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Revision 1488 - (show annotations)
Fri Apr 11 00:22:31 2008 UTC (11 years, 5 months ago) by artak
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restart parameter is changed in GMRES(m)
1 #
2 # $Id$
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.
19
20 Currently includes:
21 - Projector - to project a discontinuous
22 - Locator - to trace values in data objects at a certain location
23 - TimeIntegrationManager - to handel extraplotion in time
24 - SaddlePointProblem - solver for Saddle point problems using the inexact uszawa scheme
25
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 """
33
34 __author__="Lutz Gross, l.gross@uq.edu.au"
35 __copyright__=""" Copyright (c) 2006 by ACcESS MNRF
36 http://www.access.edu.au
37 Primary Business: Queensland, Australia"""
38 __license__="""Licensed under the Open Software License version 3.0
39 http://www.opensource.org/licenses/osl-3.0.php"""
40 __url__="http://www.iservo.edu.au/esys"
41 __version__="$Revision$"
42 __date__="$Date$"
43
44
45 import escript
46 import linearPDEs
47 import numarray
48 import util
49 import math
50
51 ##### Added by Artak
52 # from Numeric import zeros,Int,Float64
53 ###################################
54
55
56 class TimeIntegrationManager:
57 """
58 a simple mechanism to manage time dependend values.
59
60 typical usage is::
61
62 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
70 @note: currently only p=1 is supported.
71 """
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 def getValue(self):
92 out=self.__v_mem[0]
93 if len(out)==1:
94 return out[0]
95 else:
96 return out
97
98 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
133
134 class Projector:
135 """
136 The Projector is a factory which projects a discontiuous function onto a
137 continuous function on the a given domain.
138 """
139 def __init__(self, domain, reduce = True, fast=True):
140 """
141 Create a continuous function space projector for a domain.
142
143 @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 """
147 self.__pde = linearPDEs.LinearPDE(domain)
148 if fast:
149 self.__pde.setSolverMethod(linearPDEs.LinearPDE.LUMPING)
150 self.__pde.setSymmetryOn()
151 self.__pde.setReducedOrderTo(reduce)
152 self.__pde.setValue(D = 1.)
153 return
154
155 def __call__(self, input_data):
156 """
157 Projects input_data onto a continuous function
158
159 @param input_data: The input_data to be projected.
160 """
161 out=escript.Data(0.,input_data.getShape(),self.__pde.getFunctionSpaceForSolution())
162 self.__pde.setValue(Y = escript.Data(), Y_reduced = escript.Data())
163 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 class NoPDE:
191 """
192 solves the following problem for u:
193
194 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 @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 """
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 @type D: C{float}, C{int}, L{numarray.NumArray}, L{Data}
226 @param Y: right hand side
227 @type Y: C{float}, C{int}, L{numarray.NumArray}, L{Data}
228 @param q: location of constraints
229 @type q: C{float}, C{int}, L{numarray.NumArray}, L{Data}
230 @param r: value of solution at locations of constraints
231 @type r: C{float}, C{int}, L{numarray.NumArray}, L{Data}
232 """
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 @type D: C{float}, C{int}, L{numarray.NumArray}, L{Data}
260 @param Y: right hand side
261 @type Y: C{float}, C{int}, L{numarray.NumArray}, L{Data}
262 @param q: location of constraints
263 @type q: C{float}, C{int}, L{numarray.NumArray}, L{Data}
264 @param r: value of solution at locations of constraints
265 @type r: C{float}, C{int}, L{numarray.NumArray}, L{Data}
266 """
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 class Locator:
304 """
305 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 """
312
313 def __init__(self,where,x=numarray.zeros((3,))):
314 """
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
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 """
324 if isinstance(where,escript.FunctionSpace):
325 self.__function_space=where
326 else:
327 self.__function_space=escript.ContinuousFunction(where)
328 if isinstance(x, list):
329 self.__id=[]
330 for p in x:
331 self.__id.append(util.length(self.__function_space.getX()-p[:self.__function_space.getDim()]).minGlobalDataPoint())
332 else:
333 self.__id=util.length(self.__function_space.getX()-x[:self.__function_space.getDim()]).minGlobalDataPoint()
334
335 def __str__(self):
336 """
337 Returns the coordinates of the Locator as a string.
338 """
339 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
354 def getX(self):
355 """
356 Returns the exact coordinates of the Locator.
357 """
358 return self(self.getFunctionSpace().getX())
359
360 def getFunctionSpace(self):
361 """
362 Returns the function space of the Locator.
363 """
364 return self.__function_space
365
366 def getId(self,item=None):
367 """
368 Returns the identifier of the location.
369 """
370 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
378
379 def __call__(self,data):
380 """
381 Returns the value of data at the Locator of a Data object otherwise
382 the object is returned.
