        I{Templates for the Solution of Linear Systems by R. Barrett, M. Berry,
1008        T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo,
1009        C. Romine, and H. van der Vorst}.
1010    
1011        @param r: initial residual M{r=b-Ax}. C{r} is altered.
1012        @type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1013        @param x: an initial guess for the solution
1014        @type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1015        @param Aprod: returns the value Ax
1016        @type Aprod: function C{Aprod(x)} where C{x} is of the same object like
1017                     argument C{x}. The returned object needs to be of the same
1018                     type like argument C{r}.
1019        @param Msolve: solves Mx=r
1020        @type Msolve: function C{Msolve(r)} where C{r} is of the same type like
1021                      argument C{r}. The returned object needs to be of the same
1022                      type like argument C{x}.
1023        @param bilinearform: inner product C{<x,r>}
1024        @type bilinearform: function C{bilinearform(x,r)} where C{x} is of the same
1025                            type like argument C{x} and C{r} is. The returned value
1026                            is a C{float}.
1027        @param atol: absolute tolerance
1028        @type atol: non-negative C{float}
1029        @param rtol: relative tolerance
1030        @type rtol: non-negative C{float}
1031        @param iter_max: maximum number of iteration steps
1032        @type iter_max: C{int}
1033        @return: the solution approximation and the corresponding residual
1034        @rtype: C{tuple}
1035        @warning: C{r} and C{x} are altered.
1036        """
1037      #------------------------------------------------------------------      #------------------------------------------------------------------
1038      # Set up y and v for the first Lanczos vector v1.      # Set up y and v for the first Lanczos vector v1.
1039      # y  =  beta1 P' v1,  where  P = C**(-1).      # y  =  beta1 P' v1,  where  P = C**(-1).
1040      # v is really P' v1.      # v is really P' v1.
1041      #------------------------------------------------------------------      #------------------------------------------------------------------
1042      if x==None:      r1    = r
1043        x=0*b      y = Msolve(r)
1044      else:      beta1 = bilinearform(y,r)
       b += (-1)*Aprod(x)  
1045    
     r1    = b  
     y = Msolve(b)  
     beta1 = bilinearform(y,b)  
   
1046      if beta1< 0: raise NegativeNorm,"negative norm."      if beta1< 0: raise NegativeNorm,"negative norm."
1047    
1048      #  If b = 0 exactly, stop with x = 0.      #  If r = 0 exactly, stop with x
1049      if beta1==0: return x*0.      if beta1==0: return x
1050    
1051      if beta1> 0:      if beta1> 0: beta1  = math.sqrt(beta1)
       beta1  = math.sqrt(beta1)        
1052    
1053      #------------------------------------------------------------------      #------------------------------------------------------------------
1054      # Initialize quantities.      # Initialize quantities.
# Line 1008  def MINRES(b, Aprod, Msolve, bilinearfor Line 1068  def MINRES(b, Aprod, Msolve, bilinearfor
1068      ynorm2 = 0      ynorm2 = 0
1069      cs     = -1      cs     = -1
1070      sn     = 0      sn     = 0
1071      w      = b*0.      w      = r*0.
1072      w2     = b*0.      w2     = r*0.
1073      r2     = r1      r2     = r1
1074      eps    = 0.0001      eps    = 0.0001
1075    
1076      #---------------------------------------------------------------------      #---------------------------------------------------------------------
1077      # Main iteration loop.      # Main iteration loop.
1078      # --------------------------------------------------------------------      # --------------------------------------------------------------------
1079      while not stoppingcriterium(rnorm,Anorm*ynorm,'MINRES'):    #  checks ||r|| < (||A|| ||x||) * TOL      while not rnorm<=atol+rtol*Anorm*ynorm:    #  checks ||r|| < (||A|| ||x||) * TOL
1080    
1081      if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max      if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
1082          iter    = iter  +  1          iter    = iter  +  1
# Line 1035  def MINRES(b, Aprod, Msolve, bilinearfor Line 1095  def MINRES(b, Aprod, Msolve, bilinearfor
1095          #-----------------------------------------------------------------          #-----------------------------------------------------------------
1096          s = 1/beta                 # Normalize previous vector (in y).          s = 1/beta                 # Normalize previous vector (in y).
1097          v = s*y                    # v = vk if P = I          v = s*y                    # v = vk if P = I
1098        
1099          y      = Aprod(v)          y      = Aprod(v)
1100        
1101          if iter >= 2:          if iter >= 2:
1102            y = y - (beta/oldb)*r1            y = y - (beta/oldb)*r1
1103    
1104          alfa   = bilinearform(v,y)              # alphak          alfa   = bilinearform(v,y)              # alphak
1105          y      += (- alfa/beta)*r2          y      += (- alfa/beta)*r2
1106          r1     = r2          r1     = r2
1107          r2     = y          r2     = y
1108          y = Msolve(r2)          y = Msolve(r2)
# Line 1052  def MINRES(b, Aprod, Msolve, bilinearfor Line 1112  def MINRES(b, Aprod, Msolve, bilinearfor
1112    
1113          beta   = math.sqrt( beta )          beta   = math.sqrt( beta )
1114          tnorm2 = tnorm2 + alfa*alfa + oldb*oldb + beta*beta          tnorm2 = tnorm2 + alfa*alfa + oldb*oldb + beta*beta
1115            
1116          if iter==1:                 # Initialize a few things.          if iter==1:                 # Initialize a few things.
1117            gmax   = abs( alfa )      # alpha1            gmax   = abs( alfa )      # alpha1
1118            gmin   = gmax             # alpha1            gmin   = gmax             # alpha1
# Line 1060  def MINRES(b, Aprod, Msolve, bilinearfor Line 1120  def MINRES(b, Aprod, Msolve, bilinearfor
1120          # Apply previous rotation Qk-1 to get          # Apply previous rotation Qk-1 to get
1121          #   [deltak epslnk+1] = [cs  sn][dbark    0   ]          #   [deltak epslnk+1] = [cs  sn][dbark    0   ]
1122          #   [gbar k dbar k+1]   [sn -cs][alfak betak+1].          #   [gbar k dbar k+1]   [sn -cs][alfak betak+1].
1123        
1124          oldeps = epsln          oldeps = epsln
1125          delta  = cs * dbar  +  sn * alfa  # delta1 = 0         deltak          delta  = cs * dbar  +  sn * alfa  # delta1 = 0         deltak
1126          gbar   = sn * dbar  -  cs * alfa  # gbar 1 = alfa1     gbar k          gbar   = sn * dbar  -  cs * alfa  # gbar 1 = alfa1     gbar k
# Line 1070  def MINRES(b, Aprod, Msolve, bilinearfor Line 1130  def MINRES(b, Aprod, Msolve, bilinearfor
1130          # Compute the next plane rotation Qk          # Compute the next plane rotation Qk
1131    
1132          gamma  = math.sqrt(gbar*gbar+beta*beta)  # gammak          gamma  = math.sqrt(gbar*gbar+beta*beta)  # gammak
1133          gamma  = max(gamma,eps)          gamma  = max(gamma,eps)
1134          cs     = gbar / gamma             # ck          cs     = gbar / gamma             # ck
1135          sn     = beta / gamma             # sk          sn     = beta / gamma             # sk
1136          phi    = cs * phibar              # phik          phi    = cs * phibar              # phik
# Line 1078  def MINRES(b, Aprod, Msolve, bilinearfor Line 1138  def MINRES(b, Aprod, Msolve, bilinearfor
1138    
1139          # Update  x.          # Update  x.
1140    
1141          denom = 1/gamma          denom = 1/gamma
1142          w1    = w2          w1    = w2
1143          w2    = w          w2    = w
1144          w     = (v - oldeps*w1 - delta*w2) * denom          w     = (v - oldeps*w1 - delta*w2) * denom
1145          x     +=  phi*w          x     +=  phi*w
1146    
# Line 1095  def MINRES(b, Aprod, Msolve, bilinearfor Line 1155  def MINRES(b, Aprod, Msolve, bilinearfor
1155    
1156          # Estimate various norms and test for convergence.          # Estimate various norms and test for convergence.
1157    
1158          Anorm  = math.sqrt( tnorm2 )          Anorm  = math.sqrt( tnorm2 )
1159          ynorm  = math.sqrt( ynorm2 )          ynorm  = math.sqrt( ynorm2 )
1160    
1161          rnorm  = phibar          rnorm  = phibar
1162    
1163      return x      return x
1164    
1165  def TFQMR(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100):  def TFQMR(r, Aprod, x, bilinearform, atol=0, rtol=1.e-8, iter_max=100):
1166      """
1167      Solver for
1168    
1169  # TFQMR solver for linear systems    M{Ax=b}
1170  #  
1171  #    with a general operator A (more details required!).
1172  # initialization    It uses the Transpose-Free Quasi-Minimal Residual method (TFQMR).
 #  
   errtol = math.sqrt(bilinearform(b,b))  
   norm_b=errtol  
   kmax  = iter_max  
   error = []  
   
