/[escript]/trunk/escript/py_src/pdetools.py
ViewVC logotype

Annotation of /trunk/escript/py_src/pdetools.py

Parent Directory Parent Directory | Revision Log Revision Log


Revision 1475 - (hide annotations)
Mon Apr 7 05:13:21 2008 UTC (11 years, 3 months ago) by artak
File MIME type: text/x-python
File size: 39891 byte(s)
Initial version of GMRES with restart
1 ksteube 1312 #
2 jgs 121 # $Id$
3 ksteube 1312 #
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 jgs 121
17     """
18 jgs 149 Provides some tools related to PDEs.
19 jgs 121
20 jgs 149 Currently includes:
21     - Projector - to project a discontinuous
22 gross 351 - Locator - to trace values in data objects at a certain location
23     - TimeIntegrationManager - to handel extraplotion in time
24 gross 867 - SaddlePointProblem - solver for Saddle point problems using the inexact uszawa scheme
25 gross 637
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 jgs 121 """
33    
34 gross 637 __author__="Lutz Gross, l.gross@uq.edu.au"
35 elspeth 609 __copyright__=""" Copyright (c) 2006 by ACcESS MNRF
36     http://www.access.edu.au
37     Primary Business: Queensland, Australia"""
38 elspeth 614 __license__="""Licensed under the Open Software License version 3.0
39     http://www.opensource.org/licenses/osl-3.0.php"""
40 gross 637 __url__="http://www.iservo.edu.au/esys"
41     __version__="$Revision$"
42     __date__="$Date$"
43 elspeth 609
44 gross 637
45 jgs 149 import escript
46     import linearPDEs
47 jgs 121 import numarray
48 jgs 149 import util
49 ksteube 1312 import math
50 jgs 121
51 artak 1465 ##### Added by Artak
52 gross 1467 # from Numeric import zeros,Int,Float64
53 artak 1465 ###################################
54    
55    
56 gross 351 class TimeIntegrationManager:
57     """
58     a simple mechanism to manage time dependend values.
59    
60 gross 720 typical usage is::
61 gross 351
62 gross 720 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 gross 351
70 gross 720 @note: currently only p=1 is supported.
71 gross 351 """
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 gross 396 def getValue(self):
92 gross 409 out=self.__v_mem[0]
93     if len(out)==1:
94     return out[0]
95     else:
96     return out
97    
98 gross 351 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 gross 396
133 gross 351
134 jgs 121 class Projector:
135 jgs 149 """
136     The Projector is a factory which projects a discontiuous function onto a
137     continuous function on the a given domain.
138     """
139 jgs 121 def __init__(self, domain, reduce = True, fast=True):
140     """
141 jgs 149 Create a continuous function space projector for a domain.
142 jgs 121
143 jgs 149 @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 jgs 121 """
147 jgs 149 self.__pde = linearPDEs.LinearPDE(domain)
148 jgs 148 if fast:
149 jgs 149 self.__pde.setSolverMethod(linearPDEs.LinearPDE.LUMPING)
150 jgs 121 self.__pde.setSymmetryOn()
151     self.__pde.setReducedOrderTo(reduce)
152     self.__pde.setValue(D = 1.)
153 ksteube 1312 return
154 jgs 121
155     def __call__(self, input_data):
156     """
157 jgs 149 Projects input_data onto a continuous function
158 jgs 121
159 jgs 149 @param input_data: The input_data to be projected.