383 """
384 return self.getValue(data)
385
386 def getValue(self,data):
387 """
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 if data.getFunctionSpace()==self.getFunctionSpace():
393 dat=data
394 else:
395 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 o=data.getValueOfGlobalDataPoint(*i)
402 if data.getRank()==0:
403 out.append(o[0])
404 else:
405 out.append(o)
406 return out
407 else:
408 out=data.getValueOfGlobalDataPoint(*id)
409 if data.getRank()==0:
410 return out[0]
411 else:
412 return out
413 else:
414 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
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
614 v.append(r/rho)
615 g[0]=rho
616
617 while not (stoppingcriterium(rho,norm_b) or iter==iter_restart-1):
618
619 if iter >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
620
621
622 p=Msolve(Aprod(v[iter]))
623
624 v.append(p)
625
626 v_norm1=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
627
628 # Modified Gram-Schmidt
629 for j in range(iter+1):
630 h[j][iter]=bilinearform(v[j],v[iter+1])
631 v[iter+1]+=(-1.)*h[j][iter]*v[j]
632
633 h[iter+1][iter]=math.sqrt(bilinearform(v[iter+1],v[iter+1]))
634 v_norm2=h[iter+1][iter]
635
636
637 # Reorthogonalize if needed
638 if v_norm1 + 0.001*v_norm2 == v_norm1: #Brown/Hindmarsh condition (default)
639 for j in range(iter+1):
640 hr=bilinearform(v[j],v[iter+1])
641 h[j][iter]=h[j][iter]+hr #vhat
642 v[iter+1] +=(-1.)*hr*v[j]
643
644 v_norm2=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
645 h[iter+1][iter]=v_norm2
646
647 # watch out for happy breakdown
648 if v_norm2 != 0:
649 v[iter+1]=v[iter+1]/h[iter+1][iter]
650
651 # Form and store the information for the new Givens rotation
652 if iter > 0 :
653 hhat=[]
654 for i in range(iter+1) : hhat.append(h[i][iter])
655 hhat=givapp(c[0:iter],s[0:iter],hhat);
656 for i in range(iter+1) : h[i][iter]=hhat[i]
657
658 mu=math.sqrt(h[iter][iter]*h[iter][iter]+h[iter+1][iter]*h[iter+1][iter])
659 if mu!=0 :
660 c[iter]=h[iter][iter]/mu
661 s[iter]=-h[iter+1][iter]/mu
662 h[iter][iter]=c[iter]*h[iter][iter]-s[iter]*h[iter+1][iter]
663 h[iter+1][iter]=0.0
664 g[iter:iter+2]=givapp(c[iter],s[iter],g[iter:iter+2])
665
666 # Update the residual norm
667 rho=abs(g[iter+1])
668 iter+=1
669
670 # At this point either iter > iter_max or rho < tol.
671 # It's time to compute x and leave.
672
673 if iter > 0 :
674 y=numarray.zeros(iter,numarray.Float64)
675 y[iter-1] = g[iter-1] / h[iter-1][iter-1]
676 if iter > 1 :
677 i=iter-2
678 while i>=0 :
679 y[i] = ( g[i] - numarray.dot(h[i][i+1:iter], y[i+1:iter])) / h[i][i]
680 i=i-1
681 xhat=v[iter-1]*y[iter-1]
682 for i in range(iter-1):
683 xhat += v[i]*y[i]
684 else : xhat=v[0]
685
686 x += xhat
687 if iter<iter_restart-1:
688 stopped=True
689 else:
690 stopped=False
691
692 return x,stopped
693
694 def MINRES(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100):
695
696 #
697 # minres solves the system of linear equations Ax = b
698 # where A is a symmetric matrix (possibly indefinite or singular)
699 # and b is a given vector.
