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Revision 1469 - (hide annotations)
Thu Apr 3 05:16:56 2008 UTC (11 years, 5 months ago) by gross
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additional stopping criterion added
1 ksteube 1312 #
2 jgs 121 # $Id$
3 ksteube 1312 #
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 jgs 121
17     """
18 jgs 149 Provides some tools related to PDEs.
19 jgs 121
20 jgs 149 Currently includes:
21     - Projector - to project a discontinuous
22 gross 351 - Locator - to trace values in data objects at a certain location
23     - TimeIntegrationManager - to handel extraplotion in time
24 gross 867 - SaddlePointProblem - solver for Saddle point problems using the inexact uszawa scheme
25 gross 637
26     @var __author__: name of author
27     @var __copyright__: copyrights
28     @var __license__: licence agreement
29     @var __url__: url entry point on documentation
30     @var __version__: version
31     @var __date__: date of the version
32 jgs 121 """
33    
34 gross 637 __author__="Lutz Gross, l.gross@uq.edu.au"
35 elspeth 609 __copyright__=""" Copyright (c) 2006 by ACcESS MNRF
36     http://www.access.edu.au
37     Primary Business: Queensland, Australia"""
38 elspeth 614 __license__="""Licensed under the Open Software License version 3.0
39     http://www.opensource.org/licenses/osl-3.0.php"""
40 gross 637 __url__="http://www.iservo.edu.au/esys"
41     __version__="$Revision$"
42     __date__="$Date$"
43 elspeth 609
44 gross 637
45 jgs 149 import escript
46     import linearPDEs
47 jgs 121 import numarray
48 jgs 149 import util
49 ksteube 1312 import math
50 jgs 121
51 artak 1465 ##### Added by Artak
52 gross 1467 # from Numeric import zeros,Int,Float64
53 artak 1465 ###################################
54    
55    
56 gross 351 class TimeIntegrationManager:
57     """
58     a simple mechanism to manage time dependend values.
59    
60 gross 720 typical usage is::
61 gross 351
62 gross 720 dt=0.1 # time increment
63     tm=TimeIntegrationManager(inital_value,p=1)
64     while t<1.
65     v_guess=tm.extrapolate(dt) # extrapolate to t+dt
66     v=...
67     tm.checkin(dt,v)
68     t+=dt
69 gross 351
70 gross 720 @note: currently only p=1 is supported.
71 gross 351 """
72     def __init__(self,*inital_values,**kwargs):
73     """
74     sets up the value manager where inital_value is the initial value and p is order used for extrapolation
75     """
76     if kwargs.has_key("p"):
77     self.__p=kwargs["p"]
78     else:
79     self.__p=1
80     if kwargs.has_key("time"):
81     self.__t=kwargs["time"]
82     else:
83     self.__t=0.
84     self.__v_mem=[inital_values]
85     self.__order=0
86     self.__dt_mem=[]
87     self.__num_val=len(inital_values)
88    
89     def getTime(self):
90     return self.__t
91 gross 396 def getValue(self):
92 gross 409 out=self.__v_mem[0]
93     if len(out)==1:
94     return out[0]
95     else:
96     return out
97    
98 gross 351 def checkin(self,dt,*values):
99     """
100     adds new values to the manager. the p+1 last value get lost
101     """
102     o=min(self.__order+1,self.__p)
103     self.__order=min(self.__order+1,self.__p)
104     v_mem_new=[values]
105     dt_mem_new=[dt]
106     for i in range(o-1):
107     v_mem_new.append(self.__v_mem[i])
108     dt_mem_new.append(self.__dt_mem[i])
109     v_mem_new.append(self.__v_mem[o-1])
110     self.__order=o
111     self.__v_mem=v_mem_new
112     self.__dt_mem=dt_mem_new
113     self.__t+=dt
114    
115     def extrapolate(self,dt):
116     """
117     extrapolates to dt forward in time.
118     """
119     if self.__order==0:
120     out=self.__v_mem[0]
121     else:
122     out=[]
123     for i in range(self.__num_val):
124     out.append((1.+dt/self.__dt_mem[0])*self.__v_mem[0][i]-dt/self.__dt_mem[0]*self.__v_mem[1][i])
125    
126     if len(out)==0:
127     return None
128     elif len(out)==1:
129     return out[0]
130     else:
131     return out
132 gross 396
133 gross 351
134 jgs 121 class Projector:
135 jgs 149 """
136     The Projector is a factory which projects a discontiuous function onto a
137     continuous function on the a given domain.
138     """
139 jgs 121 def __init__(self, domain, reduce = True, fast=True):
140     """
141 jgs 149 Create a continuous function space projector for a domain.
142 jgs 121
143 jgs 149 @param domain: Domain of the projection.
144     @param reduce: Flag to reduce projection order (default is True)
145     @param fast: Flag to use a fast method based on matrix lumping (default is true)
146 jgs 121 """
147 jgs 149 self.__pde = linearPDEs.LinearPDE(domain)
148 jgs 148 if fast:
149 jgs 149 self.__pde.setSolverMethod(linearPDEs.LinearPDE.LUMPING)
150 jgs 121 self.__pde.setSymmetryOn()
151     self.__pde.setReducedOrderTo(reduce)
152     self.__pde.setValue(D = 1.)