160 jgs 121 """
161 gross 525 out=escript.Data(0.,input_data.getShape(),self.__pde.getFunctionSpaceForSolution())
162 gross 1122 self.__pde.setValue(Y = escript.Data(), Y_reduced = escript.Data())
163 jgs 121 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 gross 525 class NoPDE:
191     """
192     solves the following problem for u:
193 jgs 121
194 gross 525 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 gross 720 @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 gross 525 """
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 gross 720 @type D: C{float}, C{int}, L{numarray.NumArray}, L{Data}
226 gross 525 @param Y: right hand side
227 gross 720 @type Y: C{float}, C{int}, L{numarray.NumArray}, L{Data}
228 gross 525 @param q: location of constraints
229 gross 720 @type q: C{float}, C{int}, L{numarray.NumArray}, L{Data}
230 gross 525 @param r: value of solution at locations of constraints
231 gross 720 @type r: C{float}, C{int}, L{numarray.NumArray}, L{Data}
232 gross 525 """
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 gross 720 @type D: C{float}, C{int}, L{numarray.NumArray}, L{Data}
260 gross 525 @param Y: right hand side
261 gross 720 @type Y: C{float}, C{int}, L{numarray.NumArray}, L{Data}
262 gross 525 @param q: location of constraints
263 gross 720 @type q: C{float}, C{int}, L{numarray.NumArray}, L{Data}
264 gross 525 @param r: value of solution at locations of constraints
265 gross 720 @type r: C{float}, C{int}, L{numarray.NumArray}, L{Data}
266 gross 525 """
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 jgs 147 class Locator:
304     """
305 jgs 149 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 jgs 147 """
312    
313     def __init__(self,where,x=numarray.zeros((3,))):
314 jgs 149 """
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 gross 880
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 jgs 149 """
324     if isinstance(where,escript.FunctionSpace):
325 jgs 147 self.__function_space=where
326 jgs 121 else:
327 jgs 149 self.__function_space=escript.ContinuousFunction(where)
328 gross 880 if isinstance(x, list):
329     self.__id=[]
330     for p in x:
331 gross 921 self.__id.append(util.length(self.__function_space.getX()-p[:self.__function_space.getDim()]).minGlobalDataPoint())
332 gross 880 else:
333 gross 921 self.__id=util.length(self.__function_space.getX()-x[:self.__function_space.getDim()]).minGlobalDataPoint()
334 jgs 121
335 jgs 147 def __str__(self):
336 jgs 149 """
337     Returns the coordinates of the Locator as a string.
338     """
339 gross 880 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 jgs 121
354 gross 880 def getX(self):
355     """
356     Returns the exact coordinates of the Locator.
357     """
358     return self(self.getFunctionSpace().getX())
359    
360 jgs 147 def getFunctionSpace(self):
361 jgs 149 """
362     Returns the function space of the Locator.
363     """
364 jgs 147 return self.__function_space
365    
366 gross 880 def getId(self,item=None):
367 jgs 149 """
368     Returns the identifier of the location.
369     """
370 gross 880 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 jgs 121
378    
379 jgs 147 def __call__(self,data):
380 jgs 149 """
381     Returns the value of data at the Locator of a Data object otherwise
382     the object is returned.
383     """
384 jgs 147 return self.getValue(data)
385 jgs 121
386 jgs 147 def getValue(self,data):
387 jgs 149 """
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 jgs 147 if data.getFunctionSpace()==self.getFunctionSpace():
393 gross 880 dat=data
394 jgs 147 else:
395 gross 880 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 gross 921 o=data.getValueOfGlobalDataPoint(*i)
402 gross 880 if data.getRank()==0:
403     out.append(o[0])
404     else:
405     out.append(o)
406     return out
407 jgs 147 else:
408 gross 921 out=data.getValueOfGlobalDataPoint(*id)
409 gross 880 if data.getRank()==0:
410     return out[0]
411     else:
412     return out
413 jgs 147 else:
414     return data
415 jgs 149
416 ksteube 1312 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 gross 1330 class IterationHistory(object):
444 ksteube 1312 """
445 gross 1330 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 gross 1469 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 gross 1330 def PCG(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100):
498     """
499 ksteube 1312 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 gross 1330 The iteration is terminated if the C{stoppingcriterium} function return C{True}.
507 ksteube 1312
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 gross 1330 @type b: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
516 ksteube 1312 @param Aprod: returns the value Ax
517 gross 1330 @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 ksteube 1312 @param Msolve: solves Mx=r
519 gross 1330 @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 ksteube 1312 @param bilinearform: inner product C{<x,r>}
522 gross 1330 @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 ksteube 1312 @param iter_max: maximum number of iteration steps.
528     @type iter_max: C{int}
529 gross 1330 @return: the solution approximation and the corresponding residual
530     @rtype: C{tuple}
531     @warning: C{b} and C{x} are altered.
532 ksteube 1312 """
533     iter=0
534 gross 1330 if x==None:
535     x=0*b
536     else:
537     b += (-1)*Aprod(x)
538 ksteube 1312 r=b
539     rhat=Msolve(r)
540 gross 1330 d = rhat
541 ksteube 1312 rhat_dot_r = bilinearform(rhat, r)
542 gross 1330 if rhat_dot_r<0: raise NegativeNorm,"negative norm."