700 #
701 # "A" may be a dense or sparse matrix (preferably sparse!)
702 # or the name of a function such that
703 # y = A(x)
704 # returns the product y = Ax for any given vector x.
705 #
706 # "M" defines a positive-definite preconditioner M = C C'.
707 # "M" may be a dense or sparse matrix (preferably sparse!)
708 # or the name of a function such that
709 # solves the system My = x for any given vector x.
710 #
711 #
712
713 #------------------------------------------------------------------
714 # Set up y and v for the first Lanczos vector v1.
715 # y = beta1 P' v1, where P = C**(-1).
716 # v is really P' v1.
717 #------------------------------------------------------------------
718 if x==None:
719 x=0*b
720 else:
721 b += (-1)*Aprod(x)
722
723 r1 = b
724 y = Msolve(b)
725 beta1 = bilinearform(b,y)
726
727 if beta1< 0: raise NegativeNorm,"negative norm."
728
729 # If b = 0 exactly, stop with x = 0.
730 if beta1==0: return x*0.
731
732 if beta1> 0:
733 beta1 = math.sqrt(beta1)
734
735 #------------------------------------------------------------------
736 # Initialize quantities.
737 # ------------------------------------------------------------------
738 iter = 0
739 Anorm = 0
740 ynorm = 0
741 oldb = 0
742 beta = beta1
743 dbar = 0
744 epsln = 0
745 phibar = beta1
746 rhs1 = beta1
747 rhs2 = 0
748 rnorm = phibar
749 tnorm2 = 0
750 ynorm2 = 0
751 cs = -1
752 sn = 0
753 w = b*0.
754 w2 = b*0.
755 r2 = r1
756 eps = 0.0001
757
758 #---------------------------------------------------------------------
759 # Main iteration loop.
760 # --------------------------------------------------------------------
761 while not stoppingcriterium(rnorm,Anorm*ynorm): # checks ||r|| < (||A|| ||x||) * TOL
762
763 if iter >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
764 iter = iter + 1
765
766 #-----------------------------------------------------------------
767 # Obtain quantities for the next Lanczos vector vk+1, k = 1, 2,...
768 # The general iteration is similar to the case k = 1 with v0 = 0:
769 #
770 # p1 = Operator * v1 - beta1 * v0,
771 # alpha1 = v1'p1,
772 # q2 = p2 - alpha1 * v1,
773 # beta2^2 = q2'q2,
774 # v2 = (1/beta2) q2.
775 #
776 # Again, y = betak P vk, where P = C**(-1).
777 #-----------------------------------------------------------------
778 s = 1/beta # Normalize previous vector (in y).
779 v = s*y # v = vk if P = I
780
781 y = Aprod(v)
782
783 if iter >= 2:
784 y = y - (beta/oldb)*r1
785
786 alfa = bilinearform(v,y) # alphak
787 y = (- alfa/beta)*r2 + y
788 r1 = r2
789 r2 = y
790 y = Msolve(r2)
791 oldb = beta # oldb = betak
792 beta = bilinearform(r2,y) # beta = betak+1^2
793 if beta < 0: raise NegativeNorm,"negative norm."
794
795 beta = math.sqrt( beta )
796 tnorm2 = tnorm2 + alfa*alfa + oldb*oldb + beta*beta
797
798 if iter==1: # Initialize a few things.
799 gmax = abs( alfa ) # alpha1
800 gmin = gmax # alpha1
801
802 # Apply previous rotation Qk-1 to get
803 # [deltak epslnk+1] = [cs sn][dbark 0 ]
804 # [gbar k dbar k+1] [sn -cs][alfak betak+1].
805
806 oldeps = epsln
807 delta = cs * dbar + sn * alfa # delta1 = 0 deltak
808 gbar = sn * dbar - cs * alfa # gbar 1 = alfa1 gbar k
809 epsln = sn * beta # epsln2 = 0 epslnk+1
810 dbar = - cs * beta # dbar 2 = beta2 dbar k+1
811
812 # Compute the next plane rotation Qk
813
814 gamma = math.sqrt(gbar*gbar+beta*beta) # gammak
815 gamma = max(gamma,eps)
816 cs = gbar / gamma # ck
817 sn = beta / gamma # sk
818 phi = cs * phibar # phik
819 phibar = sn * phibar # phibark+1
820
821 # Update x.
