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Revision 1482 - (show annotations)
Wed Apr 9 02:29:47 2008 UTC (11 years, 5 months ago) by artak
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inital x taken into account in MINRES algorithm
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*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 def MINRES(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100):
694
695 #
696 # minres solves the system of linear equations Ax = b
697 # where A is a symmetric matrix (possibly indefinite or singular)
698 # and b is a given vector.
699 #
700 # "A" may be a dense or sparse matrix (preferably sparse!)
701 # or the name of a function such that
702 # y = A(x)
703 # returns the product y = Ax for any given vector x.
704 #
705 # "M" defines a positive-definite preconditioner M = C C'.
706 # "M" may be a dense or sparse matrix (preferably sparse!)
707 # or the name of a function such that
708 # solves the system My = x for any given vector x.
709 #
710 #
711
712 #------------------------------------------------------------------
713 # Set up y and v for the first Lanczos vector v1.
714 # y = beta1 P' v1, where P = C**(-1).
715 # v is really P' v1.
716 #------------------------------------------------------------------
717 if x==None:
718 x=0*b
719 else:
720 b += (-1)*Aprod(x)
721
722 r1 = b
723 y = Msolve(b)
724 beta1 = bilinearform(b,y)
725
726 if beta1< 0: raise NegativeNorm,"negative norm."
727
728 # If b = 0 exactly, stop with x = 0.
729 if beta1==0: return x*0.
730
731 if beta1> 0:
732 beta1 = math.sqrt(beta1) # Normalize y to get v1 later.
733
734 #------------------------------------------------------------------
735 # Initialize other quantities.
736 # ------------------------------------------------------------------
737 # Initialize
738
739 iter = 0
740 Anorm = 0
741 ynorm = 0
742 # x=x*0
743
744 oldb = 0
745 beta = beta1
746 dbar = 0
747 epsln = 0
748 phibar = beta1
749 rhs1 = beta1
750 rhs2 = 0
751 rnorm = phibar
752 tnorm2 = 0
753 ynorm2 = 0
754 cs = -1
755 sn = 0
756 w = b*0.
757 w2 = b*0.
758 r2 = r1
759 eps = 0.0001
760
761 #---------------------------------------------------------------------
762 # Main iteration loop.
763 # --------------------------------------------------------------------
764 while not stoppingcriterium(rnorm,Anorm*ynorm): # ||r|| / (||A|| ||x||)
765
766 if iter >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
767 iter = iter + 1
768
769 #-----------------------------------------------------------------
770 # Obtain quantities for the next Lanczos vector vk+1, k = 1, 2,...
771 # The general iteration is similar to the case k = 1 with v0 = 0:
772 #
773 # p1 = Operator * v1 - beta1 * v0,
774 # alpha1 = v1'p1,
775 # q2 = p2 - alpha1 * v1,
776 # beta2^2 = q2'q2,
777 # v2 = (1/beta2) q2.
778 #
779 # Again, y = betak P vk, where P = C**(-1).
780 #-----------------------------------------------------------------
781 s = 1/beta # Normalize previous vector (in y).
782 v = s*y # v = vk if P = I
783
784 y = Aprod(v)
785
786 if iter >= 2:
787 y = y - (beta/oldb)*r1
788
789 alfa = bilinearform(v,y) # alphak
790 y = (- alfa/beta)*r2 + y
791 r1 = r2
792 r2 = y
793 y = Msolve(r2)
794 oldb = beta # oldb = betak
795 beta = bilinearform(r2,y) # beta = betak+1^2
796 if beta < 0: raise NegativeNorm,"negative norm."
797
798 beta = math.sqrt( beta )
799 tnorm2 = tnorm2 + alfa*alfa + oldb*oldb + beta*beta
800
801 if iter==1: # Initialize a few things.
802 gmax = abs( alfa ) # alpha1
803 gmin = gmax # alpha1
804
805 # Apply previous rotation Qk-1 to get
806 # [deltak epslnk+1] = [cs sn][dbark 0 ]
807 # [gbar k dbar k+1] [sn -cs][alfak betak+1].