543 ksteube 1312
544 gross 1330 while not stoppingcriterium(math.sqrt(rhat_dot_r),r,x):
545     iter+=1
546 ksteube 1312 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 gross 1330 if rhat_dot_r<0: raise NegativeNorm,"negative norm."
561 ksteube 1312
562 gross 1330 return x,r
563 ksteube 1312
564 artak 1465
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 gross 1467 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 artak 1465 return vrot
581    
582 artak 1475 def GMRES(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100, iter_restart=10):
583     m=iter_restart
584 gross 1467 iter=0
585 artak 1475 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 gross 1467 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 artak 1465
599     if x==None:
600     x=0*b
601 gross 1467 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 artak 1465
606 artak 1475 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 artak 1465 v=[]
611    
612 gross 1467 rho=math.sqrt(r_dot_r)
613     v.append(r/rho)
614 artak 1465 g[0]=rho
615    
616 artak 1475 while not (stoppingcriterium(rho,norm_b) or iter==iter_restart-1):
617 artak 1465
618     if iter >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
619    
620    
621 gross 1467 p=Msolve(Aprod(v[iter]))
622 artak 1465
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 gross 1467 hhat=givapp(c[0:iter],s[0:iter],hhat);
655 artak 1465 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 gross 1467 g[iter:iter+2]=givapp(c[iter],s[iter],g[iter:iter+2])
664 artak 1465
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 gross 1467 y=numarray.zeros(iter,numarray.Float64)
674 artak 1465 y[iter-1] = g[iter-1] / h[iter-1][iter-1]
675     if iter > 1 :
676     i=iter-2
677     while i>=0 :
678 gross 1467 y[i] = ( g[i] - numarray.dot(h[i][i+1:iter], y[i+1:iter])) / h[i][i]
679 artak 1465 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 artak 1475 if iter!=iter_restart-1:
687     stopped=True
688     else:
689     stopped=False
690 artak 1465
691 artak 1475 return x,stopped
692 artak 1465
693     #############################################
694    
695 gross 1331 class ArithmeticTuple(object):
696     """
697     tuple supporting inplace update x+=y and scaling x=a*y where x,y is an ArithmeticTuple and a is a float.
698    
699     example of usage:
700    
701     from esys.escript import Data
702     from numarray import array
703     a=Data(...)
704     b=array([1.,4.])
705     x=ArithmeticTuple(a,b)
706     y=5.*x
707    
708     """
709     def __init__(self,*args):
710     """
711     initialize object with elements args.
712    
713     @param args: tuple of object that support implace add (x+=y) and scaling (x=a*y)
714     """
715     self.__items=list(args)
716    
717     def __len__(self):
718     """
719     number of items
720    
721     @return: number of items
722     @rtype: C{int}
723     """
724     return len(self.__items)
725    
726     def __getitem__(self,index):
727     """
728     get an item
729    
730     @param index: item to be returned
731     @type index: C{int}
732     @return: item with index C{index}
733     """
734     return self.__items.__getitem__(index)
735    
736     def __mul__(self,other):
737     """
738     scaling from the right
739    
740     @param other: scaling factor
741     @type other: C{float}
742     @return: itemwise self*other
743     @rtype: L{ArithmeticTuple}
744     """
745     out=[]
746     for i in range(len(self)):
747     out.append(self[i]*other)
748     return ArithmeticTuple(*tuple(out))
749    
750     def __rmul__(self,other):
751     """
752     scaling from the left
753    
754     @param other: scaling factor
755     @type other: C{float}
756     @return: itemwise other*self
757     @rtype: L{ArithmeticTuple}
758     """
759     out=[]
760     for i in range(len(self)):
761     out.append(other*self[i])
762     return ArithmeticTuple(*tuple(out))
763    
764 artak 1465 #########################
765     # Added by Artak
766     #########################
767     def __div__(self,other):
768     """
769     dividing from the right
770    
771     @param other: scaling factor
772     @type other: C{float}
773     @return: itemwise self/other
774     @rtype: L{ArithmeticTuple}
775     """
776     out=[]
777     for i in range(len(self)):
778     out.append(self[i]/other)
779     return ArithmeticTuple(*tuple(out))
780    
781     def __rdiv__(self,other):
782     """
783     dividing from the left
784    
785     @param other: scaling factor
786     @type other: C{float}
787     @return: itemwise other/self
788     @rtype: L{ArithmeticTuple}
789     """
790     out=[]
791     for i in range(len(self)):
792     out.append(other/self[i])
793     return ArithmeticTuple(*tuple(out))
794    
795     ##########################################33
796    
797 gross 1331 def __iadd__(self,other):
798     """
799     in-place add of other to self
800    
801     @param other: increment
802     @type other: C{ArithmeticTuple}
803     """
804     if len(self) != len(other):
805     raise ValueError,"tuple length must match."
