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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,Float32,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 PCG(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100):
481 """
482 Solver for
483
484 M{Ax=b}
485
486 with a symmetric and positive definite operator A (more details required!).
487 It uses the conjugate gradient method with preconditioner M providing an approximation of A.
488
489 The iteration is terminated if the C{stoppingcriterium} function return C{True}.
490
491 For details on the preconditioned conjugate gradient method see the book:
492
493 Templates for the Solution of Linear Systems by R. Barrett, M. Berry,
494 T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo,
495 C. Romine, and H. van der Vorst.
496
497 @param b: the right hand side of the liner system. C{b} is altered.
498 @type b: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
499 @param Aprod: returns the value Ax
500 @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}.
501 @param Msolve: solves Mx=r
502 @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
503 type like argument C{x}.
504 @param bilinearform: inner product C{<x,r>}
505 @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}.
506 @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.
507 @type stoppingcriterium: function that returns C{True} or C{False}
508 @param x: an initial guess for the solution. If no C{x} is given 0*b is used.
509 @type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
510 @param iter_max: maximum number of iteration steps.
511 @type iter_max: C{int}
512 @return: the solution approximation and the corresponding residual
513 @rtype: C{tuple}
514 @warning: C{b} and C{x} are altered.
515 """
516 iter=0
517 if x==None:
518 x=0*b
519 else:
520 b += (-1)*Aprod(x)
521 r=b
522 rhat=Msolve(r)
523 d = rhat
524 rhat_dot_r = bilinearform(rhat, r)
525 if rhat_dot_r<0: raise NegativeNorm,"negative norm."
526
527 while not stoppingcriterium(math.sqrt(rhat_dot_r),r,x):
528 iter+=1
529 if iter >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
530
531 q=Aprod(d)
532 alpha = rhat_dot_r / bilinearform(d, q)
533 x += alpha * d
534 r += (-alpha) * q
535
536 rhat=Msolve(r)
537 rhat_dot_r_new = bilinearform(rhat, r)
538 beta = rhat_dot_r_new / rhat_dot_r
539 rhat+=beta * d
540 d=rhat
541
542 rhat_dot_r = rhat_dot_r_new
543 if rhat_dot_r<0: raise NegativeNorm,"negative norm."
544
545 return x,r
546
547
548 ############################
549 # Added by Artak
550 #################################3
551
552 #Apply a sequence of k Givens rotations, used within gmres codes
553 # vrot=givapp(c, s, vin, k)
554 def givapp(c,s,vin,k):
555 vrot=vin
556 for i in range(k+1):
557 w1=c[i]*vrot[i]-s[i]*vrot[i+1]
558 w2=s[i]*vrot[i]+c[i]*vrot[i+1]
559 vrot[i:i+2]=w1,w2
560 return vrot
561
562 def GMRES(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100):
563
564 from numarray import dot
565
566 v0=b
567 iter=0
568 if x==None:
569 x=0*b
570 else:
571 b += (-1.)*Aprod(x)
572 r=b
573
574 rhat=Msolve(r)
575
576 rhat_dot_r = bilinearform(rhat, rhat)
577 norm_r=math.sqrt(rhat_dot_r)
578
579 if rhat_dot_r<0: raise NegativeNorm,"negative norm."