153 ksteube 1312 return
154 jgs 121
155     def __call__(self, input_data):
156     """
157 jgs 149 Projects input_data onto a continuous function
158 jgs 121
159 jgs 149 @param input_data: The input_data to be projected.
160 jgs 121 """
161 gross 525 out=escript.Data(0.,input_data.getShape(),self.__pde.getFunctionSpaceForSolution())
162 gross 1122 self.__pde.setValue(Y = escript.Data(), Y_reduced = escript.Data())
163 jgs 121 if input_data.getRank()==0:
164     self.__pde.setValue(Y = input_data)
165     out=self.__pde.getSolution()
166     elif input_data.getRank()==1:
167     for i0 in range(input_data.getShape()[0]):
168     self.__pde.setValue(Y = input_data[i0])
169     out[i0]=self.__pde.getSolution()
170     elif input_data.getRank()==2:
171     for i0 in range(input_data.getShape()[0]):
172     for i1 in range(input_data.getShape()[1]):
173     self.__pde.setValue(Y = input_data[i0,i1])
174     out[i0,i1]=self.__pde.getSolution()
175     elif input_data.getRank()==3:
176     for i0 in range(input_data.getShape()[0]):
177     for i1 in range(input_data.getShape()[1]):
178     for i2 in range(input_data.getShape()[2]):
179     self.__pde.setValue(Y = input_data[i0,i1,i2])
180     out[i0,i1,i2]=self.__pde.getSolution()
181     else:
182     for i0 in range(input_data.getShape()[0]):
183     for i1 in range(input_data.getShape()[1]):
184     for i2 in range(input_data.getShape()[2]):
185     for i3 in range(input_data.getShape()[3]):
186     self.__pde.setValue(Y = input_data[i0,i1,i2,i3])
187     out[i0,i1,i2,i3]=self.__pde.getSolution()
188     return out
189    
190 gross 525 class NoPDE:
191     """
192     solves the following problem for u:
193 jgs 121
194 gross 525 M{kronecker[i,j]*D[j]*u[j]=Y[i]}
195    
196     with constraint
197    
198     M{u[j]=r[j]} where M{q[j]>0}
199    
200     where D, Y, r and q are given functions of rank 1.
201    
202     In the case of scalars this takes the form
203    
204     M{D*u=Y}
205    
206     with constraint
207    
208     M{u=r} where M{q>0}
209    
210     where D, Y, r and q are given scalar functions.
211    
212     The constraint is overwriting any other condition.
213    
214 gross 720 @note: This class is similar to the L{linearPDEs.LinearPDE} class with A=B=C=X=0 but has the intention
215     that all input parameter are given in L{Solution} or L{ReducedSolution}. The whole
216     thing is a bit strange and I blame Robert.Woodcock@csiro.au for this.
217 gross 525 """
218     def __init__(self,domain,D=None,Y=None,q=None,r=None):
219     """
220     initialize the problem
221    
222     @param domain: domain of the PDE.
223     @type domain: L{Domain}
224     @param D: coefficient of the solution.
225 gross 720 @type D: C{float}, C{int}, L{numarray.NumArray}, L{Data}
226 gross 525 @param Y: right hand side
227 gross 720 @type Y: C{float}, C{int}, L{numarray.NumArray}, L{Data}
228 gross 525 @param q: location of constraints
229 gross 720 @type q: C{float}, C{int}, L{numarray.NumArray}, L{Data}
230 gross 525 @param r: value of solution at locations of constraints
231 gross 720 @type r: C{float}, C{int}, L{numarray.NumArray}, L{Data}
232 gross 525 """
233     self.__domain=domain
234     self.__D=D
235     self.__Y=Y
236     self.__q=q
237     self.__r=r
238     self.__u=None
239     self.__function_space=escript.Solution(self.__domain)
240     def setReducedOn(self):
241     """
242     sets the L{FunctionSpace} of the solution to L{ReducedSolution}
243     """
244     self.__function_space=escript.ReducedSolution(self.__domain)
245     self.__u=None
246    
247     def setReducedOff(self):
248     """
249     sets the L{FunctionSpace} of the solution to L{Solution}
250     """
251     self.__function_space=escript.Solution(self.__domain)
252     self.__u=None
253    
254     def setValue(self,D=None,Y=None,q=None,r=None):
255     """
256     assigns values to the parameters.
257    
258     @param D: coefficient of the solution.
259 gross 720 @type D: C{float}, C{int}, L{numarray.NumArray}, L{Data}
260 gross 525 @param Y: right hand side
261 gross 720 @type Y: C{float}, C{int}, L{numarray.NumArray}, L{Data}
262 gross 525 @param q: location of constraints
263 gross 720 @type q: C{float}, C{int}, L{numarray.NumArray}, L{Data}
264 gross 525 @param r: value of solution at locations of constraints
265 gross 720 @type r: C{float}, C{int}, L{numarray.NumArray}, L{Data}
266 gross 525 """
267     if not D==None:
268     self.__D=D
269     self.__u=None
270     if not Y==None:
271     self.__Y=Y
272     self.__u=None
273     if not q==None:
274     self.__q=q
275     self.__u=None
276     if not r==None:
277     self.__r=r
278     self.__u=None
279    
280     def getSolution(self):
281     """
282     returns the solution
283    
284     @return: the solution of the problem
285     @rtype: L{Data} object in the L{FunctionSpace} L{Solution} or L{ReducedSolution}.