822
823 denom = 1/gamma
824 w1 = w2
825 w2 = w
826 w = (v - oldeps*w1 - delta*w2) * denom
827 x = x + phi*w
828
829 # Go round again.
830
831 gmax = max(gmax,gamma)
832 gmin = min(gmin,gamma)
833 z = rhs1 / gamma
834 ynorm2 = z*z + ynorm2
835 rhs1 = rhs2 - delta*z
836 rhs2 = - epsln*z
837
838 # Estimate various norms and test for convergence.
839
840 Anorm = math.sqrt( tnorm2 )
841 ynorm = math.sqrt( ynorm2 )
842
843 rnorm = phibar
844
845 return x
846
847 #############################################
848
849 class ArithmeticTuple(object):
850 """
851 tuple supporting inplace update x+=y and scaling x=a*y where x,y is an ArithmeticTuple and a is a float.
852
853 example of usage:
854
855 from esys.escript import Data
856 from numarray import array
857 a=Data(...)
858 b=array([1.,4.])
859 x=ArithmeticTuple(a,b)
860 y=5.*x
861
862 """
863 def __init__(self,*args):
864 """
865 initialize object with elements args.
866
867 @param args: tuple of object that support implace add (x+=y) and scaling (x=a*y)
868 """
869 self.__items=list(args)
870
871 def __len__(self):
872 """
873 number of items
874
875 @return: number of items
876 @rtype: C{int}
877 """
878 return len(self.__items)
879
880 def __getitem__(self,index):
881 """
882 get an item
883
884 @param index: item to be returned
885 @type index: C{int}
886 @return: item with index C{index}
887 """
888 return self.__items.__getitem__(index)
889
890 def __mul__(self,other):
891 """
892 scaling from the right
893
894 @param other: scaling factor
895 @type other: C{float}
896 @return: itemwise self*other
897 @rtype: L{ArithmeticTuple}
898 """
899 out=[]
900 for i in range(len(self)):
901 out.append(self[i]*other)
902 return ArithmeticTuple(*tuple(out))
903
904 def __rmul__(self,other):
905 """
906 scaling from the left
907
908 @param other: scaling factor
909 @type other: C{float}
910 @return: itemwise other*self
911 @rtype: L{ArithmeticTuple}
912 """
913 out=[]
914 for i in range(len(self)):
915 out.append(other*self[i])
916 return ArithmeticTuple(*tuple(out))
917
918 #########################
919 # Added by Artak
920 #########################
921 def __div__(self,other):
922 """
923 dividing from the right
924
925 @param other: scaling factor
926 @type other: C{float}
927 @return: itemwise self/other
928 @rtype: L{ArithmeticTuple}
929 """
930 out=[]
931 for i in range(len(self)):
932 out.append(self[i]/other)
933 return ArithmeticTuple(*tuple(out))
934
935 def __rdiv__(self,other):
936 """
937 dividing from the left
938
939 @param other: scaling factor
940 @type other: C{float}
941 @return: itemwise other/self
942 @rtype: L{ArithmeticTuple}
943 """
944 out=[]
945 for i in range(len(self)):
946 out.append(other/self[i])
947 return ArithmeticTuple(*tuple(out))
948
949 ##########################################33
950
951 def __iadd__(self,other):
952 """
953 in-place add of other to self
954
955 @param other: increment
956 @type other: C{ArithmeticTuple}
957 """
958 if len(self) != len(other):
959 raise ValueError,"tuple length must match."
960 for i in range(len(self)):
961 self.__items[i]+=other[i]
962 return self
963
964 class HomogeneousSaddlePointProblem(object):
965 """
966 This provides a framwork for solving homogeneous saddle point problem of the form
967
968 Av+B^*p=f
969 Bv =0
970
971 for the unknowns v and p and given operators A and B and given right hand side f.