   if math.sqrt(bilinearform(x,x)) != 0.0:  
     r = b - Aprod(x)  
   else:  
     r = b  
1173    
1174    r=Msolve(r)    The iteration is terminated if
1175    
1176      M{|r| <= atol+rtol*|r0|}
1177    
1178      where M{r0} is the initial residual and M{|.|} is the energy norm. In fact
1179    
1180      M{|r| = sqrt( bilinearform(r,r))}
1181    
1182      @param r: initial residual M{r=b-Ax}. C{r} is altered.
1183      @type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1184      @param x: an initial guess for the solution
1185      @type x: same like C{r}
1186      @param Aprod: returns the value Ax
1187      @type Aprod: function C{Aprod(x)} where C{x} is of the same object like
1188                   argument C{x}. The returned object needs to be of the same type
1189                   like argument C{r}.
1190      @param bilinearform: inner product C{<x,r>}
1191      @type bilinearform: function C{bilinearform(x,r)} where C{x} is of the same
1192                          type like argument C{x} and C{r}. The returned value is
1193                          a C{float}.
1194      @param atol: absolute tolerance
1195      @type atol: non-negative C{float}
1196      @param rtol: relative tolerance
1197      @type rtol: non-negative C{float}
1198      @param iter_max: maximum number of iteration steps
1199      @type iter_max: C{int}
1200      @rtype: C{tuple}
1201      @warning: C{r} and C{x} are altered.
1202      """
1203    u1=0    u1=0
1204    u2=0    u2=0
1205    y1=0    y1=0
1206    y2=0    y2=0
1207    
1208    w = r    w = r
1209    y1 = r    y1 = r
1210    iter = 0    iter = 0
1211    d = 0    d = 0
1212        v = Aprod(y1)
   v = Msolve(Aprod(y1))  
1213    u1 = v    u1 = v
1214      
1215    theta = 0.0;    theta = 0.0;
1216    eta = 0.0;    eta = 0.0;
1217    tau = math.sqrt(bilinearform(r,r))    rho=bilinearform(r,r)
1218    error = [ error, tau ]    if rho < 0: raise NegativeNorm,"negative norm."
1219    rho = tau * tau    tau = math.sqrt(rho)
1220    m=1    norm_r0=tau
1221  #    while tau>atol+rtol*norm_r0:
 #  TFQMR iteration  
 #  
 #  while ( iter < kmax-1 ):  
     
   while not stoppingcriterium(tau*math.sqrt ( m + 1 ),norm_b,'TFQMR'):  
1222      if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max      if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
1223    
1224      sigma = bilinearform(r,v)      sigma = bilinearform(r,v)
1225        if sigma == 0.0: raise IterationBreakDown,'TFQMR breakdown, sigma=0'
     if ( sigma == 0.0 ):  
       raise 'TFQMR breakdown, sigma=0'  
       