808
809 oldeps = epsln
810 delta = cs * dbar + sn * alfa # delta1 = 0 deltak
811 gbar = sn * dbar - cs * alfa # gbar 1 = alfa1 gbar k
812 epsln = sn * beta # epsln2 = 0 epslnk+1
813 dbar = - cs * beta # dbar 2 = beta2 dbar k+1
814
815 # Compute the next plane rotation Qk
816
817 gamma = math.sqrt(gbar*gbar+beta*beta) # gammak
818 gamma = max(gamma,eps)
819 cs = gbar / gamma # ck
820 sn = beta / gamma # sk
821 phi = cs * phibar # phik
822 phibar = sn * phibar # phibark+1
823
824 # Update x.
825
826 denom = 1/gamma
827 w1 = w2
828 w2 = w
829 w = (v - oldeps*w1 - delta*w2) * denom
830 x = x + phi*w
831
832 # Go round again.
833
834 gmax = max(gmax,gamma)
835 gmin = min(gmin,gamma)
836 z = rhs1 / gamma
837 ynorm2 = z*z + ynorm2
838 rhs1 = rhs2 - delta*z
839 rhs2 = - epsln*z
840
841 # Estimate various norms and test for convergence.
842
843 Anorm = math.sqrt( tnorm2 )
844 ynorm = math.sqrt( ynorm2 )
845
846 rnorm = phibar
847
848 # Return final answer.
849 print iter
850 return x
851
852 #############################################
853
854 class ArithmeticTuple(object):
855 """
856 tuple supporting inplace update x+=y and scaling x=a*y where x,y is an ArithmeticTuple and a is a float.
857
858 example of usage:
859
860 from esys.escript import Data
861 from numarray import array
862 a=Data(...)
863 b=array([1.,4.])
864 x=ArithmeticTuple(a,b)
865 y=5.*x
866
867 """
868 def __init__(self,*args):
869 """
870 initialize object with elements args.
871
872 @param args: tuple of object that support implace add (x+=y) and scaling (x=a*y)
873 """
874 self.__items=list(args)
875
876 def __len__(self):
877 """
878 number of items
879
880 @return: number of items
881 @rtype: C{int}
882 """
883 return len(self.__items)
884
885 def __getitem__(self,index):
886 """
887 get an item
888
889 @param index: item to be returned
890 @type index: C{int}
891 @return: item with index C{index}
892 """
893 return self.__items.__getitem__(index)
894
895 def __mul__(self,other):
896 """
897 scaling from the right
898
899 @param other: scaling factor
900 @type other: C{float}
901 @return: itemwise self*other
902 @rtype: L{ArithmeticTuple}
903 """
904 out=[]
905 for i in range(len(self)):
906 out.append(self[i]*other)
907 return ArithmeticTuple(*tuple(out))
908
909 def __rmul__(self,other):
910 """
911 scaling from the left
912
913 @param other: scaling factor
914 @type other: C{float}
915 @return: itemwise other*self
916 @rtype: L{ArithmeticTuple}
917 """
918 out=[]
919 for i in range(len(self)):
920 out.append(other*self[i])
921 return ArithmeticTuple(*tuple(out))
922
923 #########################
924 # Added by Artak
925 #########################
926 def __div__(self,other):
927 """
928 dividing from the right
929
930 @param other: scaling factor
931 @type other: C{float}
932 @return: itemwise self/other
933 @rtype: L{ArithmeticTuple}
934 """
935 out=[]
936 for i in range(len(self)):
937 out.append(self[i]/other)
938 return ArithmeticTuple(*tuple(out))
939
940 def __rdiv__(self,other):
941 """
942 dividing from the left
943
944 @param other: scaling factor
945 @type other: C{float}
946 @return: itemwise other/self
947 @rtype: L{ArithmeticTuple}
948 """
949 out=[]
950 for i in range(len(self)):
951 out.append(other/self[i])
952 return ArithmeticTuple(*tuple(out))
953
954 ##########################################33
955
956 def __iadd__(self,other):
957 """
958 in-place add of other to self
959
960 @param other: increment
961 @type other: C{ArithmeticTuple}
962 """
963 if len(self) != len(other):
964 raise ValueError,"tuple length must match."