972 B^* is the adjoint operator of B is the given inner product.
973
974 """
975 def __init__(self,**kwargs):
976 self.setTolerance()
977 self.setToleranceReductionFactor()
978
979 def initialize(self):
980 """
981 initialize the problem (overwrite)
982 """
983 pass
984 def B(self,v):
985 """
986 returns Bv (overwrite)
987 @rtype: equal to the type of p
988
989 @note: boundary conditions on p should be zero!
990 """
991 pass
992
993 def inner(self,p0,p1):
994 """
995 returns inner product of two element p0 and p1 (overwrite)
996
997 @type p0: equal to the type of p
998 @type p1: equal to the type of p
999 @rtype: C{float}
1000
1001 @rtype: equal to the type of p
1002 """
1003 pass
1004
1005 def solve_A(self,u,p):
1006 """
1007 solves Av=f-Au-B^*p with accuracy self.getReducedTolerance() (overwrite)
1008
1009 @rtype: equal to the type of v
1010 @note: boundary conditions on v should be zero!
1011 """
1012 pass
1013
1014 def solve_prec(self,p):
1015 """
1016 provides a preconditioner for BA^{-1}B^* with accuracy self.getReducedTolerance() (overwrite)
1017
1018 @rtype: equal to the type of p
1019 """
1020 pass
1021
1022 def stoppingcriterium(self,Bv,v,p):
1023 """
1024 returns a True if iteration is terminated. (overwrite)
1025
1026 @rtype: C{bool}
1027 """
1028 pass
1029
1030 def __inner(self,p,r):
1031 return self.inner(p,r[1])
1032
1033 def __inner_p(self,p1,p2):
1034 return self.inner(p1,p2)
1035
1036 def __stoppingcriterium(self,norm_r,r,p):
1037 return self.stoppingcriterium(r[1],r[0],p)
1038
1039 def __stoppingcriterium_GMRES(self,norm_r,norm_b):
1040 return self.stoppingcriterium_GMRES(norm_r,norm_b)
1041
1042 def __stoppingcriterium_MINRES(self,norm_r,norm_Ax):
1043 return self.stoppingcriterium_MINRES(norm_r,norm_Ax)
1044
1045
1046 def setTolerance(self,tolerance=1.e-8):
1047 self.__tol=tolerance
1048 def getTolerance(self):
1049 return self.__tol
1050 def setToleranceReductionFactor(self,reduction=0.01):
1051 self.__reduction=reduction
1052 def getSubProblemTolerance(self):
1053 return self.__reduction*self.getTolerance()
1054
1055 def solve(self,v,p,max_iter=20, verbose=False, show_details=False, solver='PCG',iter_restart=10):
1056 """
1057 solves the saddle point problem using initial guesses v and p.
1058
1059 @param max_iter: maximum number of iteration steps.
1060 """
1061 self.verbose=verbose
1062 self.show_details=show_details and self.verbose
1063
1064 # assume p is known: then v=A^-1(f-B^*p)
1065 # which leads to BA^-1B^*p = BA^-1f
1066
1067 # Az=f is solved as A(z-v)=f-Av (z-v = 0 on fixed_u_mask)
1068
1069
1070 self.__z=v+self.solve_A(v,p*0)
1071
1072 Bz=self.B(self.__z)
1073 #
1074 # solve BA^-1B^*p = Bz
1075 #
1076 # note that the residual r=Bz-BA^-1B^*p = B(z-A^-1B^*p) = Bv
1077 #
1078 # with Av=Az-B^*p = f - B^*p (v=z on fixed_u_mask)
1079 # A(v-z)=Az-B^*p-Az = f -Az - B^*p (v-z=0 on fixed_u_mask)
1080 #
1081 self.iter=0
1082 if solver=='GMRES':
1083 if self.verbose: print "enter GMRES method (iter_max=%s)"%max_iter
1084 p=GMRES(Bz,self.__Aprod_GMRES,self.__Msolve_GMRES,self.__inner_p,self.__stoppingcriterium_GMRES,iter_max=max_iter, x=p*1.,iter_restart=iter_restart)
1085 # solve Au=f-B^*p
1086 # A(u-v)=f-B^*p-Av
1087 # u=v+(u-v)
1088 u=v+self.solve_A(v,p)
1089
1090 if solver=='MINRES':
1091 if self.verbose: print "enter MINRES method (iter_max=%s)"%max_iter
1092 p=MINRES(Bz,self.__Aprod_GMRES,self.__Msolve_GMRES,self.__inner_p,self.__stoppingcriterium_MINRES,iter_max=max_iter, x=p*1.)