1226    
1227      alpha = rho / sigma      alpha = rho / sigma
1228    
# Line 1162  def TFQMR(b, Aprod, Msolve, bilinearform Line 1232  def TFQMR(b, Aprod, Msolve, bilinearform
1232  #  #
1233        if ( j == 1 ):        if ( j == 1 ):
1234          y2 = y1 - alpha * v          y2 = y1 - alpha * v
1235          u2 = Msolve(Aprod(y2))          u2 = Aprod(y2)
1236    
1237        m = 2 * (iter+1) - 2 + (j+1)        m = 2 * (iter+1) - 2 + (j+1)
1238        if j==0:        if j==0:
1239           w = w - alpha * u1           w = w - alpha * u1
1240           d = y1 + ( theta * theta * eta / alpha ) * d           d = y1 + ( theta * theta * eta / alpha ) * d
1241        if j==1:        if j==1:
# Line 1180  def TFQMR(b, Aprod, Msolve, bilinearform Line 1250  def TFQMR(b, Aprod, Msolve, bilinearform
1250  #  #
1251  #  Try to terminate the iteration at each pass through the loop  #  Try to terminate the iteration at each pass through the loop
1252  #  #
1253       # if ( tau * math.sqrt ( m + 1 ) <= errtol ):      if rho == 0.0: raise IterationBreakDown,'TFQMR breakdown, rho=0'
      #   error = [ error, tau ]  
      #   total_iters = iter  
      #   break  
         
   
     if ( rho == 0.0 ):  
       raise 'TFQMR breakdown, rho=0'  
       
1254    
1255      rhon = bilinearform(r,w)      rhon = bilinearform(r,w)
1256      beta = rhon / rho;      beta = rhon / rho;
1257      rho = rhon;      rho = rhon;
1258      y1 = w + beta * y2;      y1 = w + beta * y2;
1259      u1 = Msolve(Aprod(y1))      u1 = Aprod(y1)
1260      v = u1 + beta * ( u2 + beta * v )      v = u1 + beta * ( u2 + beta * v )
1261      error = [ error, tau ]  
1262      total_iters = iter      iter += 1
       
     iter = iter + 1  
1263    
1264    return x    return x
1265    
# Line 1208  def TFQMR(b, Aprod, Msolve, bilinearform Line 1268  def TFQMR(b, Aprod, Msolve, bilinearform
1268    
1269  class ArithmeticTuple(object):  class ArithmeticTuple(object):
1270     """     """
1271     tuple supporting inplace update x+=y and scaling x=a*y where x,y is an ArithmeticTuple and a is a float.     Tuple supporting inplace update x+=y and scaling x=a*y where C{x,y} is an
1272       ArithmeticTuple and C{a} is a float.
1273    
1274     example of usage:     Example of usage::
1275    
1276     from esys.escript import Data         from esys.escript import Data
1277     from numarray import array         from numarray import array
1278     a=Data(...)         a=Data(...)
1279     b=array([1.,4.])         b=array([1.,4.])
1280     x=ArithmeticTuple(a,b)         x=ArithmeticTuple(a,b)
1281     y=5.*x         y=5.*x
1282    
1283     """     """
1284     def __init__(self,*args):     def __init__(self,*args):
1285         """         """
1286         initialize object with elements args.         Initializes object with elements C{args}.
1287    
1288         @param args: tuple of object that support implace add (x+=y) and scaling (x=a*y)         @param args: tuple of objects that support inplace add (x+=y) and
1289                        scaling (x=a*y)
1290         """         """
1291         self.__items=list(args)         self.__items=list(args)
1292    
1293     def __len__(self):     def __len__(self):
1294         """         """
1295         number of items         Returns the number of items.
1296    
1297         @return: number of items         @return: number of items
1298         @rtype: C{int}         @rtype: C{int}
# Line 1239  class ArithmeticTuple(object): Line 1301  class ArithmeticTuple(object):
1301    
1302     def __getitem__(self,index):     def __getitem__(self,index):
1303         """         """
1304         get an item         Returns item at specified position.
1305    
1306         @param index: item to be returned         @param index: index of item to be returned
1307         @type index: C{int}         @type index: C{int}
1308         @return: item with index C{index}         @return: item with index C{index}
1309         """         """
# Line 1249  class ArithmeticTuple(object): Line 1311  class ArithmeticTuple(object):
1311    
1312     def __mul__(self,other):     def __mul__(self,other):
1313         """         """
1314         scaling from the right         Scales by C{other} from the right.
1315    
1316         @param other: scaling factor         @param other: scaling factor
1317         @type other: C{float}         @type other: C{float}
# Line 1257  class ArithmeticTuple(object): Line 1319  class ArithmeticTuple(object):
1319         @rtype: L{ArithmeticTuple}         @rtype: L{ArithmeticTuple}
1320         """         """
1321         out=[]         out=[]
1322         try:           try:
1323             l=len(other)             l=len(other)
1324             if l!=len(self):             if l!=len(self):
1325                 raise ValueError,"length of of arguments don't match."                 raise ValueError,"length of arguments don't match."
1326             for i in range(l): out.append(self[i]*other[i])             for i in range(l): out.append(self[i]*other[i])
1327         except TypeError:         except TypeError:
1328         for i in range(len(self)): out.append(self[i]*other)             for i in range(len(self)): out.append(self[i]*other)
1329         return ArithmeticTuple(*tuple(out))         return ArithmeticTuple(*tuple(out))
1330    
1331     def __rmul__(self,other):     def __rmul__(self,other):
1332         """         """
1333         scaling from the left         Scales by C{other} from the left.
1334    
1335         @param other: scaling factor         @param other: scaling factor
1336         @type other: C{float}         @type other: C{float}
# Line 1276  class ArithmeticTuple(object): Line 1338  class ArithmeticTuple(object):
1338         @rtype: L{ArithmeticTuple}         @rtype: L{ArithmeticTuple}
1339         """         """
1340         out=[]         out=[]
1341         try:           try:
1342             l=len(other)             l=len(other)
1343             if l!=len(self):             if l!=len(self):
1344                 raise ValueError,"length of of arguments don't match."                 raise ValueError,"length of arguments don't match."
1345             for i in range(l): out.append(other[i]*self[i])             for i in range(l): out.append(other[i]*self[i])
1346         except TypeError:         except TypeError:
1347         for i in range(len(self)): out.append(other*self[i])             for i in range(len(self)): out.append(other*self[i])
1348         return ArithmeticTuple(*tuple(out))         return ArithmeticTuple(*tuple(out))
1349    
1350     def __div__(self,other):     def __div__(self,other):
1351         """         """
1352         dividing from the right         Scales by (1/C{other}) from the right.
1353    
1354         @param other: scaling factor         @param other: scaling factor
1355         @type other: C{float}         @type other: C{float}
# Line 1298  class ArithmeticTuple(object): Line 1360  class ArithmeticTuple(object):
1360    
1361     def __rdiv__(self,other):     def __rdiv__(self,other):
1362         """         """
1363         dividing from the left         Scales by (1/C{other}) from the left.
1364    
1365         @param other: scaling factor         @param other: scaling factor
1366         @type other: C{float}         @type other: C{float}
# Line 1306  class ArithmeticTuple(object): Line 1368  class ArithmeticTuple(object):
1368         @rtype: L{ArithmeticTuple}         @rtype: L{ArithmeticTuple}
1369         """         """
1370         out=[]         out=[]
1371         try:           try:
1372             l=len(other)             l=len(other)
1373             if l!=len(self):             if l!=len(self):
1374                 raise ValueError,"length of of arguments don't match."                 raise ValueError,"length of arguments don't match."
1375             for i in range(l): out.append(other[i]/self[i])             for i in range(l): out.append(other[i]/self[i])
1376         except TypeError:         except TypeError:
1377         for i in range(len(self)): out.append(other/self[i])             for i in range(len(self)): out.append(other/self[i])
1378         return ArithmeticTuple(*tuple(out))         return ArithmeticTuple(*tuple(out))
1379      
1380     def __iadd__(self,other):     def __iadd__(self,other):
1381         """         """
1382         in-place add of other to self         Inplace addition of C{other} to self.
1383    
1384         @param other: increment         @param other: increment
1385         @type other: C{ArithmeticTuple}         @type other: C{ArithmeticTuple}
1386         """         """
1387         if len(self) != len(other):         if len(self) != len(other):
1388             raise ValueError,"tuple length must match."             raise ValueError,"tuple lengths must match."
1389         for i in range(len(self)):         for i in range(len(self)):
1390             self.__items[i]+=other[i]             self.__items[i]+=other[i]
1391         return self         return self
1392    
1393     def __add__(self,other):     def __add__(self,other):
1394         """         """
1395         add other to self         Adds C{other} to self.
1396    
1397         @param other: increment         @param other: increment
1398         @type other: C{ArithmeticTuple}         @type other: C{ArithmeticTuple}
1399         """         """
1400         out=[]         out=[]
1401         try:           try:
1402             l=len(other)             l=len(other)
1403             if l!=len(self):             if l!=len(self):
1404                 raise ValueError,"length of of arguments don't match."                 raise ValueError,"length of arguments don't match."
1405             for i in range(l): out.append(self[i]+other[i])             for i in range(l): out.append(self[i]+other[i])
1406         except TypeError:         except TypeError:
1407         for i in range(len(self)): out.append(self[i]+other)             for i in range(len(self)): out.append(self[i]+other)
1408         return ArithmeticTuple(*tuple(out))         return ArithmeticTuple(*tuple(out))
1409    
1410     def __sub__(self,other):     def __sub__(self,other):
1411         """         """
1412         subtract other from self         Subtracts C{other} from self.
1413    
1414         @param other: increment         @param other: decrement
1415         @type other: C{ArithmeticTuple}         @type other: C{ArithmeticTuple}
1416         """         """
1417         out=[]         out=[]
1418         try:           try:
1419             l=len(other)             l=len(other)
1420             if l!=len(self):             if l!=len(self):
1421                 raise ValueError,"length of of arguments don't match."                 raise ValueError,"length of arguments don't match."
1422             for i in range(l): out.append(self[i]-other[i])             for i in range(l): out.append(self[i]-other[i])
1423         except TypeError:         except TypeError:
1424         for i in range(len(self)): out.append(self[i]-other)             for i in range(len(self)): out.append(self[i]-other)
1425         return ArithmeticTuple(*tuple(out))         return ArithmeticTuple(*tuple(out))
1426      
1427     def __isub__(self,other):     def __isub__(self,other):
1428         """         """
1429         subtract other from self         Inplace subtraction of C{other} from self.
1430    
1431         @param other: increment         @param other: decrement
1432         @type other: C{ArithmeticTuple}         @type other: C{ArithmeticTuple}
1433         """         """
1434         if len(self) != len(other):         if len(self) != len(other):
# Line 1377  class ArithmeticTuple(object): Line 1439  class ArithmeticTuple(object):
1439    
1440     def __neg__(self):     def __neg__(self):
1441         """         """
1442         negate         Negates values.
   