965 for i in range(len(self)):
966 self.__items[i]+=other[i]
967 return self
968
969 class HomogeneousSaddlePointProblem(object):
970 """
971 This provides a framwork for solving homogeneous saddle point problem of the form
972
973 Av+B^*p=f
974 Bv =0
975
976 for the unknowns v and p and given operators A and B and given right hand side f.
977 B^* is the adjoint operator of B is the given inner product.
978
979 """
980 def __init__(self,**kwargs):
981 self.setTolerance()
982 self.setToleranceReductionFactor()
983
984 def initialize(self):
985 """
986 initialize the problem (overwrite)
987 """
988 pass
989 def B(self,v):
990 """
991 returns Bv (overwrite)
992 @rtype: equal to the type of p
993
994 @note: boundary conditions on p should be zero!
995 """
996 pass
997
998 def inner(self,p0,p1):
999 """
1000 returns inner product of two element p0 and p1 (overwrite)
1001
1002 @type p0: equal to the type of p
1003 @type p1: equal to the type of p
1004 @rtype: C{float}
1005
1006 @rtype: equal to the type of p
1007 """
1008 pass
1009
1010 def solve_A(self,u,p):
1011 """
1012 solves Av=f-Au-B^*p with accuracy self.getReducedTolerance() (overwrite)
1013
1014 @rtype: equal to the type of v
1015 @note: boundary conditions on v should be zero!
1016 """
1017 pass
1018
1019 def solve_prec(self,p):
1020 """
1021 provides a preconditioner for BA^{-1}B^* with accuracy self.getReducedTolerance() (overwrite)
1022
1023 @rtype: equal to the type of p
1024 """
1025 pass
1026
1027 def stoppingcriterium(self,Bv,v,p):
1028 """
1029 returns a True if iteration is terminated. (overwrite)
1030
1031 @rtype: C{bool}
1032 """
1033 pass
1034
1035 def __inner(self,p,r):
1036 return self.inner(p,r[1])
1037
1038 def __inner_p(self,p1,p2):
1039 return self.inner(p1,p2)
1040
1041 def __stoppingcriterium(self,norm_r,r,p):
1042 return self.stoppingcriterium(r[1],r[0],p)
1043
1044 def __stoppingcriterium_GMRES(self,norm_r,norm_b):
1045 return self.stoppingcriterium_GMRES(norm_r,norm_b)
1046
1047 def __stoppingcriterium_MINRES(self,norm_r,norm_Ax):
1048 return self.stoppingcriterium_MINRES(norm_r,norm_Ax)
1049
1050
1051 def setTolerance(self,tolerance=1.e-8):
1052 self.__tol=tolerance
1053 def getTolerance(self):
1054 return self.__tol
1055 def setToleranceReductionFactor(self,reduction=0.01):
1056 self.__reduction=reduction
1057 def getSubProblemTolerance(self):
1058 return self.__reduction*self.getTolerance()
1059
1060 def solve(self,v,p,max_iter=20, verbose=False, show_details=False, solver='PCG'):
1061 """
1062 solves the saddle point problem using initial guesses v and p.
1063
1064 @param max_iter: maximum number of iteration steps.
1065 """
1066 self.verbose=verbose
1067 self.show_details=show_details and self.verbose
1068
1069 # assume p is known: then v=A^-1(f-B^*p)
1070 # which leads to BA^-1B^*p = BA^-1f
1071
1072 # Az=f is solved as A(z-v)=f-Av (z-v = 0 on fixed_u_mask)
1073
1074
1075 self.__z=v+self.solve_A(v,p*0)
1076
1077 Bz=self.B(self.__z)
1078 #
1079 # solve BA^-1B^*p = Bz
1080 #
1081 # note that the residual r=Bz-BA^-1B^*p = B(z-A^-1B^*p) = Bv
1082 #
1083 # with Av=Az-B^*p = f - B^*p (v=z on fixed_u_mask)
1084 # A(v-z)=Az-B^*p-Az = f -Az - B^*p (v-z=0 on fixed_u_mask)
1085 #
1086 self.iter=0
1087 if solver=='GMRES':
1088 if self.verbose: print "enter GMRES method (iter_max=%s)"%max_iter
1089 p=GMRES(Bz,self.__Aprod_GMRES,self.__Msolve_GMRES,self.__inner_p,self.__stoppingcriterium_GMRES,iter_max=max_iter, x=p*1.)