1093 # solve Au=f-B^*p
1094 # A(u-v)=f-B^*p-Av
1095 # u=v+(u-v)
1096 u=v+self.solve_A(v,p)
1097
1098 if solver=='PCG':
1099 if self.verbose: print "enter PCG method (iter_max=%s)"%max_iter
1100 p,r=PCG(ArithmeticTuple(self.__z*1.,Bz),self.__Aprod,self.__Msolve,self.__inner,self.__stoppingcriterium,iter_max=max_iter, x=p)
1101 u=r[0]
1102
1103 print "RESULT div(u)=",util.Lsup(self.B(u)),util.Lsup(u)
1104
1105 return u,p
1106
1107 def __Msolve(self,r):
1108 return self.solve_prec(r[1])
1109
1110 def __Msolve_GMRES(self,r):
1111 return self.solve_prec(r)
1112
1113
1114 def __Aprod(self,p):
1115 # return BA^-1B*p
1116 #solve Av =-B^*p as Av =f-Az-B^*p
1117 v=self.solve_A(self.__z,-p)
1118 return ArithmeticTuple(v, self.B(v))
1119
1120 def __Aprod_GMRES(self,p):
1121 # return BA^-1B*p
1122 #solve Av =-B^*p as Av =f-Az-B^*p
1123 v=self.solve_A(self.__z,-p)
1124 return self.B(v)
1125
1126 class SaddlePointProblem(object):
1127 """
1128 This implements a solver for a saddlepoint problem
1129
1130 M{f(u,p)=0}
1131 M{g(u)=0}
1132
1133 for u and p. The problem is solved with an inexact Uszawa scheme for p:
1134
1135 M{Q_f (u^{k+1}-u^{k}) = - f(u^{k},p^{k})}
1136 M{Q_g (p^{k+1}-p^{k}) = g(u^{k+1})}
1137
1138 where Q_f is an approximation of the Jacobiean A_f of f with respect to u and Q_f is an approximation of
1139 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'
1140 Q_g can be difficult, non-linear conjugate gradient method is applied to solve for p, so Q_g plays
1141 in fact the role of a preconditioner.
1142 """
1143 def __init__(self,verbose=False,*args):
1144 """
1145 initializes the problem
1146
1147 @param verbose: switches on the printing out some information
1148 @type verbose: C{bool}
1149 @note: this method may be overwritten by a particular saddle point problem
1150 """
1151 if not isinstance(verbose,bool):
1152 raise TypeError("verbose needs to be of type bool.")
1153 self.__verbose=verbose
1154 self.relaxation=1.
1155
1156 def trace(self,text):
1157 """
1158 prints text if verbose has been set
1159
1160 @param text: a text message
1161 @type text: C{str}
1162 """
1163 if self.__verbose: print "%s: %s"%(str(self),text)
1164
1165 def solve_f(self,u,p,tol=1.e-8):
1166 """
1167 solves
1168
1169 A_f du = f(u,p)
1170
1171 with tolerance C{tol} and return du. A_f is Jacobiean of f with respect to u.
1172
1173 @param u: current approximation of u
1174 @type u: L{escript.Data}
1175 @param p: current approximation of p
1176 @type p: L{escript.Data}
1177 @param tol: tolerance expected for du
1178 @type tol: C{float}
1179 @return: increment du
1180 @rtype: L{escript.Data}
1181 @note: this method has to be overwritten by a particular saddle point problem
1182 """
1183 pass
1184
1185 def solve_g(self,u,tol=1.e-8):
1186 """
1187 solves
1188
1189 Q_g dp = g(u)
1190
1191 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.
