/[escript]/trunk/escript/py_src/pdetools.py
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revision 1122 by gross, Tue May 1 03:21:04 2007 UTC revision 1476 by gross, Mon Apr 7 23:38:50 2008 UTC
# Line 1  Line 1 
1    #
2  # $Id$  # $Id$
3    #
4    #######################################################
5    #
6    #           Copyright 2003-2007 by ACceSS MNRF
7    #       Copyright 2007 by University of Queensland
8    #
9    #                http://esscc.uq.edu.au
10    #        Primary Business: Queensland, Australia
11    #  Licensed under the Open Software License version 3.0
12    #     http://www.opensource.org/licenses/osl-3.0.php
13    #
14    #######################################################
15    #
16    
17  """  """
18  Provides some tools related to PDEs.  Provides some tools related to PDEs.
# Line 32  import escript Line 46  import escript
46  import linearPDEs  import linearPDEs
47  import numarray  import numarray
48  import util  import util
49    import math
50    
51    ##### Added by Artak
52    # from Numeric import zeros,Int,Float64
53    ###################################
54    
55    
56  class TimeIntegrationManager:  class TimeIntegrationManager:
57    """    """
# Line 393  class Locator: Line 413  class Locator:
413          else:          else:
414             return data             return data
415    
416    class SolverSchemeException(Exception):
417       """
418       exceptions thrown by solvers
419       """
420       pass
421    
422    class IndefinitePreconditioner(SolverSchemeException):
423       """
424       the preconditioner is not positive definite.
425       """
426       pass
427    class MaxIterReached(SolverSchemeException):
428       """
429       maxium number of iteration steps is reached.
430       """
431       pass
432    class IterationBreakDown(SolverSchemeException):
433       """
434       iteration scheme econouters an incurable breakdown.
435       """
436       pass
437    class NegativeNorm(SolverSchemeException):
438       """
439       a norm calculation returns a negative norm.
440       """
441       pass
442    
443    class IterationHistory(object):
444       """
445       The IterationHistory class is used to define a stopping criterium. It keeps track of the
446       residual norms. The stoppingcriterium indicates termination if the residual norm has been reduced by
447       a given tolerance.
448       """
449       def __init__(self,tolerance=math.sqrt(util.EPSILON),verbose=False):
450          """
451          Initialization
452    
453          @param tolerance: tolerance
454          @type tolerance: positive C{float}
455          @param verbose: switches on the printing out some information
456          @type verbose: C{bool}
457          """
458          if not tolerance>0.:
459              raise ValueError,"tolerance needs to be positive."
460          self.tolerance=tolerance
461          self.verbose=verbose
462          self.history=[]
463       def stoppingcriterium(self,norm_r,r,x):
464           """
465           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.
466    
467          
468           @param norm_r: current residual norm
469           @type norm_r: non-negative C{float}
470           @param r: current residual (not used)
471           @param x: current solution approximation (not used)
472           @return: C{True} is the stopping criterium is fullfilled. Otherwise C{False} is returned.
473           @rtype: C{bool}
474    
475           """
476           self.history.append(norm_r)
477           if self.verbose: print "iter: %s:  inner(rhat,r) = %e"%(len(self.history)-1, self.history[-1])
478           return self.history[-1]<=self.tolerance * self.history[0]
479    
480       def stoppingcriterium2(self,norm_r,norm_b):
481           """
482           returns True if the C{norm_r} is C{tolerance}*C{norm_b}
483    
484          
485           @param norm_r: current residual norm
486           @type norm_r: non-negative C{float}
487           @param norm_b: norm of right hand side
488           @type norm_b: non-negative C{float}
489           @return: C{True} is the stopping criterium is fullfilled. Otherwise C{False} is returned.
490           @rtype: C{bool}
491    
492           """
493           self.history.append(norm_r)
494           if self.verbose: print "iter: %s:  norm(r) = %e"%(len(self.history)-1, self.history[-1])
495           return self.history[-1]<=self.tolerance * norm_b
496    
497    def PCG(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100):
498       """
499       Solver for
500    
501       M{Ax=b}
502    
503       with a symmetric and positive definite operator A (more details required!).
504       It uses the conjugate gradient method with preconditioner M providing an approximation of A.
505    
506       The iteration is terminated if the C{stoppingcriterium} function return C{True}.
507    
508       For details on the preconditioned conjugate gradient method see the book:
509    
510       Templates for the Solution of Linear Systems by R. Barrett, M. Berry,
511       T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo,
512       C. Romine, and H. van der Vorst.
513    
514       @param b: the right hand side of the liner system. C{b} is altered.