1090 # solve Au=f-B^*p
1091 # A(u-v)=f-B^*p-Av
1092 # u=v+(u-v)
1093 u=v+self.solve_A(v,p)
1094
1095 if solver=='MINRES':
1096 if self.verbose: print "enter MINRES method (iter_max=%s)"%max_iter
1097 p=GMRES(Bz,self.__Aprod_GMRES,self.__Msolve_GMRES,self.__inner_p,self.__stoppingcriterium_MINRES,iter_max=max_iter, x=p*1.)
1098 # solve Au=f-B^*p
1099 # A(u-v)=f-B^*p-Av
1100 # u=v+(u-v)
1101 u=v+self.solve_A(v,p)
1102
1103 if solver=='PCG':
1104 if self.verbose: print "enter PCG method (iter_max=%s)"%max_iter
1105 p,r=PCG(ArithmeticTuple(self.__z*1.,Bz),self.__Aprod,self.__Msolve,self.__inner,self.__stoppingcriterium,iter_max=max_iter, x=p)
1106 u=r[0]
1107
1108 print "RESULT div(u)=",util.Lsup(self.B(u)),util.Lsup(u)
1109
1110 return u,p
1111
1112 def __Msolve(self,r):
1113 return self.solve_prec(r[1])
1114
1115 def __Msolve_GMRES(self,r):
1116 return self.solve_prec(r)
1117
1118
1119 def __Aprod(self,p):
1120 # return BA^-1B*p
1121 #solve Av =-B^*p as Av =f-Az-B^*p
1122 v=self.solve_A(self.__z,-p)
1123 return ArithmeticTuple(v, self.B(v))
1124
1125 def __Aprod_GMRES(self,p):
1126 # return BA^-1B*p
1127 #solve Av =-B^*p as Av =f-Az-B^*p
1128 v=self.solve_A(self.__z,-p)
1129 return self.B(v)
1130
1131 class SaddlePointProblem(object):
1132 """
1133 This implements a solver for a saddlepoint problem
1134
1135 M{f(u,p)=0}
1136 M{g(u)=0}
1137
1138 for u and p. The problem is solved with an inexact Uszawa scheme for p:
1139
1140 M{Q_f (u^{k+1}-u^{k}) = - f(u^{k},p^{k})}
1141 M{Q_g (p^{k+1}-p^{k}) = g(u^{k+1})}
1142
1143 where Q_f is an approximation of the Jacobiean A_f of f with respect to u and Q_f is an approximation of
1144 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'
1145 Q_g can be difficult, non-linear conjugate gradient method is applied to solve for p, so Q_g plays
1146 in fact the role of a preconditioner.
1147 """
1148 def __init__(self,verbose=False,*args):
1149 """
1150 initializes the problem
1151
1152 @param verbose: switches on the printing out some information
1153 @type verbose: C{bool}
1154 @note: this method may be overwritten by a particular saddle point problem
1155 """
1156 if not isinstance(verbose,bool):
1157 raise TypeError("verbose needs to be of type bool.")
1158 self.__verbose=verbose
1159 self.relaxation=1.
1160
1161 def trace(self,text):
1162 """
1163 prints text if verbose has been set
1164
1165 @param text: a text message
1166 @type text: C{str}
1167 """
1168 if self.__verbose: print "%s: %s"%(str(self),text)
1169
1170 def solve_f(self,u,p,tol=1.e-8):
1171 """
1172 solves
1173
1174 A_f du = f(u,p)
1175
1176 with tolerance C{tol} and return du. A_f is Jacobiean of f with respect to u.
1177
1178 @param u: current approximation of u
1179 @type u: L{escript.Data}
1180 @param p: current approximation of p
1181 @type p: L{escript.Data}
1182 @param tol: tolerance expected for du
1183 @type tol: C{float}
1184 @return: increment du
1185 @rtype: L{escript.Data}
1186 @note: this method has to be overwritten by a particular saddle point problem
1187 """
1188 pass
1189
1190 def solve_g(self,u,tol=1.e-8):
1191 """
1192 solves
1193
1194 Q_g dp = g(u)
1195
1196 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.