1192
1193 @param u: current approximation of u
1194 @type u: L{escript.Data}
1195 @param tol: tolerance expected for dp
1196 @type tol: C{float}
1197 @return: increment dp
1198 @rtype: L{escript.Data}
1199 @note: this method has to be overwritten by a particular saddle point problem
1200 """
1201 pass
1202
1203 def inner(self,p0,p1):
1204 """
1205 inner product of p0 and p1 approximating p. Typically this returns integrate(p0*p1)
1206 @return: inner product of p0 and p1
1207 @rtype: C{float}
1208 """
1209 pass
1210
1211 subiter_max=3
1212 def solve(self,u0,p0,tolerance=1.e-6,tolerance_u=None,iter_max=100,accepted_reduction=0.995,relaxation=None):
1213 """
1214 runs the solver
1215
1216 @param u0: initial guess for C{u}
1217 @type u0: L{esys.escript.Data}
1218 @param p0: initial guess for C{p}
1219 @type p0: L{esys.escript.Data}
1220 @param tolerance: tolerance for relative error in C{u} and C{p}
1221 @type tolerance: positive C{float}
1222 @param tolerance_u: tolerance for relative error in C{u} if different from C{tolerance}
1223 @type tolerance_u: positive C{float}
1224 @param iter_max: maximum number of iteration steps.
1225 @type iter_max: C{int}
1226 @param accepted_reduction: if the norm g cannot be reduced by C{accepted_reduction} backtracking to adapt the
1227 relaxation factor. If C{accepted_reduction=None} no backtracking is used.
1228 @type accepted_reduction: positive C{float} or C{None}
1229 @param relaxation: initial relaxation factor. If C{relaxation==None}, the last relaxation factor is used.
1230 @type relaxation: C{float} or C{None}
1231 """
1232 tol=1.e-2
1233 if tolerance_u==None: tolerance_u=tolerance
1234 if not relaxation==None: self.relaxation=relaxation
1235 if accepted_reduction ==None:
1236 angle_limit=0.
1237 elif accepted_reduction>=1.:
1238 angle_limit=0.
1239 else:
1240 angle_limit=util.sqrt(1-accepted_reduction**2)
1241 self.iter=0
1242 u=u0
1243 p=p0
1244 #
1245 # initialize things:
1246 #
1247 converged=False
1248 #
1249 # start loop:
1250 #
1251 # initial search direction is g
1252 #
1253 while not converged :
1254 if self.iter>iter_max:
1255 raise ArithmeticError("no convergence after %s steps."%self.iter)
1256 f_new=self.solve_f(u,p,tol)
1257 norm_f_new = util.Lsup(f_new)
1258 u_new=u-f_new
1259 g_new=self.solve_g(u_new,tol)
1260 self.iter+=1
1261 norm_g_new = util.sqrt(self.inner(g_new,g_new))
1262 if norm_f_new==0. and norm_g_new==0.: return u, p
1263 if self.iter>1 and not accepted_reduction==None:
1264 #
1265 # did we manage to reduce the norm of G? I
1266 # if not we start a backtracking procedure
1267 #
1268 # print "new/old norm = ",norm_g_new, norm_g, norm_g_new/norm_g
1269 if norm_g_new > accepted_reduction * norm_g:
1270 sub_iter=0
1271 s=self.relaxation
1272 d=g
1273 g_last=g
1274 self.trace(" start substepping: f = %s, g = %s, relaxation = %s."%(norm_f_new, norm_g_new, s))
1275 while sub_iter < self.subiter_max and norm_g_new > accepted_reduction * norm_g:
1276 dg= g_new-g_last
1277 norm_dg=abs(util.sqrt(self.inner(dg,dg))/self.relaxation)
1278 rad=self.inner(g_new,dg)/self.relaxation
1279 # print " ",sub_iter,": rad, norm_dg:",abs(rad), norm_dg*norm_g_new * angle_limit
1280 # print " ",sub_iter,": rad, norm_dg:",rad, norm_dg, norm_g_new, norm_g
1281 if abs(rad) < norm_dg*norm_g_new * angle_limit:
1282 if sub_iter>0: self.trace(" no further improvements expected from backtracking.")