286     """
287     if self.__u==None:
288     if self.__D==None:
289     raise ValueError,"coefficient D is undefined"
290     D=escript.Data(self.__D,self.__function_space)
291     if D.getRank()>1:
292     raise ValueError,"coefficient D must have rank 0 or 1"
293     if self.__Y==None:
294     self.__u=escript.Data(0.,D.getShape(),self.__function_space)
295     else:
296     self.__u=util.quotient(self.__Y,D)
297     if not self.__q==None:
298     q=util.wherePositive(escript.Data(self.__q,self.__function_space))
299     self.__u*=(1.-q)
300     if not self.__r==None: self.__u+=q*self.__r
301     return self.__u
302    
303 jgs 147 class Locator:
304     """
305 jgs 149 Locator provides access to the values of data objects at a given
306     spatial coordinate x.
307    
308     In fact, a Locator object finds the sample in the set of samples of a
309     given function space or domain where which is closest to the given
310     point x.
311 jgs 147 """
312    
313     def __init__(self,where,x=numarray.zeros((3,))):
314 jgs 149 """
315     Initializes a Locator to access values in Data objects on the Doamin
316     or FunctionSpace where for the sample point which
317     closest to the given point x.
318 gross 880
319     @param where: function space
320     @type where: L{escript.FunctionSpace}
321     @param x: coefficient of the solution.
322     @type x: L{numarray.NumArray} or C{list} of L{numarray.NumArray}
323 jgs 149 """
324     if isinstance(where,escript.FunctionSpace):
325 jgs 147 self.__function_space=where
326 jgs 121 else:
327 jgs 149 self.__function_space=escript.ContinuousFunction(where)
328 gross 880 if isinstance(x, list):
329     self.__id=[]
330     for p in x:
331 gross 921 self.__id.append(util.length(self.__function_space.getX()-p[:self.__function_space.getDim()]).minGlobalDataPoint())
332 gross 880 else:
333 gross 921 self.__id=util.length(self.__function_space.getX()-x[:self.__function_space.getDim()]).minGlobalDataPoint()
334 jgs 121
335 jgs 147 def __str__(self):
336 jgs 149 """
337     Returns the coordinates of the Locator as a string.
338     """
339 gross 880 x=self.getX()
340     if instance(x,list):
341     out="["
342     first=True
343     for xx in x:
344     if not first:
345     out+=","
346     else:
347     first=False
348     out+=str(xx)
349     out+="]>"
350     else:
351     out=str(x)
352     return out
353 jgs 121
354 gross 880 def getX(self):
355     """
356     Returns the exact coordinates of the Locator.
357     """
358     return self(self.getFunctionSpace().getX())
359    
360 jgs 147 def getFunctionSpace(self):
361 jgs 149 """
362     Returns the function space of the Locator.
363     """
364 jgs 147 return self.__function_space
365    
366 gross 880 def getId(self,item=None):
367 jgs 149 """
368     Returns the identifier of the location.
369     """
370 gross 880 if item == None:
371     return self.__id
372     else:
373     if isinstance(self.__id,list):
374     return self.__id[item]
375     else:
376     return self.__id
377 jgs 121
378    
379 jgs 147 def __call__(self,data):
380 jgs 149 """
381     Returns the value of data at the Locator of a Data object otherwise
382     the object is returned.
383     """
384 jgs 147 return self.getValue(data)
385 jgs 121
386 jgs 147 def getValue(self,data):
387 jgs 149 """
388     Returns the value of data at the Locator if data is a Data object
389     otherwise the object is returned.
390     """
391     if isinstance(data,escript.Data):
392 jgs 147 if data.getFunctionSpace()==self.getFunctionSpace():
393 gross 880 dat=data
394 jgs 147 else:
395 gross 880 dat=data.interpolate(self.getFunctionSpace())
396     id=self.getId()
397     r=data.getRank()
398     if isinstance(id,list):
399     out=[]
400     for i in id:
401 gross 921 o=data.getValueOfGlobalDataPoint(*i)
402 gross 880 if data.getRank()==0:
403     out.append(o[0])
404     else:
405     out.append(o)
406     return out
407 jgs 147 else:
408 gross 921 out=data.getValueOfGlobalDataPoint(*id)
409 gross 880 if data.getRank()==0:
410     return out[0]
411     else:
412     return out
413 jgs 147 else:
414     return data
415 jgs 149
416 ksteube 1312 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 gross 1330 class IterationHistory(object):
444 ksteube 1312 """
445 gross 1330 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 gross 1469 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 gross 1330 def PCG(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100):
498     """
499 ksteube 1312 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 gross 1330 The iteration is terminated if the C{stoppingcriterium} function return C{True}.