580
581 h=zeros((iter_max,iter_max),Float32)
582 c=zeros(iter_max,Float32)
583 s=zeros(iter_max,Float32)
584 g=zeros(iter_max,Float32)
585 v=[]
586
587 v.append(rhat/norm_r)
588 rho=norm_r
589 g[0]=rho
590
591 while not stoppingcriterium(rho,rho,norm_r):
592
593 if iter >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
594
595
596 vhat=Aprod(v[iter])
597 p=Msolve(vhat)
598
599 v.append(p)
600
601 v_norm1=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
602
603 # Modified Gram-Schmidt
604 for j in range(iter+1):
605 h[j][iter]=bilinearform(v[j],v[iter+1])
606 v[iter+1]+=(-1.)*h[j][iter]*v[j]
607
608 h[iter+1][iter]=math.sqrt(bilinearform(v[iter+1],v[iter+1]))
609 v_norm2=h[iter+1][iter]
610
611
612 # Reorthogonalize if needed
613 if v_norm1 + 0.001*v_norm2 == v_norm1: #Brown/Hindmarsh condition (default)
614 for j in range(iter+1):
615 hr=bilinearform(v[j],v[iter+1])
616 h[j][iter]=h[j][iter]+hr #vhat
617 v[iter+1] +=(-1.)*hr*v[j]
618
619 v_norm2=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
620 h[iter+1][iter]=v_norm2
621
622 # watch out for happy breakdown
623 if v_norm2 != 0:
624 v[iter+1]=v[iter+1]/h[iter+1][iter]
625
626 # Form and store the information for the new Givens rotation
627 if iter > 0 :
628 hhat=[]
629 for i in range(iter+1) : hhat.append(h[i][iter])
630 hhat=givapp(c[0:iter],s[0:iter],hhat,iter-1);
631 for i in range(iter+1) : h[i][iter]=hhat[i]
632
633 mu=math.sqrt(h[iter][iter]*h[iter][iter]+h[iter+1][iter]*h[iter+1][iter])
634 if mu!=0 :
635 c[iter]=h[iter][iter]/mu
636 s[iter]=-h[iter+1][iter]/mu
637 h[iter][iter]=c[iter]*h[iter][iter]-s[iter]*h[iter+1][iter]
638 h[iter+1][iter]=0.0
639 g[iter:iter+2]=givapp(c[iter],s[iter],g[iter:iter+2],0)
640
641 # Update the residual norm
642 rho=abs(g[iter+1])
643 iter+=1
644
645 # At this point either iter > iter_max or rho < tol.
646 # It's time to compute x and leave.
647
648 if iter > 0 :
649 y=zeros(iter,Float32)
650 y[iter-1] = g[iter-1] / h[iter-1][iter-1]
651 if iter > 1 :
652 i=iter-2
653 while i>=0 :
654 y[i] = ( g[i] - dot(h[i][i+1:iter], y[i+1:iter])) / h[i][i]
655 i=i-1
656 xhat=v[iter-1]*y[iter-1]
657 for i in range(iter-1):
658 xhat += v[i]*y[i]
659 else : xhat=v[0]
660
661 x += xhat
662
663 return x
664
665 #############################################
666
667 class ArithmeticTuple(object):
668 """
669 tuple supporting inplace update x+=y and scaling x=a*y where x,y is an ArithmeticTuple and a is a float.
670
671 example of usage:
672
673 from esys.escript import Data
674 from numarray import array
675 a=Data(...)
676 b=array([1.,4.])
677 x=ArithmeticTuple(a,b)
678 y=5.*x
679
680 """
681 def __init__(self,*args):
682 """
683 initialize object with elements args.
684
685 @param args: tuple of object that support implace add (x+=y) and scaling (x=a*y)
686 """
687 self.__items=list(args)
688
689 def __len__(self):
690 """
691 number of items
692
693 @return: number of items
694 @rtype: C{int}
695 """
696 return len(self.__items)
697
698 def __getitem__(self,index):
699 """
700 get an item
701
702 @param index: item to be returned
703 @type index: C{int}
704 @return: item with index C{index}
705 """
706 return self.__items.__getitem__(index)
707
708 def __mul__(self,other):
709 """
710 scaling from the right
711
712 @param other: scaling factor
713 @type other: C{float}
714 @return: itemwise self*other
715 @rtype: L{ArithmeticTuple}
716 """
717 out=[]
718 for i in range(len(self)):
719 out.append(self[i]*other)
720 return ArithmeticTuple(*tuple(out))
721
722 def __rmul__(self,other):
723 """
724 scaling from the left
725
726 @param other: scaling factor
727 @type other: C{float}
728 @return: itemwise other*self
729 @rtype: L{ArithmeticTuple}
730 """
731 out=[]
732 for i in range(len(self)):
733 out.append(other*self[i])
734 return ArithmeticTuple(*tuple(out))
735
736 #########################
737 # Added by Artak
738 #########################
739 def __div__(self,other):
740 """
741 dividing from the right
742
743 @param other: scaling factor
744 @type other: C{float}
745 @return: itemwise self/other
746 @rtype: L{ArithmeticTuple}
747 """
748 out=[]
749 for i in range(len(self)):
750 out.append(self[i]/other)
751 return ArithmeticTuple(*tuple(out))
752
753 def __rdiv__(self,other):
754 """
755 dividing from the left
756
757 @param other: scaling factor
758 @type other: C{float}
759 @return: itemwise other/self
760 @rtype: L{ArithmeticTuple}
761 """
762 out=[]
763 for i in range(len(self)):
764 out.append(other/self[i])
765 return ArithmeticTuple(*tuple(out))
766
767 ##########################################33
768
769 def __iadd__(self,other):
770 """
771 in-place add of other to self
772
773 @param other: increment
774 @type other: C{ArithmeticTuple}
775 """
776 if len(self) != len(other):
777 raise ValueError,"tuple length must match."