1283 break
1284 r=self.relaxation
1285 self.relaxation= - rad/norm_dg**2
1286 s+=self.relaxation
1287 #####
1288 # a=g_new+self.relaxation*dg/r
1289 # print "predicted new norm = ",util.sqrt(self.inner(a,a)),util.sqrt(self.inner(g_new,g_new)), self.relaxation
1290 #####
1291 g_last=g_new
1292 p+=self.relaxation*d
1293 f_new=self.solve_f(u,p,tol)
1294 u_new=u-f_new
1295 g_new=self.solve_g(u_new,tol)
1296 self.iter+=1
1297 norm_f_new = util.Lsup(f_new)
1298 norm_g_new = util.sqrt(self.inner(g_new,g_new))
1299 # print " ",sub_iter," new g norm",norm_g_new
1300 self.trace(" %s th sub-step: f = %s, g = %s, relaxation = %s."%(sub_iter, norm_f_new, norm_g_new, s))
1301 #
1302 # can we expect reduction of g?
1303 #
1304 # u_last=u_new
1305 sub_iter+=1
1306 self.relaxation=s
1307 #
1308 # check for convergence:
1309 #
1310 norm_u_new = util.Lsup(u_new)
1311 p_new=p+self.relaxation*g_new
1312 norm_p_new = util.sqrt(self.inner(p_new,p_new))
1313 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))
1314
1315 if self.iter>1:
1316 dg2=g_new-g
1317 df2=f_new-f
1318 norm_dg2=util.sqrt(self.inner(dg2,dg2))
1319 norm_df2=util.Lsup(df2)
1320 # print norm_g_new, norm_g, norm_dg, norm_p, tolerance
1321 tol_eq_g=tolerance*norm_dg2/(norm_g*abs(self.relaxation))*norm_p_new
1322 tol_eq_f=tolerance_u*norm_df2/norm_f*norm_u_new
1323 if norm_g_new <= tol_eq_g and norm_f_new <= tol_eq_f:
1324 converged=True
1325 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
1326 self.trace("convergence after %s steps."%self.iter)
1327 return u,p
1328 # def solve(self,u0,p0,tolerance=1.e-6,iter_max=10,self.relaxation=1.):
1329 # tol=1.e-2
1330 # iter=0
1331 # converged=False
1332 # u=u0*1.
1333 # p=p0*1.
1334 # while not converged and iter<iter_max:
1335 # du=self.solve_f(u,p,tol)
1336 # u-=du
1337 # norm_du=util.Lsup(du)
1338 # norm_u=util.Lsup(u)
1339 #
1340 # dp=self.relaxation*self.solve_g(u,tol)
1341 # p+=dp
1342 # norm_dp=util.sqrt(self.inner(dp,dp))
1343 # norm_p=util.sqrt(self.inner(p,p))
1344 # 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)
1345 # iter+=1
1346 #
1347 # converged = (norm_du <= tolerance*norm_u) and (norm_dp <= tolerance*norm_p)
1348 # if converged:
1349 # print "convergence after %s steps."%iter
1350 # else:
1351 # raise ArithmeticError("no convergence after %s steps."%iter)
1352 #
1353 # return u,p
1354
1355 def MaskFromBoundaryTag(function_space,*tags):
1356 """
1357 create a mask on the given function space which one for samples
1358 that touch the boundary tagged by tags.
1359
1360 usage: m=MaskFromBoundaryTag(Solution(domain),"left", "right")
1361
1362 @param function_space: a given function space
1363 @type function_space: L{escript.FunctionSpace}
1364 @param tags: boundray tags
1365 @type tags: C{str}
1366 @return: a mask which marks samples used by C{function_space} that are touching the
1367 boundary tagged by any of the given tags.
1368 @rtype: L{escript.Data} of rank 0
1369 """
1370 pde=linearPDEs.LinearPDE(function_space.getDomain(),numEquations=1, numSolutions=1)
1371 d=escript.Scalar(0.,escript.FunctionOnBoundary(function_space.getDomain()))
1372 for t in tags: d.setTaggedValue(t,1.)
1373 pde.setValue(y=d)
1374 out=util.whereNonZero(pde.getRightHandSide())
1375 if out.getFunctionSpace() == function_space:
1376 return out
1377 else:
1378 return util.whereNonZero(util.interpolate(out,function_space))
1379
1380
1381

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