1197
1198 @param u: current approximation of u
1199 @type u: L{escript.Data}
1200 @param tol: tolerance expected for dp
1201 @type tol: C{float}
1202 @return: increment dp
1203 @rtype: L{escript.Data}
1204 @note: this method has to be overwritten by a particular saddle point problem
1205 """
1206 pass
1207
1208 def inner(self,p0,p1):
1209 """
1210 inner product of p0 and p1 approximating p. Typically this returns integrate(p0*p1)
1211 @return: inner product of p0 and p1
1212 @rtype: C{float}
1213 """
1214 pass
1215
1216 subiter_max=3
1217 def solve(self,u0,p0,tolerance=1.e-6,tolerance_u=None,iter_max=100,accepted_reduction=0.995,relaxation=None):
1218 """
1219 runs the solver
1220
1221 @param u0: initial guess for C{u}
1222 @type u0: L{esys.escript.Data}
1223 @param p0: initial guess for C{p}
1224 @type p0: L{esys.escript.Data}
1225 @param tolerance: tolerance for relative error in C{u} and C{p}
1226 @type tolerance: positive C{float}
1227 @param tolerance_u: tolerance for relative error in C{u} if different from C{tolerance}
1228 @type tolerance_u: positive C{float}
1229 @param iter_max: maximum number of iteration steps.
1230 @type iter_max: C{int}
1231 @param accepted_reduction: if the norm g cannot be reduced by C{accepted_reduction} backtracking to adapt the
1232 relaxation factor. If C{accepted_reduction=None} no backtracking is used.
1233 @type accepted_reduction: positive C{float} or C{None}
1234 @param relaxation: initial relaxation factor. If C{relaxation==None}, the last relaxation factor is used.
1235 @type relaxation: C{float} or C{None}
1236 """
1237 tol=1.e-2
1238 if tolerance_u==None: tolerance_u=tolerance
1239 if not relaxation==None: self.relaxation=relaxation
1240 if accepted_reduction ==None:
1241 angle_limit=0.
1242 elif accepted_reduction>=1.:
1243 angle_limit=0.
1244 else:
1245 angle_limit=util.sqrt(1-accepted_reduction**2)
1246 self.iter=0
1247 u=u0
1248 p=p0
1249 #
1250 # initialize things:
1251 #
1252 converged=False
1253 #
1254 # start loop:
1255 #
1256 # initial search direction is g
1257 #
1258 while not converged :
1259 if self.iter>iter_max:
1260 raise ArithmeticError("no convergence after %s steps."%self.iter)
1261 f_new=self.solve_f(u,p,tol)
1262 norm_f_new = util.Lsup(f_new)
1263 u_new=u-f_new
1264 g_new=self.solve_g(u_new,tol)
1265 self.iter+=1
1266 norm_g_new = util.sqrt(self.inner(g_new,g_new))
1267 if norm_f_new==0. and norm_g_new==0.: return u, p
1268 if self.iter>1 and not accepted_reduction==None:
1269 #
1270 # did we manage to reduce the norm of G? I
1271 # if not we start a backtracking procedure
1272 #
1273 # print "new/old norm = ",norm_g_new, norm_g, norm_g_new/norm_g
1274 if norm_g_new > accepted_reduction * norm_g:
1275 sub_iter=0
1276 s=self.relaxation
1277 d=g
1278 g_last=g
1279 self.trace(" start substepping: f = %s, g = %s, relaxation = %s."%(norm_f_new, norm_g_new, s))
1280 while sub_iter < self.subiter_max and norm_g_new > accepted_reduction * norm_g:
1281 dg= g_new-g_last
1282 norm_dg=abs(util.sqrt(self.inner(dg,dg))/self.relaxation)
1283 rad=self.inner(g_new,dg)/self.relaxation
1284 # print " ",sub_iter,": rad, norm_dg:",abs(rad), norm_dg*norm_g_new * angle_limit
1285 # print " ",sub_iter,": rad, norm_dg:",rad, norm_dg, norm_g_new, norm_g
1286 if abs(rad) < norm_dg*norm_g_new * angle_limit:
1287 if sub_iter>0: self.trace(" no further improvements expected from backtracking.")