507 ksteube 1312
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 gross 1330 @type b: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
516 ksteube 1312 @param Aprod: returns the value Ax
517 gross 1330 @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 ksteube 1312 @param Msolve: solves Mx=r
519 gross 1330 @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 ksteube 1312 @param bilinearform: inner product C{<x,r>}
522 gross 1330 @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 ksteube 1312 @param iter_max: maximum number of iteration steps.
528     @type iter_max: C{int}
529 gross 1330 @return: the solution approximation and the corresponding residual
530     @rtype: C{tuple}
531     @warning: C{b} and C{x} are altered.
532 ksteube 1312 """
533     iter=0
534 gross 1330 if x==None:
535     x=0*b
536     else:
537     b += (-1)*Aprod(x)
538 ksteube 1312 r=b
539     rhat=Msolve(r)
540 gross 1330 d = rhat
541 ksteube 1312 rhat_dot_r = bilinearform(rhat, r)
542 gross 1330 if rhat_dot_r<0: raise NegativeNorm,"negative norm."
543 ksteube 1312
544 gross 1330 while not stoppingcriterium(math.sqrt(rhat_dot_r),r,x):
545     iter+=1
546 ksteube 1312 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 gross 1330 if rhat_dot_r<0: raise NegativeNorm,"negative norm."
561 ksteube 1312
562 gross 1330 return x,r
563 ksteube 1312
564 artak 1465
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 gross 1467 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 artak 1465 return vrot
581    
582     def GMRES(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100):
583 gross 1467 iter=0
584     r=Msolve(b)
585     r_dot_r = bilinearform(r, r)
586     if r_dot_r<0: raise NegativeNorm,"negative norm."
587     norm_b=math.sqrt(r_dot_r)
588 artak 1465
589     if x==None:
590     x=0*b
591 gross 1467 else:
592     r=Msolve(b-Aprod(x))
593     r_dot_r = bilinearform(r, r)
594     if r_dot_r<0: raise NegativeNorm,"negative norm."
595 artak 1465
596 gross 1467 h=numarray.zeros((iter_max,iter_max),numarray.Float64)
597     c=numarray.zeros(iter_max,numarray.Float64)
598     s=numarray.zeros(iter_max,numarray.Float64)
599     g=numarray.zeros(iter_max,numarray.Float64)
600 artak 1465 v=[]
601    
602 gross 1467 rho=math.sqrt(r_dot_r)
603     v.append(r/rho)
604 artak 1465 g[0]=rho
605    
606 gross 1467 while not stoppingcriterium(rho,norm_b):
607 artak 1465
608     if iter >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
609    
610    
611 gross 1467 p=Msolve(Aprod(v[iter]))
612 artak 1465
613     v.append(p)
614    
615     v_norm1=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
616    
617     # Modified Gram-Schmidt
618     for j in range(iter+1):
619     h[j][iter]=bilinearform(v[j],v[iter+1])
620     v[iter+1]+=(-1.)*h[j][iter]*v[j]
621    
622     h[iter+1][iter]=math.sqrt(bilinearform(v[iter+1],v[iter+1]))
623     v_norm2=h[iter+1][iter]
624    
625    
626     # Reorthogonalize if needed
627     if v_norm1 + 0.001*v_norm2 == v_norm1: #Brown/Hindmarsh condition (default)
628     for j in range(iter+1):
629     hr=bilinearform(v[j],v[iter+1])
630     h[j][iter]=h[j][iter]+hr #vhat
631     v[iter+1] +=(-1.)*hr*v[j]
632    
633     v_norm2=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
634     h[iter+1][iter]=v_norm2
635    
636     # watch out for happy breakdown
637     if v_norm2 != 0:
638     v[iter+1]=v[iter+1]/h[iter+1][iter]
639    
640     # Form and store the information for the new Givens rotation
641     if iter > 0 :
642     hhat=[]
643     for i in range(iter+1) : hhat.append(h[i][iter])
644 gross 1467 hhat=givapp(c[0:iter],s[0:iter],hhat);
645 artak 1465 for i in range(iter+1) : h[i][iter]=hhat[i]
646    
647     mu=math.sqrt(h[iter][iter]*h[iter][iter]+h[iter+1][iter]*h[iter+1][iter])
648     if mu!=0 :
649     c[iter]=h[iter][iter]/mu
650     s[iter]=-h[iter+1][iter]/mu
651     h[iter][iter]=c[iter]*h[iter][iter]-s[iter]*h[iter+1][iter]
652     h[iter+1][iter]=0.0
653 gross 1467 g[iter:iter+2]=givapp(c[iter],s[iter],g[iter:iter+2])
654 artak 1465
655     # Update the residual norm
656     rho=abs(g[iter+1])
657     iter+=1
658    
659     # At this point either iter > iter_max or rho < tol.
660     # It's time to compute x and leave.