778 for i in range(len(self)):
779 self.__items[i]+=other[i]
780 return self
781
782 class HomogeneousSaddlePointProblem(object):
783 """
784 This provides a framwork for solving homogeneous saddle point problem of the form
785
786 Av+B^*p=f
787 Bv =0
788
789 for the unknowns v and p and given operators A and B and given right hand side f.
790 B^* is the adjoint operator of B is the given inner product.
791
792 """
793 def __init__(self,**kwargs):
794 self.setTolerance()
795 self.setToleranceReductionFactor()
796
797 def initialize(self):
798 """
799 initialize the problem (overwrite)
800 """
801 pass
802 def B(self,v):
803 """
804 returns Bv (overwrite)
805 @rtype: equal to the type of p
806
807 @note: boundary conditions on p should be zero!
808 """
809 pass
810
811 def inner(self,p0,p1):
812 """
813 returns inner product of two element p0 and p1 (overwrite)
814
815 @type p0: equal to the type of p
816 @type p1: equal to the type of p
817 @rtype: C{float}
818
819 @rtype: equal to the type of p
820 """
821 pass
822
823 def solve_A(self,u,p):
824 """
825 solves Av=f-Au-B^*p with accuracy self.getReducedTolerance() (overwrite)
826
827 @rtype: equal to the type of v
828 @note: boundary conditions on v should be zero!
829 """
830 pass
831
832 def solve_prec(self,p):
833 """
834 provides a preconditioner for BA^{-1}B^* with accuracy self.getReducedTolerance() (overwrite)
835
836 @rtype: equal to the type of p
837 """
838 pass
839
840 def stoppingcriterium(self,Bv,v,p):
841 """
842 returns a True if iteration is terminated. (overwrite)
843
844 @rtype: C{bool}
845 """
846 pass
847
848 def __inner(self,p,r):
849 return self.inner(p,r[1])
850
851 def __inner_p(self,p1,p2):
852 return self.inner(p1,p2)
853
854 def __stoppingcriterium(self,norm_r,r,p):
855 return self.stoppingcriterium(r[1],r[0],p)
856
857 def __stoppingcriterium_GMRES(self,norm_r,rho,r):
858 return self.stoppingcriterium_GMRES(rho,r)
859
860 def setTolerance(self,tolerance=1.e-8):
861 self.__tol=tolerance
862 def getTolerance(self):
863 return self.__tol
864 def setToleranceReductionFactor(self,reduction=0.01):
865 self.__reduction=reduction
866 def getSubProblemTolerance(self):
867 return self.__reduction*self.getTolerance()
868
869 def solve(self,v,p,max_iter=20, verbose=False, show_details=False, solver='GMRES'):
870 """
871 solves the saddle point problem using initial guesses v and p.
872
873 @param max_iter: maximum number of iteration steps.
