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Revision 1481 - (show annotations)
Wed Apr 9 00:45:47 2008 UTC (11 years, 5 months ago) by artak
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initial version of 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 # Initialize
713
714 iter = 0
715 Anorm = 0
716 ynorm = 0
717 x=x*0
718 #------------------------------------------------------------------
719 # Set up y and v for the first Lanczos vector v1.
720 # y = beta1 P' v1, where P = C**(-1).
721 # v is really P' v1.
722 #------------------------------------------------------------------
723 r1 = b
724 y = Msolve(b)
725 beta1 = bilinearform(b,y)
726
727 if beta1< 0: raise NegativeNorm,"negative norm."
728
729 # If b = 0 exactly, stop with x = 0.
730 if beta1==0: return x*0.
731
732 if beta1> 0:
733 beta1 = math.sqrt(beta1) # Normalize y to get v1 later.
734
735 #------------------------------------------------------------------
736 # Initialize other quantities.
737 # ------------------------------------------------------------------
738 oldb = 0
739 beta = beta1
740 dbar = 0
741 epsln = 0
742 phibar = beta1
743 rhs1 = beta1
744 rhs2 = 0
745 rnorm = phibar
746 tnorm2 = 0
747 ynorm2 = 0
748 cs = -1
749 sn = 0
750 w = b*0.
751 w2 = b*0.
752 r2 = r1
753 eps = 0.0001
754
755 #---------------------------------------------------------------------
756 # Main iteration loop.
757 # --------------------------------------------------------------------
758 while not stoppingcriterium(rnorm,Anorm*ynorm): # ||r|| / (||A|| ||x||)
759
760 if iter >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
761 iter = iter + 1
762
763 #-----------------------------------------------------------------
764 # Obtain quantities for the next Lanczos vector vk+1, k = 1, 2,...
765 # The general iteration is similar to the case k = 1 with v0 = 0:
766 #
767 # p1 = Operator * v1 - beta1 * v0,
768 # alpha1 = v1'p1,
769 # q2 = p2 - alpha1 * v1,
770 # beta2^2 = q2'q2,
771 # v2 = (1/beta2) q2.
772 #
773 # Again, y = betak P vk, where P = C**(-1).
774 #-----------------------------------------------------------------
775 s = 1/beta # Normalize previous vector (in y).
776 v = s*y # v = vk if P = I
777
778 y = Aprod(v)
779
780 if iter >= 2:
781 y = y - (beta/oldb)*r1
782
783 alfa = bilinearform(v,y) # alphak
784 y = (- alfa/beta)*r2 + y
785 r1 = r2
786 r2 = y
787 y = Msolve(r2)
788 oldb = beta # oldb = betak
789 beta = bilinearform(r2,y) # beta = betak+1^2
790 if beta < 0: raise NegativeNorm,"negative norm."
791
792 beta = math.sqrt( beta )
793 tnorm2 = tnorm2 + alfa*alfa + oldb*oldb + beta*beta
794
795 if iter==1: # Initialize a few things.
796 gmax = abs( alfa ) # alpha1
797 gmin = gmax # alpha1
798
799 # Apply previous rotation Qk-1 to get
800 # [deltak epslnk+1] = [cs sn][dbark 0 ]
801 # [gbar k dbar k+1] [sn -cs][alfak betak+1].
802
803 oldeps = epsln
804 delta = cs * dbar + sn * alfa # delta1 = 0 deltak
805 gbar = sn * dbar - cs * alfa # gbar 1 = alfa1 gbar k
806 epsln = sn * beta # epsln2 = 0 epslnk+1
807 dbar = - cs * beta # dbar 2 = beta2 dbar k+1
808
809 # Compute the next plane rotation Qk
810
811 gamma = math.sqrt(gbar*gbar+beta*beta) # gammak
812 gamma = max(gamma,eps)
813 cs = gbar / gamma # ck
814 sn = beta / gamma # sk
815 phi = cs * phibar # phik
816 phibar = sn * phibar # phibark+1
817
818 # Update x.