1443         """         """
1444         out=[]         out=[]
1445         for i in range(len(self)):         for i in range(len(self)):
# Line 1388  class ArithmeticTuple(object): Line 1449  class ArithmeticTuple(object):
1449    
1450  class HomogeneousSaddlePointProblem(object):  class HomogeneousSaddlePointProblem(object):
1451        """        """
1452        This provides a framwork for solving linear homogeneous saddle point problem of the form        This class provides a framework for solving linear homogeneous saddle
1453          point problems of the form::
              Av+B^*p=f  
              Bv    =0  
1454    
1455        for the unknowns v and p and given operators A and B and given right hand side f.            M{Av+B^*p=f}
1456        B^* is the adjoint operator of B is the given inner product.            M{Bv     =0}
1457    
1458          for the unknowns M{v} and M{p} and given operators M{A} and M{B} and
1459          given right hand side M{f}. M{B^*} is the adjoint operator of M{B}.
1460        """        """
1461        def __init__(self,**kwargs):        def __init__(self,**kwargs):
1462          self.setTolerance()          self.setTolerance()
1463          self.setToleranceReductionFactor()          self.setAbsoluteTolerance()
1464            self.setSubProblemTolerance()
1465    
1466          #=============================================================
1467        def initialize(self):        def initialize(self):
1468          """          """
1469          initialize the problem (overwrite)          Initializes the problem (overwrite).
1470          """          """
1471          pass          pass
1472        def B(self,v):  
1473          def inner_pBv(self,p,v):
1474           """           """
1475           returns Bv (overwrite)           Returns inner product of element p and Bv (overwrite).
          @rtype: equal to the type of p  
1476    
1477           @note: boundary conditions on p should be zero!           @param p: a pressure increment
1478             @param v: a residual
1479             @return: inner product of element p and Bv
1480             @rtype: C{float}
1481             @note: used if PCG is applied.
1482           """           """
1483           pass           raise NotImplementedError,"no inner product for p implemented."
1484    
1485        def inner(self,p0,p1):        def inner_p(self,p0,p1):
1486           """           """
1487           returns inner product of two element p0 and p1  (overwrite)           Returns inner product of p0 and p1 (overwrite).
1488            
1489           @type p0: equal to the type of p           @param p0: a pressure
1490           @type p1: equal to the type of p           @param p1: a pressure
1491             @return: inner product of p0 and p1
1492           @rtype: C{float}           @rtype: C{float}
1493             """
1494             raise NotImplementedError,"no inner product for p implemented."
1495      
1496          def norm_v(self,v):
1497             """
1498             Returns the norm of v (overwrite).
1499    
1500           @rtype: equal to the type of p           @param v: a velovity
1501             @return: norm of v
1502             @rtype: non-negative C{float}
1503             """
1504             raise NotImplementedError,"no norm of v implemented."
1505    
1506    
1507          def getV(self, p, v0):
1508           """           """
1509           pass           return the value for v for a given p (overwrite)
1510    
1511             @param p: a pressure
1512             @param v0: a initial guess for the value v to return.
1513             @return: v given as M{v= A^{-1} (f-B^*p)}
1514             """
1515             raise NotImplementedError,"no v calculation implemented."
1516            
1517          def norm_Bv(self,v):
1518            """
1519            Returns Bv (overwrite).
1520    
1521            @rtype: equal to the type of p
1522            @note: boundary conditions on p should be zero!
1523            """
1524            raise NotImplementedError, "no operator B implemented."
1525    
1526        def solve_A(self,u,p):        def solve_AinvBt(self,p):
1527           """           """
1528           solves Av=f-Au-B^*p with accuracy self.getReducedTolerance() (overwrite)           Solves M{Av=B^*p} with accuracy L{self.getSubProblemTolerance()}
1529             (overwrite).
1530    
1531           @rtype: equal to the type of v           @param p: a pressure increment
1532             @return: the solution of M{Av=B^*p}
1533           @note: boundary conditions on v should be zero!           @note: boundary conditions on v should be zero!
1534           """           """
1535           pass           raise NotImplementedError,"no operator A implemented."
1536    
1537        def solve_prec(self,p):        def solve_precB(self,v):
1538           """           """
1539           provides a preconditioner for BA^{-1}B^* with accuracy self.getReducedTolerance() (overwrite)           Provides a preconditioner for M{BA^{-1}B^*} with accuracy
1540             L{self.getSubProblemTolerance()} (overwrite).
1541    
1542           @rtype: equal to the type of p           @rtype: equal to the type of p
1543             @note: boundary conditions on p should be zero!
1544           """           """
1545           pass           raise NotImplementedError,"no preconditioner for Schur complement implemented."
1546          #=============================================================
1547          def __Aprod_PCG(self,p):
1548              return self.solve_AinvBt(p)
1549    
1550          def __inner_PCG(self,p,v):
1551             return self.inner_pBv(p,v)
1552    
1553          def __Msolve_PCG(self,v):
1554              return self.solve_precB(v)
1555          #=============================================================
1556          def __Aprod_GMRES(self,p):
1557              return self.solve_precB(self.solve_AinvBt(p))
1558          def __inner_GMRES(self,p0,p1):
1559             return self.inner_p(p0,p1)
1560          #=============================================================
1561          def norm_p(self,p):
1562              """
1563              calculates the norm of C{p}
1564              
1565              @param p: a pressure
1566              @return: the norm of C{p} using the inner product for pressure
1567              @rtype: C{float}
1568              """
1569              f=self.inner_p(p,p)
1570              if f<0: raise ValueError,"negative pressure norm."
1571              return math.sqrt(f)
1572              
1573    
1574        def stoppingcriterium(self,Bv,v,p):        def solve(self,v,p,max_iter=20, verbose=False, show_details=False, usePCG=True, iter_restart=20, max_correction_steps=10):
1575           """           """
1576           returns a True if iteration is terminated. (overwrite)           Solves the saddle point problem using initial guesses v and p.