874 """
875 self.verbose=verbose
876 self.show_details=show_details and self.verbose
877
878 # Az=f is solved as A(z-v)=f-Av (z-v = 0 on fixed_u_mask)
879
880
881 self.__z=v+self.solve_A(v,p*0)
882
883 Bz=self.B(self.__z)
884 #
885 # solve BA^-1B^*p = Bz
886 #
887 # note that the residual r=Bz-BA^-1B^*p = B(z-A^-1B^*p) = Bv
888 #
889 # with Av=Az-B^*p = f - B^*p (v=z on fixed_u_mask)
890 # A(v-z)=Az-B^*p-Az = f -Az - B^*p (v-z=0 on fixed_u_mask)
891 #
892 self.iter=0
893 if solver=='GMRES':
894 if self.verbose: print "enter GMRES method (iter_max=%s)"%max_iter
895 p=GMRES(Bz,self.__Aprod_GMRES,self.__Msolve_GMRES,self.__inner_p,self.__stoppingcriterium_GMRES,iter_max=max_iter, x=p*1)
896 u=v+self.solve_A(v,p)
897
898 else:
899 if self.verbose: print "enter PCG method (iter_max=%s)"%max_iter
900 p,r=PCG(ArithmeticTuple(self.__z*1.,Bz),self.__Aprod,self.__Msolve,self.__inner,self.__stoppingcriterium,iter_max=max_iter, x=p*1)
901 u=r[0]
902 u=v+self.solve_A(v,p) # Lutz to check !!!!!
903
904 return u,p
905
906 def __Msolve(self,r):
907 return self.solve_prec(r[1])
908
909 def __Msolve_GMRES(self,r):
910 return self.solve_prec(r)
911
912
913 def __Aprod(self,p):
914 # return BA^-1B*p
915 #solve Av =-B^*p as Av =f-Az-B^*p
916 v=self.solve_A(self.__z,p)
917 return ArithmeticTuple(v, self.B(v))
918
919 def __Aprod_GMRES(self,p):
920 # return BA^-1B*p
921 #solve Av =-B^*p as Av =f-Az-B^*p
922 v=self.solve_A(self.__z,p)
923 return self.B(v)
924
925 class SaddlePointProblem(object):
926 """
927 This implements a solver for a saddlepoint problem
928
929 M{f(u,p)=0}
930 M{g(u)=0}
931
932 for u and p. The problem is solved with an inexact Uszawa scheme for p:
933
934 M{Q_f (u^{k+1}-u^{k}) = - f(u^{k},p^{k})}
935 M{Q_g (p^{k+1}-p^{k}) = g(u^{k+1})}
936
937 where Q_f is an approximation of the Jacobiean A_f of f with respect to u and Q_f is an approximation of
938 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'
939 Q_g can be difficult, non-linear conjugate gradient method is applied to solve for p, so Q_g plays
940 in fact the role of a preconditioner.
941 """
942 def __init__(self,verbose=False,*args):
943 """
944 initializes the problem
945
946 @param verbose: switches on the printing out some information
947 @type verbose: C{bool}
948 @note: this method may be overwritten by a particular saddle point problem
949 """
950 if not isinstance(verbose,bool):
951 raise TypeError("verbose needs to be of type bool.")
952 self.__verbose=verbose
953 self.relaxation=1.
954
955 def trace(self,text):
956 """
957 prints text if verbose has been set
958
959 @param text: a text message
960 @type text: C{str}
961 """
962 if self.__verbose: print "%s: %s"%(str(self),text)
963
964 def solve_f(self,u,p,tol=1.e-8):
965 """
966 solves
967
968 A_f du = f(u,p)
969
970 with tolerance C{tol} and return du. A_f is Jacobiean of f with respect to u.
971
972 @param u: current approximation of u
973 @type u: L{escript.Data}
974 @param p: current approximation of p
975 @type p: L{escript.Data}
976 @param tol: tolerance expected for du
977 @type tol: C{float}
978 @return: increment du
979 @rtype: L{escript.Data}
980 @note: this method has to be overwritten by a particular saddle point problem
981 """
982 pass
983
984 def solve_g(self,u,tol=1.e-8):
985 """
986 solves
987
988 Q_g dp = g(u)
989
990 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.