819
820 denom = 1/gamma
821 w1 = w2
822 w2 = w
823 w = (v - oldeps*w1 - delta*w2) * denom
824 x = x + phi*w
825
826 # Go round again.
827
828 gmax = max(gmax,gamma)
829 gmin = min(gmin,gamma)
830 z = rhs1 / gamma
831 ynorm2 = z*z + ynorm2
832 rhs1 = rhs2 - delta*z
833 rhs2 = - epsln*z
834
835 # Estimate various norms and test for convergence.
836
837 Anorm = math.sqrt( tnorm2 )
838 ynorm = math.sqrt( ynorm2 )
839
840 rnorm = phibar
841
842 # Return final answer.
843 return x
844
845 #############################################
846
847 class ArithmeticTuple(object):
848 """
849 tuple supporting inplace update x+=y and scaling x=a*y where x,y is an ArithmeticTuple and a is a float.
850
851 example of usage:
852
853 from esys.escript import Data
854 from numarray import array
855 a=Data(...)
856 b=array([1.,4.])
857 x=ArithmeticTuple(a,b)
858 y=5.*x
859
860 """
861 def __init__(self,*args):
862 """
863 initialize object with elements args.
864
865 @param args: tuple of object that support implace add (x+=y) and scaling (x=a*y)
866 """
867 self.__items=list(args)
868
869 def __len__(self):
870 """
871 number of items
872
873 @return: number of items
874 @rtype: C{int}
875 """
876 return len(self.__items)
877
878 def __getitem__(self,index):
879 """
880 get an item
881
882 @param index: item to be returned
883 @type index: C{int}
884 @return: item with index C{index}
885 """
886 return self.__items.__getitem__(index)
887
888 def __mul__(self,other):
889 """
890 scaling from the right
891
892 @param other: scaling factor
893 @type other: C{float}
894 @return: itemwise self*other
895 @rtype: L{ArithmeticTuple}
896 """
897 out=[]
898 for i in range(len(self)):
899 out.append(self[i]*other)
900 return ArithmeticTuple(*tuple(out))
901
902 def __rmul__(self,other):
903 """
904 scaling from the left
905
906 @param other: scaling factor
907 @type other: C{float}
908 @return: itemwise other*self
909 @rtype: L{ArithmeticTuple}
910 """
911 out=[]
912 for i in range(len(self)):
913 out.append(other*self[i])
914 return ArithmeticTuple(*tuple(out))
915
916 #########################
917 # Added by Artak
918 #########################
919 def __div__(self,other):
920 """
921 dividing from the right
922
923 @param other: scaling factor
924 @type other: C{float}
925 @return: itemwise self/other
926 @rtype: L{ArithmeticTuple}
927 """
928 out=[]
929 for i in range(len(self)):
930 out.append(self[i]/other)
931 return ArithmeticTuple(*tuple(out))
932
933 def __rdiv__(self,other):
934 """
935 dividing from the left
936
937 @param other: scaling factor
938 @type other: C{float}
939 @return: itemwise other/self
940 @rtype: L{ArithmeticTuple}
941 """
942 out=[]
943 for i in range(len(self)):
944 out.append(other/self[i])
945 return ArithmeticTuple(*tuple(out))
946
947 ##########################################33
948
949 def __iadd__(self,other):
950 """
951 in-place add of other to self
952
953 @param other: increment
954 @type other: C{ArithmeticTuple}
955 """
956 if len(self) != len(other):
957 raise ValueError,"tuple length must match."
958 for i in range(len(self)):
959 self.__items[i]+=other[i]
960 return self
961
962 class HomogeneousSaddlePointProblem(object):
963 """
964 This provides a framwork for solving homogeneous saddle point problem of the form
965
966 Av+B^*p=f
967 Bv =0
968
969 for the unknowns v and p and given operators A and B and given right hand side f.
970 B^* is the adjoint operator of B is the given inner product.
971
972 """
973 def __init__(self,**kwargs):
974 self.setTolerance()
975 self.setToleranceReductionFactor()
976
977 def initialize(self):
978 """
979 initialize the problem (overwrite)
980 """
981 pass
982 def B(self,v):
983 """
984 returns Bv (overwrite)
985 @rtype: equal to the type of p
986
987 @note: boundary conditions on p should be zero!