661    
662     if iter > 0 :
663 gross 1467 y=numarray.zeros(iter,numarray.Float64)
664 artak 1465 y[iter-1] = g[iter-1] / h[iter-1][iter-1]
665     if iter > 1 :
666     i=iter-2
667     while i>=0 :
668 gross 1467 y[i] = ( g[i] - numarray.dot(h[i][i+1:iter], y[i+1:iter])) / h[i][i]
669 artak 1465 i=i-1
670     xhat=v[iter-1]*y[iter-1]
671     for i in range(iter-1):
672     xhat += v[i]*y[i]
673     else : xhat=v[0]
674    
675     x += xhat
676    
677     return x
678    
679     #############################################
680    
681 gross 1331 class ArithmeticTuple(object):
682     """
683     tuple supporting inplace update x+=y and scaling x=a*y where x,y is an ArithmeticTuple and a is a float.
684    
685     example of usage:
686    
687     from esys.escript import Data
688     from numarray import array
689     a=Data(...)
690     b=array([1.,4.])
691     x=ArithmeticTuple(a,b)
692     y=5.*x
693    
694     """
695     def __init__(self,*args):
696     """
697     initialize object with elements args.
698    
699     @param args: tuple of object that support implace add (x+=y) and scaling (x=a*y)
700     """
701     self.__items=list(args)
702    
703     def __len__(self):
704     """
705     number of items
706    
707     @return: number of items
708     @rtype: C{int}
709     """
710     return len(self.__items)
711    
712     def __getitem__(self,index):
713     """
714     get an item
715    
716     @param index: item to be returned
717     @type index: C{int}
718     @return: item with index C{index}
719     """
720     return self.__items.__getitem__(index)
721    
722     def __mul__(self,other):
723     """
724     scaling from the right
725    
726     @param other: scaling factor
727     @type other: C{float}
728     @return: itemwise self*other
729     @rtype: L{ArithmeticTuple}
730     """
731     out=[]
732     for i in range(len(self)):
733     out.append(self[i]*other)
734     return ArithmeticTuple(*tuple(out))
735    
736     def __rmul__(self,other):
737     """
738     scaling from the left
739    
740     @param other: scaling factor
741     @type other: C{float}
742     @return: itemwise other*self
743     @rtype: L{ArithmeticTuple}
744     """
745     out=[]
746     for i in range(len(self)):
747     out.append(other*self[i])
748     return ArithmeticTuple(*tuple(out))
749    
750 artak 1465 #########################
751     # Added by Artak
752     #########################
753     def __div__(self,other):
754     """
755     dividing from the right
756    
757     @param other: scaling factor
758     @type other: C{float}
759     @return: itemwise self/other
760     @rtype: L{ArithmeticTuple}
761     """
762     out=[]
763     for i in range(len(self)):
764     out.append(self[i]/other)
765     return ArithmeticTuple(*tuple(out))
766    
767     def __rdiv__(self,other):
768     """
769     dividing from the left
770    
771     @param other: scaling factor
772     @type other: C{float}
773     @return: itemwise other/self
774     @rtype: L{ArithmeticTuple}
775     """
776     out=[]
777     for i in range(len(self)):
778     out.append(other/self[i])
779     return ArithmeticTuple(*tuple(out))
780    
781     ##########################################33
782    
783 gross 1331 def __iadd__(self,other):
784     """
785     in-place add of other to self
786    
787     @param other: increment
788     @type other: C{ArithmeticTuple}
789     """
790     if len(self) != len(other):
791     raise ValueError,"tuple length must match."
792     for i in range(len(self)):
793     self.__items[i]+=other[i]
794     return self
795    
796 gross 1414 class HomogeneousSaddlePointProblem(object):
797     """
798     This provides a framwork for solving homogeneous saddle point problem of the form
799    
800     Av+B^*p=f
801     Bv =0
802    
803     for the unknowns v and p and given operators A and B and given right hand side f.
804     B^* is the adjoint operator of B is the given inner product.
805    
806     """
807     def __init__(self,**kwargs):
808     self.setTolerance()
809     self.setToleranceReductionFactor()
810    
811     def initialize(self):
812     """
813     initialize the problem (overwrite)
814     """
815     pass
816     def B(self,v):
817     """
818     returns Bv (overwrite)
819     @rtype: equal to the type of p
820    
821     @note: boundary conditions on p should be zero!
822     """
823     pass
824    
825     def inner(self,p0,p1):
826     """
827     returns inner product of two element p0 and p1 (overwrite)
828    
829     @type p0: equal to the type of p
830     @type p1: equal to the type of p
831     @rtype: C{float}
832    
833     @rtype: equal to the type of p
834     """
835     pass
836    
837     def solve_A(self,u,p):
838     """
839     solves Av=f-Au-B^*p with accuracy self.getReducedTolerance() (overwrite)
840    
841     @rtype: equal to the type of v
842     @note: boundary conditions on v should be zero!