1577    
1578           @rtype: C{bool}           @param v: initial guess for velocity
1579             @param p: initial guess for pressure
1580             @type v: L{Data}
1581             @type p: L{Data}
1582             @param usePCG: indicates the usage of the PCG rather than GMRES scheme.
1583             @param max_iter: maximum number of iteration steps per correction
1584                              attempt
1585             @param verbose: if True, shows information on the progress of the
1586                             saddlepoint problem solver.
1587             @param show_details: if True, shows details of the sub problem solver
1588             @param iter_restart: restart the iteration after C{iter_restart} steps
1589                                  (only used if useUzaw=False)
1590             @type usePCG: C{bool}
1591             @type max_iter: C{int}
1592             @type verbose: C{bool}
1593             @type show_details: C{bool}
1594             @type iter_restart: C{int}
1595             @rtype: C{tuple} of L{Data} objects
1596           """           """
1597           pass           self.verbose=verbose
1598                         self.show_details=show_details and self.verbose
1599        def __inner(self,p,r):           rtol=self.getTolerance()
1600           return self.inner(p,r[1])           atol=self.getAbsoluteTolerance()
1601             correction_step=0
1602             converged=False
1603             while not converged:
1604                  # calculate velocity for current pressure:
1605                  v=self.getV(p,v)
1606                  #
1607                  norm_v=self.norm_v(v)
1608                  norm_Bv=self.norm_Bv(v)
1609                  ATOL=norm_v*rtol+atol
1610                  if self.verbose: print "saddle point solver: norm v= %e, norm_Bv= %e, tolerance = %e."%(norm_v, norm_Bv,ATOL)
1611                  if not ATOL>0: raise ValueError,"overall absolute tolerance needs to be positive."
1612                  if norm_Bv <= ATOL:
1613                     converged=True
1614                  else:
1615                     correction_step+=1
1616                     if correction_step>max_correction_steps:
1617                          raise CorrectionFailed,"Given up after %d correction steps."%correction_step
1618                     dp=self.solve_precB(v)
1619                     if usePCG:
1620                       norm2=self.inner_pBv(dp,v)
1621                       if norm2<0: raise ValueError,"negative PCG norm."
1622                       norm2=math.sqrt(norm2)
1623                     else:
1624                       norm2=self.norm_p(dp)
1625                     ATOL_ITER=ATOL/norm_Bv*norm2
1626                     if self.verbose: print "saddle point solver: tolerance for solver: %e"%ATOL_ITER
1627                     if usePCG:
1628                           p,v0,a_norm=PCG(v,self.__Aprod_PCG,p,self.__Msolve_PCG,self.__inner_PCG,atol=ATOL_ITER, rtol=0.,iter_max=max_iter, verbose=self.verbose)
1629                     else:
1630                           p=GMRES(dp,self.__Aprod_GMRES, p, self.__inner_GMRES,atol=ATOL_ITER, rtol=0.,iter_max=max_iter, iter_restart=iter_restart, verbose=self.verbose)
1631             if self.verbose: print "saddle point solver: tolerance reached."
1632         return v,p
1633    
1634        def __inner_p(self,p1,p2):        #========================================================================
1635           return self.inner(p1,p2)        def setTolerance(self,tolerance=1.e-4):
1636                   """
1637        def __inner_a(self,a1,a2):           Sets the relative tolerance for (v,p).
          return self.inner_a(a1,a2)  
1638    
1639        def __inner_a1(self,a1,a2):           @param tolerance: tolerance to be used
1640           return self.inner(a1[1],a2[1])           @type tolerance: non-negative C{float}
1641             """
1642             if tolerance<0:
1643                 raise ValueError,"tolerance must be positive."
1644             self.__rtol=tolerance
1645             self.setSubProblemTolerance()
1646    
1647        def __stoppingcriterium(self,norm_r,r,p):        def getTolerance(self):
1648            return self.stoppingcriterium(r[1],r[0],p)           """
1649             Returns the relative tolerance.
1650    
1651        def __stoppingcriterium2(self,norm_r,norm_b,solver='GMRES',TOL=None):           @return: relative tolerance
1652            return self.stoppingcriterium2(norm_r,norm_b,solver,TOL)           @rtype: C{float}
1653             """
1654             return self.__rtol
1655    
1656        def setTolerance(self,tolerance=1.e-8):        def setAbsoluteTolerance(self,tolerance=0.):
1657                self.__tol=tolerance           """
1658        def getTolerance(self):           Sets the absolute tolerance.
               return self.__tol  
       def setToleranceReductionFactor(self,reduction=0.01):  
               self.__reduction=reduction  
       def getSubProblemTolerance(self):  
               return self.__reduction*self.getTolerance()  
1659    
1660        def solve(self,v,p,max_iter=20, verbose=False, show_details=False, solver='PCG',iter_restart=20):           @param tolerance: tolerance to be used
1661                """           @type tolerance: non-negative C{float}
1662                solves the saddle point problem using initial guesses v and p.           """
1663             if tolerance<0:
1664                @param max_iter: maximum number of iteration steps.               raise ValueError,"tolerance must be non-negative."
1665                """           self.__atol=tolerance
1666                self.verbose=verbose  
1667                self.show_details=show_details and self.verbose        def getAbsoluteTolerance(self):
1668             """
1669                # assume p is known: then v=A^-1(f-B^*p)           Returns the absolute tolerance.
1670                # which leads to BA^-1B^*p = BA^-1f    
1671             @return: absolute tolerance
1672            # Az=f is solved as A(z-v)=f-Av (z-v = 0 on fixed_u_mask)                 @rtype: C{float}
1673            self.__z=v+self.solve_A(v,p*0)           """
1674                Bz=self.B(self.__z)           return self.__atol
               #  
           #   solve BA^-1B^*p = Bz  
               #  
               #  
               #  
               self.iter=0  
           if solver=='GMRES':        
                 if self.verbose: print "enter GMRES method (iter_max=%s)"%max_iter  
                 p=GMRES(Bz,self.__Aprod2,self.__Msolve2,self.__inner_p,self.__stoppingcriterium2,iter_max=max_iter, x=p*1.,iter_restart=iter_restart)  
                 # solve Au=f-B^*p  
                 #       A(u-v)=f-B^*p-Av  
                 #       u=v+(u-v)  
         u=v+self.solve_A(v,p)  
   