988 """
989 pass
990
991 def inner(self,p0,p1):
992 """
993 returns inner product of two element p0 and p1 (overwrite)
994
995 @type p0: equal to the type of p
996 @type p1: equal to the type of p
997 @rtype: C{float}
998
999 @rtype: equal to the type of p
1000 """
1001 pass
1002
1003 def solve_A(self,u,p):
1004 """
1005 solves Av=f-Au-B^*p with accuracy self.getReducedTolerance() (overwrite)
1006
1007 @rtype: equal to the type of v
1008 @note: boundary conditions on v should be zero!
1009 """
1010 pass
1011
1012 def solve_prec(self,p):
1013 """
1014 provides a preconditioner for BA^{-1}B^* with accuracy self.getReducedTolerance() (overwrite)
1015
1016 @rtype: equal to the type of p
1017 """
1018 pass
1019
1020 def stoppingcriterium(self,Bv,v,p):
1021 """
1022 returns a True if iteration is terminated. (overwrite)
1023
1024 @rtype: C{bool}
1025 """
1026 pass
1027
1028 def __inner(self,p,r):
1029 return self.inner(p,r[1])
1030
1031 def __inner_p(self,p1,p2):
1032 return self.inner(p1,p2)
1033
1034 def __stoppingcriterium(self,norm_r,r,p):
1035 return self.stoppingcriterium(r[1],r[0],p)
1036
1037 def __stoppingcriterium_GMRES(self,norm_r,norm_b):
1038 return self.stoppingcriterium_GMRES(norm_r,norm_b)
1039
1040 def __stoppingcriterium_MINRES(self,norm_r,norm_Ax):
1041 return self.stoppingcriterium_MINRES(norm_r,norm_Ax)
1042
1043
1044 def setTolerance(self,tolerance=1.e-8):
1045 self.__tol=tolerance
1046 def getTolerance(self):
1047 return self.__tol
1048 def setToleranceReductionFactor(self,reduction=0.01):
1049 self.__reduction=reduction
1050 def getSubProblemTolerance(self):
1051 return self.__reduction*self.getTolerance()
1052
1053 def solve(self,v,p,max_iter=20, verbose=False, show_details=False, solver='PCG'):
1054 """
1055 solves the saddle point problem using initial guesses v and p.
1056
1057 @param max_iter: maximum number of iteration steps.
1058 """
1059 self.verbose=verbose
1060 self.show_details=show_details and self.verbose
1061
1062 # assume p is known: then v=A^-1(f-B^*p)
1063 # which leads to BA^-1B^*p = BA^-1f
1064
1065 # Az=f is solved as A(z-v)=f-Av (z-v = 0 on fixed_u_mask)
1066
1067
1068 self.__z=v+self.solve_A(v,p*0)
1069
1070 Bz=self.B(self.__z)
1071 #
1072 # solve BA^-1B^*p = Bz
1073 #
1074 # note that the residual r=Bz-BA^-1B^*p = B(z-A^-1B^*p) = Bv
1075 #
1076 # with Av=Az-B^*p = f - B^*p (v=z on fixed_u_mask)
1077 # A(v-z)=Az-B^*p-Az = f -Az - B^*p (v-z=0 on fixed_u_mask)
1078 #
1079 self.iter=0
1080 if solver=='GMRES':
1081 if self.verbose: print "enter GMRES method (iter_max=%s)"%max_iter
1082 p=GMRES(Bz,self.__Aprod_GMRES,self.__Msolve_GMRES,self.__inner_p,self.__stoppingcriterium_GMRES,iter_max=max_iter, x=p*1.)
1083 # solve Au=f-B^*p
1084 # A(u-v)=f-B^*p-Av
1085 # u=v+(u-v)
1086 u=v+self.solve_A(v,p)
1087
1088 if solver=='MINRES':
1089 if self.verbose: print "enter MINRES method (iter_max=%s)"%max_iter
1090 p=GMRES(Bz,self.__Aprod_GMRES,self.__Msolve_GMRES,self.__inner_p,self.__stoppingcriterium_MINRES,iter_max=max_iter, x=p*1.)