843     """
844     pass
845    
846     def solve_prec(self,p):
847     """
848     provides a preconditioner for BA^{-1}B^* with accuracy self.getReducedTolerance() (overwrite)
849    
850     @rtype: equal to the type of p
851     """
852     pass
853    
854     def stoppingcriterium(self,Bv,v,p):
855     """
856     returns a True if iteration is terminated. (overwrite)
857    
858     @rtype: C{bool}
859     """
860     pass
861    
862     def __inner(self,p,r):
863     return self.inner(p,r[1])
864    
865 artak 1465 def __inner_p(self,p1,p2):
866     return self.inner(p1,p2)
867    
868 gross 1414 def __stoppingcriterium(self,norm_r,r,p):
869     return self.stoppingcriterium(r[1],r[0],p)
870    
871 gross 1467 def __stoppingcriterium_GMRES(self,norm_r,norm_b):
872     return self.stoppingcriterium_GMRES(norm_r,norm_b)
873 artak 1465
874 gross 1414 def setTolerance(self,tolerance=1.e-8):
875     self.__tol=tolerance
876     def getTolerance(self):
877     return self.__tol
878     def setToleranceReductionFactor(self,reduction=0.01):
879     self.__reduction=reduction
880     def getSubProblemTolerance(self):
881     return self.__reduction*self.getTolerance()
882    
883 artak 1465 def solve(self,v,p,max_iter=20, verbose=False, show_details=False, solver='GMRES'):
884 gross 1414 """
885     solves the saddle point problem using initial guesses v and p.
886    
887     @param max_iter: maximum number of iteration steps.
888     """
889     self.verbose=verbose
890     self.show_details=show_details and self.verbose
891    
892 gross 1469 # assume p is known: then v=A^-1(f-B^*p)
893     # which leads to BA^-1B^*p = BA^-1f
894    
895 gross 1414 # Az=f is solved as A(z-v)=f-Av (z-v = 0 on fixed_u_mask)
896    
897 artak 1465
898 gross 1414 self.__z=v+self.solve_A(v,p*0)
899 artak 1465
900 gross 1414 Bz=self.B(self.__z)
901     #
902     # solve BA^-1B^*p = Bz
903     #
904     # note that the residual r=Bz-BA^-1B^*p = B(z-A^-1B^*p) = Bv
905     #
906     # with Av=Az-B^*p = f - B^*p (v=z on fixed_u_mask)
907     # A(v-z)=Az-B^*p-Az = f -Az - B^*p (v-z=0 on fixed_u_mask)
908     #
909     self.iter=0
910 artak 1465 if solver=='GMRES':
911     if self.verbose: print "enter GMRES method (iter_max=%s)"%max_iter
912 gross 1467 p=GMRES(Bz,self.__Aprod_GMRES,self.__Msolve_GMRES,self.__inner_p,self.__stoppingcriterium_GMRES,iter_max=max_iter, x=p*1.)
913     # solve Au=f-B^*p
914     # A(u-v)=f-B^*p-Av
915     # u=v+(u-v)
916 artak 1465 u=v+self.solve_A(v,p)
917    
918     else:
919     if self.verbose: print "enter PCG method (iter_max=%s)"%max_iter
920 gross 1467 p,r=PCG(ArithmeticTuple(self.__z*1.,Bz),self.__Aprod,self.__Msolve,self.__inner,self.__stoppingcriterium,iter_max=max_iter, x=p)
921 artak 1465 u=r[0]
922 gross 1469 print "div(u)=",util.Lsup(self.B(u)),util.Lsup(u)
923 gross 1414
924 artak 1465 return u,p
925    
926 gross 1414 def __Msolve(self,r):
927     return self.solve_prec(r[1])
928    
929 artak 1465 def __Msolve_GMRES(self,r):
930     return self.solve_prec(r)
931    
932    
933 gross 1414 def __Aprod(self,p):
934     # return BA^-1B*p
935     #solve Av =-B^*p as Av =f-Az-B^*p
936 gross 1469 v=self.solve_A(self.__z,-p)
937     return ArithmeticTuple(v, self.B(v))
938 gross 1414
939 artak 1465 def __Aprod_GMRES(self,p):
940     # return BA^-1B*p
941     #solve Av =-B^*p as Av =f-Az-B^*p
942 gross 1469 v=self.solve_A(self.__z,-p)
943     return self.B(v)
944 gross 1414
945 gross 867 class SaddlePointProblem(object):
946     """
947     This implements a solver for a saddlepoint problem
948    
949 gross 877 M{f(u,p)=0}
950     M{g(u)=0}
951 gross 867
952     for u and p. The problem is solved with an inexact Uszawa scheme for p:
953    
954 ksteube 990 M{Q_f (u^{k+1}-u^{k}) = - f(u^{k},p^{k})}
955 gross 877 M{Q_g (p^{k+1}-p^{k}) = g(u^{k+1})}
956 gross 867
957     where Q_f is an approximation of the Jacobiean A_f of f with respect to u and Q_f is an approximation of
958     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'
959     Q_g can be difficult, non-linear conjugate gradient method is applied to solve for p, so Q_g plays
960     in fact the role of a preconditioner.
961     """
962     def __init__(self,verbose=False,*args):
963     """
964     initializes the problem
965    
966 ksteube 990 @param verbose: switches on the printing out some information
967 gross 867 @type verbose: C{bool}
968     @note: this method may be overwritten by a particular saddle point problem
969     """
970 gross 1107 if not isinstance(verbose,bool):
971     raise TypeError("verbose needs to be of type bool.")