           if solver=='TFQMR':        
                 if self.verbose: print "enter TFQMR method (iter_max=%s)"%max_iter  
                 p=TFQMR(Bz,self.__Aprod2,self.__Msolve2,self.__inner_p,self.__stoppingcriterium2,iter_max=max_iter, x=p*1.)  
                 # solve Au=f-B^*p  
                 #       A(u-v)=f-B^*p-Av  
                 #       u=v+(u-v)  
         u=v+self.solve_A(v,p)  
   
           if solver=='MINRES':        
                 if self.verbose: print "enter MINRES method (iter_max=%s)"%max_iter  
                 p=MINRES(Bz,self.__Aprod2,self.__Msolve2,self.__inner_p,self.__stoppingcriterium2,iter_max=max_iter, x=p*1.)  
                 # solve Au=f-B^*p  
                 #       A(u-v)=f-B^*p-Av  
                 #       u=v+(u-v)  
         u=v+self.solve_A(v,p)  
                 
           if solver=='GMRESC':        
                 if self.verbose: print "enter GMRES coupled method (iter_max=%s)"%max_iter  
                 p0=self.solve_prec1(Bz)  
             #alfa=(1/self.vol)*util.integrate(util.interpolate(p,escript.Function(self.domain)))  
                 #p-=alfa  
                 x=GMRES(ArithmeticTuple(self.__z*1.,p0*1),self.__Anext,self.__Mempty,self.__inner_a,self.__stoppingcriterium2,iter_max=max_iter, x=ArithmeticTuple(v*1,p*1),iter_restart=20)  
                 #x=NewtonGMRES(ArithmeticTuple(self.__z*1.,p0*1),self.__Aprod_Newton2,self.__Mempty,self.__inner_a,self.__stoppingcriterium2,iter_max=max_iter, x=ArithmeticTuple(v*1,p*1),atol=0,rtol=self.getTolerance())  
   