1288 break
1289 r=self.relaxation
1290 self.relaxation= - rad/norm_dg**2
1291 s+=self.relaxation
1292 #####
1293 # a=g_new+self.relaxation*dg/r
1294 # print "predicted new norm = ",util.sqrt(self.inner(a,a)),util.sqrt(self.inner(g_new,g_new)), self.relaxation
1295 #####
1296 g_last=g_new
1297 p+=self.relaxation*d
1298 f_new=self.solve_f(u,p,tol)
1299 u_new=u-f_new
1300 g_new=self.solve_g(u_new,tol)
1301 self.iter+=1
1302 norm_f_new = util.Lsup(f_new)
1303 norm_g_new = util.sqrt(self.inner(g_new,g_new))
1304 # print " ",sub_iter," new g norm",norm_g_new
1305 self.trace(" %s th sub-step: f = %s, g = %s, relaxation = %s."%(sub_iter, norm_f_new, norm_g_new, s))
1306 #
1307 # can we expect reduction of g?
1308 #
1309 # u_last=u_new
1310 sub_iter+=1
1311 self.relaxation=s
1312 #
1313 # check for convergence:
1314 #
1315 norm_u_new = util.Lsup(u_new)
1316 p_new=p+self.relaxation*g_new
1317 norm_p_new = util.sqrt(self.inner(p_new,p_new))
1318 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))
1319
1320 if self.iter>1:
1321 dg2=g_new-g
1322 df2=f_new-f
1323 norm_dg2=util.sqrt(self.inner(dg2,dg2))
1324 norm_df2=util.Lsup(df2)
1325 # print norm_g_new, norm_g, norm_dg, norm_p, tolerance
1326 tol_eq_g=tolerance*norm_dg2/(norm_g*abs(self.relaxation))*norm_p_new
1327 tol_eq_f=tolerance_u*norm_df2/norm_f*norm_u_new
1328 if norm_g_new <= tol_eq_g and norm_f_new <= tol_eq_f:
1329 converged=True
1330 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
1331 self.trace("convergence after %s steps."%self.iter)
1332 return u,p
1333 # def solve(self,u0,p0,tolerance=1.e-6,iter_max=10,self.relaxation=1.):
1334 # tol=1.e-2
1335 # iter=0
1336 # converged=False
1337 # u=u0*1.
1338 # p=p0*1.
1339 # while not converged and iter<iter_max:
1340 # du=self.solve_f(u,p,tol)
1341 # u-=du
1342 # norm_du=util.Lsup(du)
1343 # norm_u=util.Lsup(u)
1344 #
1345 # dp=self.relaxation*self.solve_g(u,tol)
1346 # p+=dp
1347 # norm_dp=util.sqrt(self.inner(dp,dp))
1348 # norm_p=util.sqrt(self.inner(p,p))
1349 # 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)
1350 # iter+=1
1351 #
1352 # converged = (norm_du <= tolerance*norm_u) and (norm_dp <= tolerance*norm_p)
1353 # if converged:
1354 # print "convergence after %s steps."%iter
1355 # else:
1356 # raise ArithmeticError("no convergence after %s steps."%iter)
1357 #
1358 # return u,p
1359
1360 def MaskFromBoundaryTag(function_space,*tags):
1361 """
1362 create a mask on the given function space which one for samples
1363 that touch the boundary tagged by tags.
1364
1365 usage: m=MaskFromBoundaryTag(Solution(domain),"left", "right")
1366
1367 @param function_space: a given function space
1368 @type function_space: L{escript.FunctionSpace}
1369 @param tags: boundray tags
1370 @type tags: C{str}
1371 @return: a mask which marks samples used by C{function_space} that are touching the
1372 boundary tagged by any of the given tags.
1373 @rtype: L{escript.Data} of rank 0
1374 """
1375 pde=linearPDEs.LinearPDE(function_space.getDomain(),numEquations=1, numSolutions=1)
1376 d=escript.Scalar(0.,escript.FunctionOnBoundary(function_space.getDomain()))
1377 for t in tags: d.setTaggedValue(t,1.)
1378 pde.setValue(y=d)
1379 out=util.whereNonZero(pde.getRightHandSide())
1380 if out.getFunctionSpace() == function_space:
1381 return out
1382 else:
1383 return util.whereNonZero(util.interpolate(out,function_space))
1384
1385
1386

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