972 gross 1106 self.__verbose=verbose
973 gross 877 self.relaxation=1.
974 gross 867
975     def trace(self,text):
976     """
977     prints text if verbose has been set
978    
979 ksteube 990 @param text: a text message
980 gross 867 @type text: C{str}
981     """
982     if self.__verbose: print "%s: %s"%(str(self),text)
983    
984 gross 873 def solve_f(self,u,p,tol=1.e-8):
985 gross 867 """
986     solves
987    
988     A_f du = f(u,p)
989    
990     with tolerance C{tol} and return du. A_f is Jacobiean of f with respect to u.
991    
992     @param u: current approximation of u
993     @type u: L{escript.Data}
994     @param p: current approximation of p
995     @type p: L{escript.Data}
996 gross 873 @param tol: tolerance expected for du
997 gross 867 @type tol: C{float}
998     @return: increment du
999     @rtype: L{escript.Data}
1000     @note: this method has to be overwritten by a particular saddle point problem
1001     """
1002     pass
1003    
1004 gross 873 def solve_g(self,u,tol=1.e-8):
1005 gross 867 """
1006     solves
1007    
1008     Q_g dp = g(u)
1009    
1010     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.
1011    
1012     @param u: current approximation of u
1013     @type u: L{escript.Data}
1014 gross 873 @param tol: tolerance expected for dp
1015     @type tol: C{float}
1016 gross 867 @return: increment dp
1017     @rtype: L{escript.Data}
1018     @note: this method has to be overwritten by a particular saddle point problem
1019     """
1020     pass
1021    
1022     def inner(self,p0,p1):
1023     """
1024     inner product of p0 and p1 approximating p. Typically this returns integrate(p0*p1)
1025     @return: inner product of p0 and p1
1026     @rtype: C{float}
1027     """
1028     pass
1029    
1030 gross 877 subiter_max=3
1031     def solve(self,u0,p0,tolerance=1.e-6,tolerance_u=None,iter_max=100,accepted_reduction=0.995,relaxation=None):
1032     """
1033     runs the solver
1034 gross 873
1035 gross 877 @param u0: initial guess for C{u}
1036     @type u0: L{esys.escript.Data}
1037     @param p0: initial guess for C{p}
1038     @type p0: L{esys.escript.Data}
1039     @param tolerance: tolerance for relative error in C{u} and C{p}
1040     @type tolerance: positive C{float}
1041     @param tolerance_u: tolerance for relative error in C{u} if different from C{tolerance}
1042     @type tolerance_u: positive C{float}
1043     @param iter_max: maximum number of iteration steps.
1044     @type iter_max: C{int}
1045     @param accepted_reduction: if the norm g cannot be reduced by C{accepted_reduction} backtracking to adapt the
1046     relaxation factor. If C{accepted_reduction=None} no backtracking is used.
1047     @type accepted_reduction: positive C{float} or C{None}
1048     @param relaxation: initial relaxation factor. If C{relaxation==None}, the last relaxation factor is used.
1049     @type relaxation: C{float} or C{None}
1050     """
1051     tol=1.e-2
1052     if tolerance_u==None: tolerance_u=tolerance
1053     if not relaxation==None: self.relaxation=relaxation
1054     if accepted_reduction ==None:
1055     angle_limit=0.
1056     elif accepted_reduction>=1.:
1057     angle_limit=0.
1058     else:
1059     angle_limit=util.sqrt(1-accepted_reduction**2)
1060     self.iter=0
1061     u=u0
1062     p=p0
1063     #
1064     # initialize things:
1065     #
1066     converged=False
1067     #
1068     # start loop:
1069     #
1070     # initial search direction is g
1071     #
1072     while not converged :
1073     if self.iter>iter_max:
1074     raise ArithmeticError("no convergence after %s steps."%self.iter)
1075     f_new=self.solve_f(u,p,tol)
1076     norm_f_new = util.Lsup(f_new)
1077     u_new=u-f_new
1078     g_new=self.solve_g(u_new,tol)
1079     self.iter+=1
1080     norm_g_new = util.sqrt(self.inner(g_new,g_new))
1081     if norm_f_new==0. and norm_g_new==0.: return u, p
1082     if self.iter>1 and not accepted_reduction==None:
1083     #
1084     # did we manage to reduce the norm of G? I
1085     # if not we start a backtracking procedure
1086     #
1087     # print "new/old norm = ",norm_g_new, norm_g, norm_g_new/norm_g
1088     if norm_g_new > accepted_reduction * norm_g:
1089     sub_iter=0
1090     s=self.relaxation
1091     d=g
1092     g_last=g
1093     self.trace(" start substepping: f = %s, g = %s, relaxation = %s."%(norm_f_new, norm_g_new, s))
1094     while sub_iter < self.subiter_max and norm_g_new > accepted_reduction * norm_g:
1095     dg= g_new-g_last
1096     norm_dg=abs(util.sqrt(self.inner(dg,dg))/self.relaxation)
1097     rad=self.inner(g_new,dg)/self.relaxation
1098     # print " ",sub_iter,": rad, norm_dg:",abs(rad), norm_dg*norm_g_new * angle_limit
1099     # print " ",sub_iter,": rad, norm_dg:",rad, norm_dg, norm_g_new, norm_g
1100     if abs(rad) < norm_dg*norm_g_new * angle_limit:
1101     if sub_iter>0: self.trace(" no further improvements expected from backtracking.")