                 # solve Au=f-B^*p  
                 #       A(u-v)=f-B^*p-Av  
                 #       u=v+(u-v)  
             p=x[1]  
         u=v+self.solve_A(v,p)        
         #p=x[1]  
         #u=x[0]  
   
               if solver=='PCG':  
                 #   note that the residual r=Bz-BA^-1B^*p = B(z-A^-1B^*p) = Bv  
                 #  
                 #   with                    Av=Az-B^*p = f - B^*p (v=z on fixed_u_mask)  
                 #                           A(v-z)= f -Az - B^*p (v-z=0 on fixed_u_mask)  
                 if self.verbose: print "enter PCG method (iter_max=%s)"%max_iter  
                 p,r=PCG(ArithmeticTuple(self.__z*1.,Bz),self.__Aprod,self.__Msolve,self.__inner,self.__stoppingcriterium,iter_max=max_iter, x=p)  
             u=r[0]    
                 # print "DIFF=",util.integrate(p)  
   
               # print "RESULT div(u)=",util.Lsup(self.B(u)),util.Lsup(u)  
   
           return u,p  
   
       def __Msolve(self,r):  
           return self.solve_prec(r[1])  
   
       def __Msolve2(self,r):  
           return self.solve_prec(r*1)  
   
       def __Mempty(self,r):  
           return r  
   
   
       def __Aprod(self,p):  
           # return BA^-1B*p  
           #solve Av =B^*p as Av =f-Az-B^*(-p)  
           v=self.solve_A(self.__z,-p)  
           return ArithmeticTuple(v, self.B(v))  
   
       def __Anext(self,x):  
           # return next v,p  
           #solve Av =-B^*p as Av =f-Az-B^*p  
   
       pc=x[1]  
           v=self.solve_A(self.__z,-pc)  
       p=self.solve_prec1(self.B(v))  
   
           return ArithmeticTuple(v,p)  
   
   
       def __Aprod2(self,p):  
           # return BA^-1B*p  
           #solve Av =B^*p as Av =f-Az-B^*(-p)  
       v=self.solve_A(self.__z,-p)  
           return self.B(v)  
   
       def __Aprod_Newton(self,p):  
           # return BA^-1B*p - Bz  
           #solve Av =-B^*p as Av =f-Az-B^*p  
       v=self.solve_A(self.__z,-p)  
           return self.B(v-self.__z)  
   