515       @type b: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
516       @param Aprod: returns the value Ax
517       @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}.
518       @param Msolve: solves Mx=r
519       @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
520    type like argument C{x}.
521       @param bilinearform: inner product C{<x,r>}
522       @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}.
523       @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.
524       @type stoppingcriterium: function that returns C{True} or C{False}
525       @param x: an initial guess for the solution. If no C{x} is given 0*b is used.
526       @type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
527       @param iter_max: maximum number of iteration steps.
528       @type iter_max: C{int}
529       @return: the solution approximation and the corresponding residual
530       @rtype: C{tuple}
531       @warning: C{b} and C{x} are altered.
532       """
533       iter=0
534       if x==None:
535          x=0*b
536       else:
537          b += (-1)*Aprod(x)
538       r=b
539       rhat=Msolve(r)
540       d = rhat
541       rhat_dot_r = bilinearform(rhat, r)
542       if rhat_dot_r<0: raise NegativeNorm,"negative norm."
543    
544       while not stoppingcriterium(math.sqrt(rhat_dot_r),r,x):
545           iter+=1
546           if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
547    
548           q=Aprod(d)
549           alpha = rhat_dot_r / bilinearform(d, q)
550           x += alpha * d
551           r += (-alpha) * q
552    
553           rhat=Msolve(r)
554           rhat_dot_r_new = bilinearform(rhat, r)
555           beta = rhat_dot_r_new / rhat_dot_r
556           rhat+=beta * d
557           d=rhat
558    
559           rhat_dot_r = rhat_dot_r_new
560           if rhat_dot_r<0: raise NegativeNorm,"negative norm."
561    
562       return x,r
563    
564    
565    ############################
566    # Added by Artak
567    #################################3
568    
569    #Apply a sequence of k Givens rotations, used within gmres codes
570    # vrot=givapp(c, s, vin, k)
571    def givapp(c,s,vin):
572        vrot=vin # warning: vin is altered!!!!
573        if isinstance(c,float):
574            vrot=[c*vrot[0]-s*vrot[1],s*vrot[0]+c*vrot[1]]
575        else:
576            for i in range(len(c)):
577                w1=c[i]*vrot[i]-s[i]*vrot[i+1]
578            w2=s[i]*vrot[i]+c[i]*vrot[i+1]
579                vrot[i:i+2]=w1,w2
580        return vrot
581    
582    def GMRES(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100, iter_restart=10):
583       m=iter_restart
584       iter=0
585       while True:
586          if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
587          x,stopped=GMRESm(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=x, iter_max=iter_max-iter, iter_restart=m)
588          iter+=iter_restart
589          if stopped: break
590       return x
591    
592    def GMRESm(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100, iter_restart=10):
593       iter=0
594       r=Msolve(b)
595       r_dot_r = bilinearform(r, r)
596       if r_dot_r<0: raise NegativeNorm,"negative norm."
597       norm_b=math.sqrt(r_dot_r)
598    
599       if x==None:
600          x=0*b
601       else:
602          r=Msolve(b-Aprod(x))
603          r_dot_r = bilinearform(r, r)
604          if r_dot_r<0: raise NegativeNorm,"negative norm."