1091 # solve Au=f-B^*p
1092 # A(u-v)=f-B^*p-Av
1093 # u=v+(u-v)
1094 u=v+self.solve_A(v,p)
1095
1096 if solver=='PCG':
1097 if self.verbose: print "enter PCG method (iter_max=%s)"%max_iter
1098 p,r=PCG(ArithmeticTuple(self.__z*1.,Bz),self.__Aprod,self.__Msolve,self.__inner,self.__stoppingcriterium,iter_max=max_iter, x=p)
1099 u=r[0]
1100
1101 print "RESULT div(u)=",util.Lsup(self.B(u)),util.Lsup(u)
1102
1103 return u,p
1104
1105 def __Msolve(self,r):
1106 return self.solve_prec(r[1])
1107
1108 def __Msolve_GMRES(self,r):
1109 return self.solve_prec(r)
1110
1111
1112 def __Aprod(self,p):
1113 # return BA^-1B*p
1114 #solve Av =-B^*p as Av =f-Az-B^*p
1115 v=self.solve_A(self.__z,-p)
1116 return ArithmeticTuple(v, self.B(v))
1117
1118 def __Aprod_GMRES(self,p):
1119 # return BA^-1B*p
1120 #solve Av =-B^*p as Av =f-Az-B^*p
1121 v=self.solve_A(self.__z,-p)
1122 return self.B(v)
1123
1124 class SaddlePointProblem(object):
1125 """
1126 This implements a solver for a saddlepoint problem
1127
1128 M{f(u,p)=0}
1129 M{g(u)=0}
1130
1131 for u and p. The problem is solved with an inexact Uszawa scheme for p:
1132
1133 M{Q_f (u^{k+1}-u^{k}) = - f(u^{k},p^{k})}
1134 M{Q_g (p^{k+1}-p^{k}) = g(u^{k+1})}
1135
1136 where Q_f is an approximation of the Jacobiean A_f of f with respect to u and Q_f is an approximation of
1137 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'
1138 Q_g can be difficult, non-linear conjugate gradient method is applied to solve for p, so Q_g plays
1139 in fact the role of a preconditioner.
1140 """
1141 def __init__(self,verbose=False,*args):
1142 """
1143 initializes the problem
1144
1145 @param verbose: switches on the printing out some information
1146 @type verbose: C{bool}
1147 @note: this method may be overwritten by a particular saddle point problem
1148 """
1149 if not isinstance(verbose,bool):
1150 raise TypeError("verbose needs to be of type bool.")
1151 self.__verbose=verbose
1152 self.relaxation=1.
1153
1154 def trace(self,text):
1155 """
1156 prints text if verbose has been set
1157
1158 @param text: a text message
1159 @type text: C{str}
1160 """
1161 if self.__verbose: print "%s: %s"%(str(self),text)
1162
1163 def solve_f(self,u,p,tol=1.e-8):
1164 """
1165 solves
1166
1167 A_f du = f(u,p)
1168
1169 with tolerance C{tol} and return du. A_f is Jacobiean of f with respect to u.
1170
1171 @param u: current approximation of u
1172 @type u: L{escript.Data}
1173 @param p: current approximation of p
1174 @type p: L{escript.Data}
1175 @param tol: tolerance expected for du
1176 @type tol: C{float}
1177 @return: increment du
1178 @rtype: L{escript.Data}
1179 @note: this method has to be overwritten by a particular saddle point problem
1180 """
1181 pass
1182
1183 def solve_g(self,u,tol=1.e-8):
1184 """
1185 solves
1186
1187 Q_g dp = g(u)
1188
1189 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.