991
992 @param u: current approximation of u
993 @type u: L{escript.Data}
994 @param tol: tolerance expected for dp
995 @type tol: C{float}
996 @return: increment dp
997 @rtype: L{escript.Data}
998 @note: this method has to be overwritten by a particular saddle point problem
999 """
1000 pass
1001
1002 def inner(self,p0,p1):
1003 """
1004 inner product of p0 and p1 approximating p. Typically this returns integrate(p0*p1)
1005 @return: inner product of p0 and p1
1006 @rtype: C{float}
1007 """
1008 pass
1009
1010 subiter_max=3
1011 def solve(self,u0,p0,tolerance=1.e-6,tolerance_u=None,iter_max=100,accepted_reduction=0.995,relaxation=None):
1012 """
1013 runs the solver
1014
1015 @param u0: initial guess for C{u}
1016 @type u0: L{esys.escript.Data}
1017 @param p0: initial guess for C{p}
1018 @type p0: L{esys.escript.Data}
1019 @param tolerance: tolerance for relative error in C{u} and C{p}
1020 @type tolerance: positive C{float}
1021 @param tolerance_u: tolerance for relative error in C{u} if different from C{tolerance}
1022 @type tolerance_u: positive C{float}
1023 @param iter_max: maximum number of iteration steps.
1024 @type iter_max: C{int}
1025 @param accepted_reduction: if the norm g cannot be reduced by C{accepted_reduction} backtracking to adapt the
1026 relaxation factor. If C{accepted_reduction=None} no backtracking is used.
1027 @type accepted_reduction: positive C{float} or C{None}
1028 @param relaxation: initial relaxation factor. If C{relaxation==None}, the last relaxation factor is used.
1029 @type relaxation: C{float} or C{None}
1030 """
1031 tol=1.e-2
1032 if tolerance_u==None: tolerance_u=tolerance
1033 if not relaxation==None: self.relaxation=relaxation
1034 if accepted_reduction ==None:
1035 angle_limit=0.
1036 elif accepted_reduction>=1.:
1037 angle_limit=0.
1038 else:
1039 angle_limit=util.sqrt(1-accepted_reduction**2)
1040 self.iter=0
1041 u=u0
1042 p=p0
1043 #
1044 # initialize things:
1045 #
1046 converged=False
1047 #
1048 # start loop:
1049 #
1050 # initial search direction is g
1051 #
1052 while not converged :
1053 if self.iter>iter_max:
1054 raise ArithmeticError("no convergence after %s steps."%self.iter)
1055 f_new=self.solve_f(u,p,tol)
1056 norm_f_new = util.Lsup(f_new)
1057 u_new=u-f_new
1058 g_new=self.solve_g(u_new,tol)
1059 self.iter+=1
1060 norm_g_new = util.sqrt(self.inner(g_new,g_new))
1061 if norm_f_new==0. and norm_g_new==0.: return u, p
1062 if self.iter>1 and not accepted_reduction==None:
1063 #
1064 # did we manage to reduce the norm of G? I
1065 # if not we start a backtracking procedure
1066 #
1067 # print "new/old norm = ",norm_g_new, norm_g, norm_g_new/norm_g
1068 if norm_g_new > accepted_reduction * norm_g:
1069 sub_iter=0
1070 s=self.relaxation
1071 d=g
1072 g_last=g
1073 self.trace(" start substepping: f = %s, g = %s, relaxation = %s."%(norm_f_new, norm_g_new, s))
1074 while sub_iter < self.subiter_max and norm_g_new > accepted_reduction * norm_g:
1075 dg= g_new-g_last
1076 norm_dg=abs(util.sqrt(self.inner(dg,dg))/self.relaxation)
1077 rad=self.inner(g_new,dg)/self.relaxation
1078 # print " ",sub_iter,": rad, norm_dg:",abs(rad), norm_dg*norm_g_new * angle_limit
1079 # print " ",sub_iter,": rad, norm_dg:",rad, norm_dg, norm_g_new, norm_g
1080 if abs(rad) < norm_dg*norm_g_new * angle_limit:
1081 if sub_iter>0: self.trace(" no further improvements expected from backtracking.")