605      
606       h=numarray.zeros((iter_restart,iter_restart),numarray.Float64)
607       c=numarray.zeros(iter_restart,numarray.Float64)
608       s=numarray.zeros(iter_restart,numarray.Float64)
609       g=numarray.zeros(iter_restart,numarray.Float64)
610       v=[]
611    
612       rho=math.sqrt(r_dot_r)
613       v.append(r/rho)
614       g[0]=rho
615    
616       while not (stoppingcriterium(rho,norm_b) or iter==iter_restart-1):
617    
618        if iter  >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
619    
620        
621        p=Msolve(Aprod(v[iter]))
622    
623        v.append(p)
624    
625        v_norm1=math.sqrt(bilinearform(v[iter+1], v[iter+1]))  
626    
627    # Modified Gram-Schmidt
628        for j in range(iter+1):
629          h[j][iter]=bilinearform(v[j],v[iter+1])  
630          v[iter+1]+=(-1.)*h[j][iter]*v[j]
631          
632        h[iter+1][iter]=math.sqrt(bilinearform(v[iter+1],v[iter+1]))
633        v_norm2=h[iter+1][iter]
634    
635    
636    # Reorthogonalize if needed
637        if v_norm1 + 0.001*v_norm2 == v_norm1:   #Brown/Hindmarsh condition (default)
638         for j in range(iter+1):
639            hr=bilinearform(v[j],v[iter+1])
640                h[j][iter]=h[j][iter]+hr #vhat
641                v[iter+1] +=(-1.)*hr*v[j]
642    
643         v_norm2=math.sqrt(bilinearform(v[iter+1], v[iter+1]))  
644         h[iter+1][iter]=v_norm2
645    
646    #   watch out for happy breakdown
647            if v_norm2 != 0:
648             v[iter+1]=v[iter+1]/h[iter+1][iter]
649    
650    #   Form and store the information for the new Givens rotation
651        if iter > 0 :
652            hhat=[]
653            for i in range(iter+1) : hhat.append(h[i][iter])
654            hhat=givapp(c[0:iter],s[0:iter],hhat);
655                for i in range(iter+1) : h[i][iter]=hhat[i]
656    
657        mu=math.sqrt(h[iter][iter]*h[iter][iter]+h[iter+1][iter]*h[iter+1][iter])
658        if mu!=0 :
659            c[iter]=h[iter][iter]/mu
660            s[iter]=-h[iter+1][iter]/mu
661            h[iter][iter]=c[iter]*h[iter][iter]-s[iter]*h[iter+1][iter]
662            h[iter+1][iter]=0.0
663            g[iter:iter+2]=givapp(c[iter],s[iter],g[iter:iter+2])
664    
665    # Update the residual norm
666            rho=abs(g[iter+1])
667        iter+=1
668    
669    # At this point either iter > iter_max or rho < tol.
670    # It's time to compute x and leave.        
671    
672       if iter > 0 :
673         y=numarray.zeros(iter,numarray.Float64)    
674         y[iter-1] = g[iter-1] / h[iter-1][iter-1]
675         if iter > 1 :  
676            i=iter-2  
677            while i>=0 :
678              y[i] = ( g[i] - numarray.dot(h[i][i+1:iter], y[i+1:iter])) / h[i][i]
679              i=i-1
680         xhat=v[iter-1]*y[iter-1]
681         for i in range(iter-1):
682        xhat += v[i]*y[i]
683       else : xhat=v[0]
684        
685       x += xhat
686       if iter!=iter_restart-1:
687          stopped=True
688       else:
689          stopped=False
690    
691       return x,stopped
692        
693    #############################################
694    
695    class ArithmeticTuple(object):
696       """
697       tuple supporting inplace update x+=y and scaling x=a*y where x,y is an ArithmeticTuple and a is a float.
698    
699       example of usage:
700    
701       from esys.escript import Data
702       from numarray import array
703       a=Data(...)
704       b=array([1.,4.])
705       x=ArithmeticTuple(a,b)
706       y=5.*x
707    
708       """
709       def __init__(self,*args):
710           """
711           initialize object with elements args.
712    
713           @param args: tuple of object that support implace add (x+=y) and scaling (x=a*y)
714           """
715           self.__items=list(args)
716    
717       def __len__(self):
718           """
719           number of items
720    
721           @return: number of items
722           @rtype: C{int}
723           """
724           return len(self.__items)
725    
726       def __getitem__(self,index):
727           """
728           get an item
729    
730           @param index: item to be returned
731           @type index: C{int}
732           @return: item with index C{index}
733           """
734           return self.__items.__getitem__(index)
735    
736       def __mul__(self,other):
737           """
738           scaling from the right
739    
740           @param other: scaling factor
741           @type other: C{float}
742           @return: itemwise self*other
743           @rtype: L{ArithmeticTuple}
744           """
745           out=[]
746           for i in range(len(self)):
747               out.append(self[i]*other)
748           return ArithmeticTuple(*tuple(out))
749    
750       def __rmul__(self,other):
751           """
752           scaling from the left
753    
754           @param other: scaling factor
755           @type other: C{float}
756           @return: itemwise other*self
757           @rtype: L{ArithmeticTuple}
758           """
759           out=[]
760           for i in range(len(self)):
761               out.append(other*self[i])
762           return ArithmeticTuple(*tuple(out))
763    
764    #########################
765    # Added by Artak
766    #########################
767       def __div__(self,other):
768           """
769           dividing from the right
770    
771           @param other: scaling factor
772           @type other: C{float}
773           @return: itemwise self/other
774           @rtype: L{ArithmeticTuple}
775           """
776           out=[]
777           for i in range(len(self)):
778               out.append(self[i]/other)
779           return ArithmeticTuple(*tuple(out))
780    
781       def __rdiv__(self,other):
782           """
783           dividing from the left
784    
785           @param other: scaling factor
786           @type other: C{float}
787           @return: itemwise other/self
788           @rtype: L{ArithmeticTuple}
789           """
790           out=[]
791           for i in range(len(self)):
792               out.append(other/self[i])
793           return ArithmeticTuple(*tuple(out))
794      
795    ##########################################33
796    
797       def __iadd__(self,other):
798           """
799           in-place add of other to self
800    
801           @param other: increment
802           @type other: C{ArithmeticTuple}
803           """
804           if len(self) != len(other):
805               raise ValueError,"tuple length must match."