       def __Aprod_Newton2(self,x):  
           # return BA^-1B*p - Bz  
           #solve Av =-B^*p as Av =f-Az-B^*p  
           pc=x[1]  
       v=self.solve_A(self.__z,-pc)  
           p=self.solve_prec1(self.B(v-self.__z))  
           return ArithmeticTuple(v,p)  
1675    
1676          def setSubProblemTolerance(self,rtol=None):
1677             """
1678             Sets the relative tolerance to solve the subproblem(s).
1679    
1680             @param rtol: relative tolerence
1681             @type rtol: positive C{float}
1682             """
1683             if rtol == None:
1684                  rtol=max(200.*util.EPSILON,self.getTolerance()**2)
1685             if rtol<=0:
1686                 raise ValueError,"tolerance must be positive."
1687             self.__sub_tol=rtol
1688    
1689          def getSubProblemTolerance(self):
1690             """
1691             Returns the subproblem reduction factor.
1692    
1693             @return: subproblem reduction factor
1694             @rtype: C{float}
1695             """
1696             return self.__sub_tol
1697    
1698  def MaskFromBoundaryTag(domain,*tags):  def MaskFromBoundaryTag(domain,*tags):
1699     """     """
1700     creates a mask on the Solution(domain) function space which one for samples     Creates a mask on the Solution(domain) function space where the value is
1701     that touch the boundary tagged by tags.     one for samples that touch the boundary tagged by tags.
1702    
1703     usage: m=MaskFromBoundaryTag(domain,"left", "right")     Usage: m=MaskFromBoundaryTag(domain, "left", "right")
1704    
1705     @param domain: a given domain     @param domain: domain to be used
1706     @type domain: L{escript.Domain}     @type domain: L{escript.Domain}
1707     @param tags: boundray tags     @param tags: boundary tags
1708     @type tags: C{str}     @type tags: C{str}
1709     @return: a mask which marks samples that are touching the boundary tagged by any of the given tags.     @return: a mask which marks samples that are touching the boundary tagged
1710                by any of the given tags
1711     @rtype: L{escript.Data} of rank 0     @rtype: L{escript.Data} of rank 0
1712     """     """
1713     pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)     pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)
# Line 1621  def MaskFromBoundaryTag(domain,*tags): Line 1715  def MaskFromBoundaryTag(domain,*tags):
1715     for t in tags: d.setTaggedValue(t,1.)     for t in tags: d.setTaggedValue(t,1.)
1716     pde.setValue(y=d)     pde.setValue(y=d)
1717     return util.whereNonZero(pde.getRightHandSide())     return util.whereNonZero(pde.getRightHandSide())
1718  #============================================================================================================================================  
1719    #==============================================================================
1720  class SaddlePointProblem(object):  class SaddlePointProblem(object):
1721     """     """
1722     This implements a solver for a saddlepoint problem     This implements a solver for a saddle point problem
1723    
1724     M{f(u,p)=0}     M{f(u,p)=0}
1725     M{g(u)=0}     M{g(u)=0}
# Line 1634  class SaddlePointProblem(object): Line 1729  class SaddlePointProblem(object):
1729     M{Q_f (u^{k+1}-u^{k}) = - f(u^{k},p^{k})}     M{Q_f (u^{k+1}-u^{k}) = - f(u^{k},p^{k})}
1730     M{Q_g (p^{k+1}-p^{k}) =   g(u^{k+1})}     M{Q_g (p^{k+1}-p^{k}) =   g(u^{k+1})}
1731    
1732     where Q_f is an approximation of the Jacobiean A_f of f with respect to u  and Q_f is an approximation of     where Q_f is an approximation of the Jacobian A_f of f with respect to u and
1733     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_f is an approximation of A_g A_f^{-1} A_g with A_g is the Jacobian of g
1734     Q_g can be difficult, non-linear conjugate gradient method is applied to solve for p, so Q_g plays     with respect to p. As a the construction of a 'proper' Q_g can be difficult,
1735       non-linear conjugate gradient method is applied to solve for p, so Q_g plays
1736     in fact the role of a preconditioner.     in fact the role of a preconditioner.
1737     """     """
1738     def __init__(self,verbose=False,*args):     def __init__(self,verbose=False,*args):
1739         """         """
1740         initializes the problem         Initializes the problem.
1741    
1742         @param verbose: switches on the printing out some information         @param verbose: if True, some information is printed in the process
1743         @type verbose: C{bool}         @type verbose: C{bool}
1744         @note: this method may be overwritten by a particular saddle point problem         @note: this method may be overwritten by a particular saddle point
1745                  problem
1746         """         """
1747         print "SaddlePointProblem should not be used anymore!"         print "SaddlePointProblem should not be used anymore!"
1748         if not isinstance(verbose,bool):         if not isinstance(verbose,bool):
# Line 1656  class SaddlePointProblem(object): Line 1753  class SaddlePointProblem(object):
1753    
1754     def trace(self,text):     def trace(self,text):
1755         """         """
1756         prints text if verbose has been set         Prints C{text} if verbose has been set.
1757    
1758         @param text: a text message         @param text: a text message
1759         @type text: C{str}         @type text: C{str}
# Line 1665  class SaddlePointProblem(object): Line 1762  class SaddlePointProblem(object):
1762    
1763     def solve_f(self,u,p,tol=1.e-8):     def solve_f(self,u,p,tol=1.e-8):
1764         """         """
1765         solves         Solves
1766    
1767         A_f du = f(u,p)         A_f du = f(u,p)
1768    
1769         with tolerance C{tol} and return du. A_f is Jacobiean of f with respect to u.         with tolerance C{tol} and returns du. A_f is Jacobian of f with respect
1770           to u.
1771    
1772         @param u: current approximation of u         @param u: current approximation of u
1773         @type u: L{escript.Data}         @type u: L{escript.Data}
# Line 1679  class SaddlePointProblem(object): Line 1777  class SaddlePointProblem(object):
1777         @type tol: C{float}         @type tol: C{float}
1778         @return: increment du         @return: increment du
1779         @rtype: L{escript.Data}         @rtype: L{escript.Data}
1780         @note: this method has to be overwritten by a particular saddle point problem         @note: this method has to be overwritten by a particular saddle point
1781                  problem
1782         """         """
1783         pass         pass
1784    
1785     def solve_g(self,u,tol=1.e-8):     def solve_g(self,u,tol=1.e-8):
1786         """         """
1787         solves         Solves
1788    
1789         Q_g dp = g(u)         Q_g dp = g(u)
1790    
1791         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.         where Q_g is a preconditioner for A_g A_f^{-1} A_g with A_g is the
1792           Jacobian of g with respect to p.
1793    
1794         @param u: current approximation of u         @param u: current approximation of u
1795         @type u: L{escript.Data}         @type u: L{escript.Data}
# Line 1697  class SaddlePointProblem(object): Line 1797  class SaddlePointProblem(object):
1797         @type tol: C{float}         @type tol: C{float}
1798         @return: increment dp         @return: increment dp
1799         @rtype: L{escript.Data}         @rtype: L{escript.Data}
1800         @note: this method has to be overwritten by a particular saddle point problem         @note: this method has to be overwritten by a particular saddle point
1801                  problem
1802         """         """
1803         pass         pass
1804    
1805     def inner(self,p0,p1):     def inner(self,p0,p1):
1806         """         """
1807         inner product of p0 and p1 approximating p. Typically this returns integrate(p0*p1)         Inner product of p0 and p1 approximating p. Typically this returns
1808           C{integrate(p0*p1)}.
1809         @return: inner product of p0 and p1         @return: inner product of p0 and p1
1810         @rtype: C{float}         @rtype: C{float}
1811         """         """
# Line 1712  class SaddlePointProblem(object): Line 1814  class SaddlePointProblem(object):
1814     subiter_max=3     subiter_max=3
1815     def solve(self,u0,p0,tolerance=1.e-6,tolerance_u=None,iter_max=100,accepted_reduction=0.995,relaxation=None):     def solve(self,u0,p0,tolerance=1.e-6,tolerance_u=None,iter_max=100,accepted_reduction=0.995,relaxation=None):
1816          """          """
1817          runs the solver          Runs the solver.
1818    
1819          @param u0: initial guess for C{u}          @param u0: initial guess for C{u}
1820          @type u0: L{esys.escript.Data}          @type u0: L{esys.escript.Data}
# Line 1720  class SaddlePointProblem(object): Line 1822  class SaddlePointProblem(object):
1822          @type p0: L{esys.escript.Data}          @type p0: L{esys.escript.Data}
1823          @param tolerance: tolerance for relative error in C{u} and C{p}          @param tolerance: tolerance for relative error in C{u} and C{p}
1824          @type tolerance: positive C{float}          @type tolerance: positive C{float}
1825          @param tolerance_u: tolerance for relative error in C{u} if different from C{tolerance}          @param tolerance_u: tolerance for relative error in C{u} if different
1826                                from C{tolerance}
1827          @type tolerance_u: positive C{float}          @type tolerance_u: positive C{float}
1828          @param iter_max: maximum number of iteration steps.          @param iter_max: maximum number of iteration steps
1829          @type iter_max: C{int}          @type iter_max: C{int}
1830          @param accepted_reduction: if the norm  g cannot be reduced by C{accepted_reduction} backtracking to adapt the          @param accepted_reduction: if the norm g cannot be reduced by
1831                                     relaxation factor. If C{accepted_reduction=None} no backtracking is used.                                     C{accepted_reduction} backtracking to adapt
1832                                       the relaxation factor. If
1833                                       C{accepted_reduction=None} no backtracking
1834                                       is used.
1835          @type accepted_reduction: positive C{float} or C{None}          @type accepted_reduction: positive C{float} or C{None}
1836          @param relaxation: initial relaxation factor. If C{relaxation==None}, the last relaxation factor is used.          @param relaxation: initial relaxation factor. If C{relaxation==None},
1837                               the last relaxation factor is used.
1838          @type relaxation: C{float} or C{None}          @type relaxation: C{float} or C{None}
1839          """          """
1840          tol=1.e-2          tol=1.e-2
# Line 1826  class SaddlePointProblem(object): Line 1933  class SaddlePointProblem(object):
1933              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              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
1934          self.trace("convergence after %s steps."%self.iter)          self.trace("convergence after %s steps."%self.iter)
1935          return u,p          return u,p
1936    

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