1082 break
1083 r=self.relaxation
1084 self.relaxation= - rad/norm_dg**2
1085 s+=self.relaxation
1086 #####
1087 # a=g_new+self.relaxation*dg/r
1088 # print "predicted new norm = ",util.sqrt(self.inner(a,a)),util.sqrt(self.inner(g_new,g_new)), self.relaxation
1089 #####
1090 g_last=g_new
1091 p+=self.relaxation*d
1092 f_new=self.solve_f(u,p,tol)
1093 u_new=u-f_new
1094 g_new=self.solve_g(u_new,tol)
1095 self.iter+=1
1096 norm_f_new = util.Lsup(f_new)
1097 norm_g_new = util.sqrt(self.inner(g_new,g_new))
1098 # print " ",sub_iter," new g norm",norm_g_new
1099 self.trace(" %s th sub-step: f = %s, g = %s, relaxation = %s."%(sub_iter, norm_f_new, norm_g_new, s))
1100 #
1101 # can we expect reduction of g?
1102 #
1103 # u_last=u_new
1104 sub_iter+=1
1105 self.relaxation=s
1106 #
1107 # check for convergence:
1108 #
1109 norm_u_new = util.Lsup(u_new)
1110 p_new=p+self.relaxation*g_new
1111 norm_p_new = util.sqrt(self.inner(p_new,p_new))
1112 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))
1113
1114 if self.iter>1:
1115 dg2=g_new-g
1116 df2=f_new-f
1117 norm_dg2=util.sqrt(self.inner(dg2,dg2))
1118 norm_df2=util.Lsup(df2)
1119 # print norm_g_new, norm_g, norm_dg, norm_p, tolerance
1120 tol_eq_g=tolerance*norm_dg2/(norm_g*abs(self.relaxation))*norm_p_new
1121 tol_eq_f=tolerance_u*norm_df2/norm_f*norm_u_new
1122 if norm_g_new <= tol_eq_g and norm_f_new <= tol_eq_f:
1123 converged=True
1124 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
1125 self.trace("convergence after %s steps."%self.iter)
1126 return u,p
1127 # def solve(self,u0,p0,tolerance=1.e-6,iter_max=10,self.relaxation=1.):
1128 # tol=1.e-2
1129 # iter=0
1130 # converged=False
1131 # u=u0*1.
1132 # p=p0*1.
1133 # while not converged and iter<iter_max:
1134 # du=self.solve_f(u,p,tol)
1135 # u-=du
1136 # norm_du=util.Lsup(du)
1137 # norm_u=util.Lsup(u)
1138 #
1139 # dp=self.relaxation*self.solve_g(u,tol)
1140 # p+=dp
1141 # norm_dp=util.sqrt(self.inner(dp,dp))
1142 # norm_p=util.sqrt(self.inner(p,p))
1143 # 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)
1144 # iter+=1
1145 #
1146 # converged = (norm_du <= tolerance*norm_u) and (norm_dp <= tolerance*norm_p)
1147 # if converged:
1148 # print "convergence after %s steps."%iter
1149 # else:
1150 # raise ArithmeticError("no convergence after %s steps."%iter)
1151 #
1152 # return u,p
1153
1154 def MaskFromBoundaryTag(function_space,*tags):
1155 """
1156 create a mask on the given function space which one for samples
1157 that touch the boundary tagged by tags.
1158
1159 usage: m=MaskFromBoundaryTag(Solution(domain),"left", "right")
1160
1161 @param function_space: a given function space
1162 @type function_space: L{escript.FunctionSpace}
1163 @param tags: boundray tags
1164 @type tags: C{str}
1165 @return: a mask which marks samples used by C{function_space} that are touching the
1166 boundary tagged by any of the given tags.
1167 @rtype: L{escript.Data} of rank 0
1168 """
1169 pde=linearPDEs.LinearPDE(function_space.getDomain(),numEquations=1, numSolutions=1)
1170 d=escript.Scalar(0.,escript.FunctionOnBoundary(function_space.getDomain()))
1171 for t in tags: d.setTaggedValue(t,1.)
1172 pde.setValue(y=d)
1173 out=util.whereNonZero(pde.getRightHandSide())
1174 if out.getFunctionSpace() == function_space:
1175 return out
1176 else:
1177 return util.whereNonZero(util.interpolate(out,function_space))
1178
1179
1180

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