806     for i in range(len(self)):
807     self.__items[i]+=other[i]
808     return self
809    
810 gross 1414 class HomogeneousSaddlePointProblem(object):
811     """
812     This provides a framwork for solving homogeneous saddle point problem of the form
813    
814     Av+B^*p=f
815     Bv =0
816    
817     for the unknowns v and p and given operators A and B and given right hand side f.
818     B^* is the adjoint operator of B is the given inner product.
819    
820     """
821     def __init__(self,**kwargs):
822     self.setTolerance()
823     self.setToleranceReductionFactor()
824    
825     def initialize(self):
826     """
827     initialize the problem (overwrite)
828     """
829     pass
830     def B(self,v):
831     """
832     returns Bv (overwrite)
833     @rtype: equal to the type of p
834    
835     @note: boundary conditions on p should be zero!
836     """
837     pass
838    
839     def inner(self,p0,p1):
840     """
841     returns inner product of two element p0 and p1 (overwrite)
842    
843     @type p0: equal to the type of p
844     @type p1: equal to the type of p
845     @rtype: C{float}
846    
847     @rtype: equal to the type of p
848     """
849     pass
850    
851     def solve_A(self,u,p):
852     """
853     solves Av=f-Au-B^*p with accuracy self.getReducedTolerance() (overwrite)
854    
855     @rtype: equal to the type of v
856     @note: boundary conditions on v should be zero!
857     """
858     pass
859    
860     def solve_prec(self,p):
861     """
862     provides a preconditioner for BA^{-1}B^* with accuracy self.getReducedTolerance() (overwrite)
863    
864     @rtype: equal to the type of p
865     """
866     pass
867    
868     def stoppingcriterium(self,Bv,v,p):
869     """
870     returns a True if iteration is terminated. (overwrite)
871    
872     @rtype: C{bool}
873     """
874     pass
875    
876     def __inner(self,p,r):
877     return self.inner(p,r[1])
878    
879 artak 1465 def __inner_p(self,p1,p2):
880     return self.inner(p1,p2)
881    
882 gross 1414 def __stoppingcriterium(self,norm_r,r,p):
883     return self.stoppingcriterium(r[1],r[0],p)
884    
885 gross 1467 def __stoppingcriterium_GMRES(self,norm_r,norm_b):
886     return self.stoppingcriterium_GMRES(norm_r,norm_b)
887 artak 1465
888 gross 1414 def setTolerance(self,tolerance=1.e-8):
889     self.__tol=tolerance
890     def getTolerance(self):
891     return self.__tol
892     def setToleranceReductionFactor(self,reduction=0.01):
893     self.__reduction=reduction
894     def getSubProblemTolerance(self):
895     return self.__reduction*self.getTolerance()
896    
897 artak 1465 def solve(self,v,p,max_iter=20, verbose=False, show_details=False, solver='GMRES'):
898 gross 1414 """
899     solves the saddle point problem using initial guesses v and p.
900    
901     @param max_iter: maximum number of iteration steps.
902     """
903     self.verbose=verbose
904     self.show_details=show_details and self.verbose
905    
906 gross 1469 # assume p is known: then v=A^-1(f-B^*p)
907     # which leads to BA^-1B^*p = BA^-1f
908    
909 gross 1414 # Az=f is solved as A(z-v)=f-Av (z-v = 0 on fixed_u_mask)
910    
911 artak 1465
912 gross 1414 self.__z=v+self.solve_A(v,p*0)
913 artak 1465
914 gross 1414 Bz=self.B(self.__z)
915     #
916     # solve BA^-1B^*p = Bz
917     #
918     # note that the residual r=Bz-BA^-1B^*p = B(z-A^-1B^*p) = Bv
919     #
920     # with Av=Az-B^*p = f - B^*p (v=z on fixed_u_mask)
921     # A(v-z)=Az-B^*p-Az = f -Az - B^*p (v-z=0 on fixed_u_mask)
922     #
923     self.iter=0
924 artak 1465 if solver=='GMRES':
925     if self.verbose: print "enter GMRES method (iter_max=%s)"%max_iter
926 gross 1467 p=GMRES(Bz,self.__Aprod_GMRES,self.__Msolve_GMRES,self.__inner_p,self.__stoppingcriterium_GMRES,iter_max=max_iter, x=p*1.)