1190
1191 @param u: current approximation of u
1192 @type u: L{escript.Data}
1193 @param tol: tolerance expected for dp
1194 @type tol: C{float}
1195 @return: increment dp
1196 @rtype: L{escript.Data}
1197 @note: this method has to be overwritten by a particular saddle point problem
1198 """
1199 pass
1200
1201 def inner(self,p0,p1):
1202 """
1203 inner product of p0 and p1 approximating p. Typically this returns integrate(p0*p1)
1204 @return: inner product of p0 and p1
1205 @rtype: C{float}
1206 """
1207 pass
1208
1209 subiter_max=3
1210 def solve(self,u0,p0,tolerance=1.e-6,tolerance_u=None,iter_max=100,accepted_reduction=0.995,relaxation=None):
1211 """
1212 runs the solver
1213
1214 @param u0: initial guess for C{u}
1215 @type u0: L{esys.escript.Data}
1216 @param p0: initial guess for C{p}
1217 @type p0: L{esys.escript.Data}
1218 @param tolerance: tolerance for relative error in C{u} and C{p}
1219 @type tolerance: positive C{float}
1220 @param tolerance_u: tolerance for relative error in C{u} if different from C{tolerance}
1221 @type tolerance_u: positive C{float}
1222 @param iter_max: maximum number of iteration steps.
1223 @type iter_max: C{int}
1224 @param accepted_reduction: if the norm g cannot be reduced by C{accepted_reduction} backtracking to adapt the
1225 relaxation factor. If C{accepted_reduction=None} no backtracking is used.
1226 @type accepted_reduction: positive C{float} or C{None}
1227 @param relaxation: initial relaxation factor. If C{relaxation==None}, the last relaxation factor is used.
1228 @type relaxation: C{float} or C{None}
1229 """
1230 tol=1.e-2
1231 if tolerance_u==None: tolerance_u=tolerance
1232 if not relaxation==None: self.relaxation=relaxation
1233 if accepted_reduction ==None:
1234 angle_limit=0.
1235 elif accepted_reduction>=1.:
1236 angle_limit=0.
1237 else:
1238 angle_limit=util.sqrt(1-accepted_reduction**2)
1239 self.iter=0
1240 u=u0
1241 p=p0
1242 #
1243 # initialize things:
1244 #
1245 converged=False
1246 #
1247 # start loop:
1248 #
1249 # initial search direction is g
1250 #
1251 while not converged :
1252 if self.iter>iter_max:
1253 raise ArithmeticError("no convergence after %s steps."%self.iter)
1254 f_new=self.solve_f(u,p,tol)
1255 norm_f_new = util.Lsup(f_new)
1256 u_new=u-f_new
1257 g_new=self.solve_g(u_new,tol)
1258 self.iter+=1
1259 norm_g_new = util.sqrt(self.inner(g_new,g_new))
1260 if norm_f_new==0. and norm_g_new==0.: return u, p
1261 if self.iter>1 and not accepted_reduction==None:
1262 #
1263 # did we manage to reduce the norm of G? I
1264 # if not we start a backtracking procedure
1265 #
1266 # print "new/old norm = ",norm_g_new, norm_g, norm_g_new/norm_g
1267 if norm_g_new > accepted_reduction * norm_g:
1268 sub_iter=0
1269 s=self.relaxation
1270 d=g
1271 g_last=g
1272 self.trace(" start substepping: f = %s, g = %s, relaxation = %s."%(norm_f_new, norm_g_new, s))
1273 while sub_iter < self.subiter_max and norm_g_new > accepted_reduction * norm_g:
1274 dg= g_new-g_last
1275 norm_dg=abs(util.sqrt(self.inner(dg,dg))/self.relaxation)
1276 rad=self.inner(g_new,dg)/self.relaxation
1277 # print " ",sub_iter,": rad, norm_dg:",abs(rad), norm_dg*norm_g_new * angle_limit
1278 # print " ",sub_iter,": rad, norm_dg:",rad, norm_dg, norm_g_new, norm_g
1279 if abs(rad) < norm_dg*norm_g_new * angle_limit:
1280 if sub_iter>0: self.trace(" no further improvements expected from backtracking.")