806           for i in range(len(self)):
807               self.__items[i]+=other[i]
808           return self
809    
810    class HomogeneousSaddlePointProblem(object):
811          """
812          This provides a framwork for solving homogeneous saddle point problem of the form
813    
814                 Av+B^*p=f
815                 Bv    =0
816    
817          for the unknowns v and p and given operators A and B and given right hand side f.
818          B^* is the adjoint operator of B is the given inner product.
819    
820          """
821          def __init__(self,**kwargs):
822            self.setTolerance()
823            self.setToleranceReductionFactor()
824    
825          def initialize(self):
826            """
827            initialize the problem (overwrite)
828            """
829            pass
830          def B(self,v):
831             """
832             returns Bv (overwrite)
833             @rtype: equal to the type of p
834    
835             @note: boundary conditions on p should be zero!
836             """
837             pass
838    
839          def inner(self,p0,p1):
840             """
841             returns inner product of two element p0 and p1  (overwrite)
842            
843             @type p0: equal to the type of p
844             @type p1: equal to the type of p
845             @rtype: C{float}
846    
847             @rtype: equal to the type of p
848             """
849             pass
850    
851          def solve_A(self,u,p):
852             """
853             solves Av=f-Au-B^*p with accuracy self.getReducedTolerance() (overwrite)
854    
855             @rtype: equal to the type of v
856             @note: boundary conditions on v should be zero!
857             """
858             pass
859    
860          def solve_prec(self,p):
861             """
862             provides a preconditioner for BA^{-1}B^* with accuracy self.getReducedTolerance() (overwrite)
863    
864             @rtype: equal to the type of p
865             """
866             pass
867    
868          def stoppingcriterium(self,Bv,v,p):
869             """
870             returns a True if iteration is terminated. (overwrite)
871    
872             @rtype: C{bool}
873             """
874             pass
875                
876          def __inner(self,p,r):
877             return self.inner(p,r[1])
878    
879          def __inner_p(self,p1,p2):
880             return self.inner(p1,p2)
881    
882          def __stoppingcriterium(self,norm_r,r,p):
883              return self.stoppingcriterium(r[1],r[0],p)
884    
885          def __stoppingcriterium_GMRES(self,norm_r,norm_b):
886              return self.stoppingcriterium_GMRES(norm_r,norm_b)
887    
888          def setTolerance(self,tolerance=1.e-8):
889                  self.__tol=tolerance
890          def getTolerance(self):
891                  return self.__tol
892          def setToleranceReductionFactor(self,reduction=0.01):
893                  self.__reduction=reduction
894          def getSubProblemTolerance(self):
895                  return self.__reduction*self.getTolerance()
896    
897          def solve(self,v,p,max_iter=20, verbose=False, show_details=False, solver='PCG'):
898                  """
899                  solves the saddle point problem using initial guesses v and p.
900    
901                  @param max_iter: maximum number of iteration steps.
902                  """
903                  self.verbose=verbose
904                  self.show_details=show_details and self.verbose
905    
906                  # assume p is known: then v=A^-1(f-B^*p)
907                  # which leads to BA^-1B^*p = BA^-1f  
908    
909              # Az=f is solved as A(z-v)=f-Av (z-v = 0 on fixed_u_mask)      
910    
911              
912              self.__z=v+self.solve_A(v,p*0)
913    
914                  Bz=self.B(self.__z)
915                  #
916              #   solve BA^-1B^*p = Bz
917                  #
918                  #   note that the residual r=Bz-BA^-1B^*p = B(z-A^-1B^*p) = Bv
919                  #
920                  #   with                    Av=Az-B^*p = f - B^*p (v=z on fixed_u_mask)
921                  #                           A(v-z)=Az-B^*p-Az = f -Az - B^*p (v-z=0 on fixed_u_mask)
922                  #
923                  self.iter=0
924              if solver=='GMRES':      
925                    if self.verbose: print "enter GMRES method (iter_max=%s)"%max_iter
926                    p=GMRES(Bz,self.__Aprod_GMRES,self.__Msolve_GMRES,self.__inner_p,self.__stoppingcriterium_GMRES,iter_max=max_iter, x=p*1.)