1102     break
1103     r=self.relaxation
1104     self.relaxation= - rad/norm_dg**2
1105     s+=self.relaxation
1106     #####
1107     # a=g_new+self.relaxation*dg/r
1108     # print "predicted new norm = ",util.sqrt(self.inner(a,a)),util.sqrt(self.inner(g_new,g_new)), self.relaxation
1109     #####
1110     g_last=g_new
1111     p+=self.relaxation*d
1112     f_new=self.solve_f(u,p,tol)
1113     u_new=u-f_new
1114     g_new=self.solve_g(u_new,tol)
1115     self.iter+=1
1116     norm_f_new = util.Lsup(f_new)
1117     norm_g_new = util.sqrt(self.inner(g_new,g_new))
1118     # print " ",sub_iter," new g norm",norm_g_new
1119     self.trace(" %s th sub-step: f = %s, g = %s, relaxation = %s."%(sub_iter, norm_f_new, norm_g_new, s))
1120     #
1121     # can we expect reduction of g?
1122     #
1123     # u_last=u_new
1124     sub_iter+=1
1125     self.relaxation=s
1126     #
1127     # check for convergence:
1128     #
1129     norm_u_new = util.Lsup(u_new)
1130     p_new=p+self.relaxation*g_new
1131     norm_p_new = util.sqrt(self.inner(p_new,p_new))
1132 ksteube 1125 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))
1133 gross 873
1134 gross 877 if self.iter>1:
1135     dg2=g_new-g
1136     df2=f_new-f
1137     norm_dg2=util.sqrt(self.inner(dg2,dg2))
1138     norm_df2=util.Lsup(df2)
1139     # print norm_g_new, norm_g, norm_dg, norm_p, tolerance
1140     tol_eq_g=tolerance*norm_dg2/(norm_g*abs(self.relaxation))*norm_p_new
1141     tol_eq_f=tolerance_u*norm_df2/norm_f*norm_u_new
1142     if norm_g_new <= tol_eq_g and norm_f_new <= tol_eq_f:
1143     converged=True
1144     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
1145     self.trace("convergence after %s steps."%self.iter)
1146     return u,p
1147     # def solve(self,u0,p0,tolerance=1.e-6,iter_max=10,self.relaxation=1.):
1148     # tol=1.e-2
1149     # iter=0
1150     # converged=False
1151     # u=u0*1.
1152     # p=p0*1.
1153     # while not converged and iter<iter_max:
1154     # du=self.solve_f(u,p,tol)
1155     # u-=du
1156     # norm_du=util.Lsup(du)
1157     # norm_u=util.Lsup(u)
1158     #
1159     # dp=self.relaxation*self.solve_g(u,tol)
1160     # p+=dp
1161     # norm_dp=util.sqrt(self.inner(dp,dp))
1162     # norm_p=util.sqrt(self.inner(p,p))
1163     # print iter,"-th step rel. errror u,p= (%s,%s) (%s,%s)(%s,%s)"%(norm_du,norm_dp,norm_du/norm_u,norm_dp/norm_p,norm_u,norm_p)
1164     # iter+=1
1165     #
1166     # converged = (norm_du <= tolerance*norm_u) and (norm_dp <= tolerance*norm_p)
1167     # if converged:
1168     # print "convergence after %s steps."%iter
1169     # else:
1170     # raise ArithmeticError("no convergence after %s steps."%iter)
1171     #
1172     # return u,p
1173 gross 873
1174 ksteube 1312 def MaskFromBoundaryTag(function_space,*tags):
1175     """
1176     create a mask on the given function space which one for samples
1177     that touch the boundary tagged by tags.
1178    
1179     usage: m=MaskFromBoundaryTag(Solution(domain),"left", "right")
1180    
1181     @param function_space: a given function space
1182     @type function_space: L{escript.FunctionSpace}
1183     @param tags: boundray tags
1184     @type tags: C{str}
1185     @return: a mask which marks samples used by C{function_space} that are touching the
1186     boundary tagged by any of the given tags.
1187     @rtype: L{escript.Data} of rank 0
1188     """
1189     pde=linearPDEs.LinearPDE(function_space.getDomain(),numEquations=1, numSolutions=1)
1190     d=escript.Scalar(0.,escript.FunctionOnBoundary(function_space.getDomain()))
1191     for t in tags: d.setTaggedValue(t,1.)
1192     pde.setValue(y=d)
1193     out=util.whereNonZero(pde.getRightHandSide())
1194     if out.getFunctionSpace() == function_space:
1195     return out
1196     else:
1197     return util.whereNonZero(util.interpolate(out,function_space))
1198    
1199 gross 1414
1200 artak 1465

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