1281 break
1282 r=self.relaxation
1283 self.relaxation= - rad/norm_dg**2
1284 s+=self.relaxation
1285 #####
1286 # a=g_new+self.relaxation*dg/r
1287 # print "predicted new norm = ",util.sqrt(self.inner(a,a)),util.sqrt(self.inner(g_new,g_new)), self.relaxation
1288 #####
1289 g_last=g_new
1290 p+=self.relaxation*d
1291 f_new=self.solve_f(u,p,tol)
1292 u_new=u-f_new
1293 g_new=self.solve_g(u_new,tol)
1294 self.iter+=1
1295 norm_f_new = util.Lsup(f_new)
1296 norm_g_new = util.sqrt(self.inner(g_new,g_new))
1297 # print " ",sub_iter," new g norm",norm_g_new
1298 self.trace(" %s th sub-step: f = %s, g = %s, relaxation = %s."%(sub_iter, norm_f_new, norm_g_new, s))
1299 #
1300 # can we expect reduction of g?
1301 #
1302 # u_last=u_new
1303 sub_iter+=1
1304 self.relaxation=s
1305 #
1306 # check for convergence:
1307 #
1308 norm_u_new = util.Lsup(u_new)
1309 p_new=p+self.relaxation*g_new
1310 norm_p_new = util.sqrt(self.inner(p_new,p_new))
1311 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))
1312
1313 if self.iter>1:
1314 dg2=g_new-g
1315 df2=f_new-f
1316 norm_dg2=util.sqrt(self.inner(dg2,dg2))
1317 norm_df2=util.Lsup(df2)
1318 # print norm_g_new, norm_g, norm_dg, norm_p, tolerance
1319 tol_eq_g=tolerance*norm_dg2/(norm_g*abs(self.relaxation))*norm_p_new
1320 tol_eq_f=tolerance_u*norm_df2/norm_f*norm_u_new
1321 if norm_g_new <= tol_eq_g and norm_f_new <= tol_eq_f:
1322 converged=True
1323 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
1324 self.trace("convergence after %s steps."%self.iter)
1325 return u,p
1326 # def solve(self,u0,p0,tolerance=1.e-6,iter_max=10,self.relaxation=1.):
1327 # tol=1.e-2
1328 # iter=0
1329 # converged=False
1330 # u=u0*1.
1331 # p=p0*1.
1332 # while not converged and iter<iter_max:
1333 # du=self.solve_f(u,p,tol)
1334 # u-=du
1335 # norm_du=util.Lsup(du)
1336 # norm_u=util.Lsup(u)
1337 #
1338 # dp=self.relaxation*self.solve_g(u,tol)
1339 # p+=dp
1340 # norm_dp=util.sqrt(self.inner(dp,dp))
1341 # norm_p=util.sqrt(self.inner(p,p))
1342 # 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)
1343 # iter+=1
1344 #
1345 # converged = (norm_du <= tolerance*norm_u) and (norm_dp <= tolerance*norm_p)
1346 # if converged:
1347 # print "convergence after %s steps."%iter
1348 # else:
1349 # raise ArithmeticError("no convergence after %s steps."%iter)
1350 #
1351 # return u,p
1352
1353 def MaskFromBoundaryTag(function_space,*tags):
1354 """
1355 create a mask on the given function space which one for samples
1356 that touch the boundary tagged by tags.
1357
1358 usage: m=MaskFromBoundaryTag(Solution(domain),"left", "right")
1359
1360 @param function_space: a given function space
1361 @type function_space: L{escript.FunctionSpace}
1362 @param tags: boundray tags
1363 @type tags: C{str}
1364 @return: a mask which marks samples used by C{function_space} that are touching the
1365 boundary tagged by any of the given tags.
1366 @rtype: L{escript.Data} of rank 0
1367 """
1368 pde=linearPDEs.LinearPDE(function_space.getDomain(),numEquations=1, numSolutions=1)
1369 d=escript.Scalar(0.,escript.FunctionOnBoundary(function_space.getDomain()))
1370 for t in tags: d.setTaggedValue(t,1.)
1371 pde.setValue(y=d)
1372 out=util.whereNonZero(pde.getRightHandSide())
1373 if out.getFunctionSpace() == function_space:
1374 return out
1375 else:
1376 return util.whereNonZero(util.interpolate(out,function_space))
1377
1378
1379

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