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Revision 1469 - (show 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 #
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):
583 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
589 if x==None:
590 x=0*b
591 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
596 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 v=[]
601
602 rho=math.sqrt(r_dot_r)
603 v.append(r/rho)
604 g[0]=rho
605
606 while not stoppingcriterium(rho,norm_b):
607
608 if iter >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
609
610
611 p=Msolve(Aprod(v[iter]))
612
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 hhat=givapp(c[0:iter],s[0:iter],hhat);
645 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 g[iter:iter+2]=givapp(c[iter],s[iter],g[iter:iter+2])
654
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 y=numarray.zeros(iter,numarray.Float64)
664 y[iter-1] = g[iter-1] / h[iter-1][iter-1]
665 if iter > 1 :
666 i=iter-2
667 while i>=0 :
668 y[i] = ( g[i] - numarray.dot(h[i][i+1:iter], y[i+1:iter])) / h[i][i]
669 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 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 #########################
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 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 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 def __inner_p(self,p1,p2):
866 return self.inner(p1,p2)
867
868 def __stoppingcriterium(self,norm_r,r,p):
869 return self.stoppingcriterium(r[1],r[0],p)
870
871 def __stoppingcriterium_GMRES(self,norm_r,norm_b):
872 return self.stoppingcriterium_GMRES(norm_r,norm_b)
873
874 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 def solve(self,v,p,max_iter=20, verbose=False, show_details=False, solver='GMRES'):
884 """
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 # assume p is known: then v=A^-1(f-B^*p)
893 # which leads to BA^-1B^*p = BA^-1f
894
895 # Az=f is solved as A(z-v)=f-Av (z-v = 0 on fixed_u_mask)
896
897
898 self.__z=v+self.solve_A(v,p*0)
899
900 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 if solver=='GMRES':
911 if self.verbose: print "enter GMRES method (iter_max=%s)"%max_iter
912 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 u=v+self.solve_A(v,p)
917
918 else:
919 if self.verbose: print "enter PCG method (iter_max=%s)"%max_iter
920 p,r=PCG(ArithmeticTuple(self.__z*1.,Bz),self.__Aprod,self.__Msolve,self.__inner,self.__stoppingcriterium,iter_max=max_iter, x=p)
921 u=r[0]
922 print "div(u)=",util.Lsup(self.B(u)),util.Lsup(u)
923
924 return u,p
925
926 def __Msolve(self,r):
927 return self.solve_prec(r[1])
928
929 def __Msolve_GMRES(self,r):
930 return self.solve_prec(r)
931
932
933 def __Aprod(self,p):
934 # return BA^-1B*p
935 #solve Av =-B^*p as Av =f-Az-B^*p
936 v=self.solve_A(self.__z,-p)
937 return ArithmeticTuple(v, self.B(v))
938
939 def __Aprod_GMRES(self,p):
940 # return BA^-1B*p
941 #solve Av =-B^*p as Av =f-Az-B^*p
942 v=self.solve_A(self.__z,-p)
943 return self.B(v)
944
945 class SaddlePointProblem(object):
946 """
947 This implements a solver for a saddlepoint problem
948
949 M{f(u,p)=0}
950 M{g(u)=0}
951
952 for u and p. The problem is solved with an inexact Uszawa scheme for p:
953
954 M{Q_f (u^{k+1}-u^{k}) = - f(u^{k},p^{k})}
955 M{Q_g (p^{k+1}-p^{k}) = g(u^{k+1})}
956
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 @param verbose: switches on the printing out some information
967 @type verbose: C{bool}
968 @note: this method may be overwritten by a particular saddle point problem
969 """
970 if not isinstance(verbose,bool):
971 raise TypeError("verbose needs to be of type bool.")
972 self.__verbose=verbose
973 self.relaxation=1.
974
975 def trace(self,text):
976 """
977 prints text if verbose has been set
978
979 @param text: a text message
980 @type text: C{str}
981 """
982 if self.__verbose: print "%s: %s"%(str(self),text)
983
984 def solve_f(self,u,p,tol=1.e-8):
985 """
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 @param tol: tolerance expected for du
997 @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 def solve_g(self,u,tol=1.e-8):
1005 """
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 @param tol: tolerance expected for dp
1015 @type tol: C{float}
1016 @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 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
1035 @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 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
1134 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
1174 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
1200

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