927                    # solve Au=f-B^*p
928                    #       A(u-v)=f-B^*p-Av
929                    #       u=v+(u-v)
930            u=v+self.solve_A(v,p)
931        
932                  else:
933                    if self.verbose: print "enter PCG method (iter_max=%s)"%max_iter
934                    p,r=PCG(ArithmeticTuple(self.__z*1.,Bz),self.__Aprod,self.__Msolve,self.__inner,self.__stoppingcriterium,iter_max=max_iter, x=p)
935                u=r[0]  
936                  print "RESULT div(u)=",util.Lsup(self.B(u)),util.Lsup(u)
937    
938              return u,p
939    
940          def __Msolve(self,r):
941              return self.solve_prec(r[1])
942    
943          def __Msolve_GMRES(self,r):
944              return self.solve_prec(r)
945    
946    
947          def __Aprod(self,p):
948              # return BA^-1B*p
949              #solve Av =-B^*p as Av =f-Az-B^*p
950              v=self.solve_A(self.__z,-p)
951              return ArithmeticTuple(v, self.B(v))
952    
953          def __Aprod_GMRES(self,p):
954              # return BA^-1B*p
955              #solve Av =-B^*p as Av =f-Az-B^*p
956          v=self.solve_A(self.__z,-p)
957              return self.B(v)
958    
959  class SaddlePointProblem(object):  class SaddlePointProblem(object):
960     """     """
961     This implements a solver for a saddlepoint problem     This implements a solver for a saddlepoint problem
# Line 580  class SaddlePointProblem(object): Line 1143  class SaddlePointProblem(object):
1143              norm_u_new = util.Lsup(u_new)              norm_u_new = util.Lsup(u_new)
1144              p_new=p+self.relaxation*g_new              p_new=p+self.relaxation*g_new
1145              norm_p_new = util.sqrt(self.inner(p_new,p_new))              norm_p_new = util.sqrt(self.inner(p_new,p_new))
1146              self.trace("%s th step: f/u = %s, g/p = %s, relaxation = %s."%(self.iter,norm_f_new/norm_u_new, norm_g_new/norm_p_new, self.relaxation))              self.trace("%s th step: f = %s, f/u = %s, g = %s, g/p = %s, relaxation = %s."%(self.iter, norm_f_new ,norm_f_new/norm_u_new, norm_g_new, norm_g_new/norm_p_new, self.relaxation))
1147    
1148              if self.iter>1:              if self.iter>1:
1149                 dg2=g_new-g                 dg2=g_new-g
# Line 622  class SaddlePointProblem(object): Line 1185  class SaddlePointProblem(object):
1185  #  #
1186  #      return u,p  #      return u,p
1187                        
1188  # vim: expandtab shiftwidth=4:  def MaskFromBoundaryTag(function_space,*tags):
1189       """
1190       create a mask on the given function space which one for samples
1191       that touch the boundary tagged by tags.
1192    
1193       usage: m=MaskFromBoundaryTag(Solution(domain),"left", "right")
1194    
1195       @param function_space: a given function space
1196       @type function_space: L{escript.FunctionSpace}
1197       @param tags: boundray tags
1198       @type tags: C{str}
1199       @return: a mask which marks samples used by C{function_space} that are touching the
1200                boundary tagged by any of the given tags.
1201       @rtype: L{escript.Data} of rank 0
1202       """
1203       pde=linearPDEs.LinearPDE(function_space.getDomain(),numEquations=1, numSolutions=1)
1204       d=escript.Scalar(0.,escript.FunctionOnBoundary(function_space.getDomain()))
1205       for t in tags: d.setTaggedValue(t,1.)
1206       pde.setValue(y=d)
1207       out=util.whereNonZero(pde.getRightHandSide())
1208       if out.getFunctionSpace() == function_space:
1209          return out
1210       else:
1211          return util.whereNonZero(util.interpolate(out,function_space))
1212    
1213    
1214    

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