927     # solve Au=f-B^*p
928     # A(u-v)=f-B^*p-Av
929     # u=v+(u-v)
930 artak 1465 u=v+self.solve_A(v,p)
931    
932     else:
933     if self.verbose: print "enter PCG method (iter_max=%s)"%max_iter
934 gross 1467 p,r=PCG(ArithmeticTuple(self.__z*1.,Bz),self.__Aprod,self.__Msolve,self.__inner,self.__stoppingcriterium,iter_max=max_iter, x=p)
935 artak 1465 u=r[0]
936 artak 1475 print "RESULT div(u)=",util.Lsup(self.B(u)),util.Lsup(u)
937 gross 1414
938 artak 1465 return u,p
939    
940 gross 1414 def __Msolve(self,r):
941     return self.solve_prec(r[1])
942    
943 artak 1465 def __Msolve_GMRES(self,r):
944     return self.solve_prec(r)
945    
946    
947 gross 1414 def __Aprod(self,p):
948     # return BA^-1B*p
949     #solve Av =-B^*p as Av =f-Az-B^*p
950 gross 1469 v=self.solve_A(self.__z,-p)
951     return ArithmeticTuple(v, self.B(v))
952 gross 1414
953 artak 1465 def __Aprod_GMRES(self,p):
954     # return BA^-1B*p
955     #solve Av =-B^*p as Av =f-Az-B^*p
956 gross 1469 v=self.solve_A(self.__z,-p)
957     return self.B(v)
958 gross 1414
959 gross 867 class SaddlePointProblem(object):
960     """
961     This implements a solver for a saddlepoint problem
962    
963 gross 877 M{f(u,p)=0}
964     M{g(u)=0}
965 gross 867
966     for u and p. The problem is solved with an inexact Uszawa scheme for p:
967    
968 ksteube 990 M{Q_f (u^{k+1}-u^{k}) = - f(u^{k},p^{k})}
969 gross 877 M{Q_g (p^{k+1}-p^{k}) = g(u^{k+1})}
970 gross 867
971     where Q_f is an approximation of the Jacobiean A_f of f with respect to u and Q_f is an approximation of
972     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'
973     Q_g can be difficult, non-linear conjugate gradient method is applied to solve for p, so Q_g plays
974     in fact the role of a preconditioner.
975     """
976     def __init__(self,verbose=False,*args):
977     """
978     initializes the problem
979    
980 ksteube 990 @param verbose: switches on the printing out some information
981 gross 867 @type verbose: C{bool}
982     @note: this method may be overwritten by a particular saddle point problem
983     """
984 gross 1107 if not isinstance(verbose,bool):
985     raise TypeError("verbose needs to be of type bool.")
986 gross 1106 self.__verbose=verbose
987 gross 877 self.relaxation=1.
988 gross 867
989     def trace(self,text):
990     """
991     prints text if verbose has been set
992    
993 ksteube 990 @param text: a text message
994 gross 867 @type text: C{str}
995     """
996     if self.__verbose: print "%s: %s"%(str(self),text)
997    
998 gross 873 def solve_f(self,u,p,tol=1.e-8):
999 gross 867 """
1000     solves
1001    
1002     A_f du = f(u,p)
1003    
1004     with tolerance C{tol} and return du. A_f is Jacobiean of f with respect to u.
1005    
1006     @param u: current approximation of u
1007     @type u: L{escript.Data}
1008     @param p: current approximation of p
1009     @type p: L{escript.Data}
1010 gross 873 @param tol: tolerance expected for du
1011 gross 867 @type tol: C{float}
1012     @return: increment du
1013     @rtype: L{escript.Data}
1014     @note: this method has to be overwritten by a particular saddle point problem
1015     """
1016     pass
1017    
1018 gross 873 def solve_g(self,u,tol=1.e-8):
1019 gross 867 """
1020     solves
1021    
1022     Q_g dp = g(u)
1023    
1024     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.
1025    
1026     @param u: current approximation of u
1027     @type u: L{escript.Data}
1028 gross 873 @param tol: tolerance expected for dp
1029     @type tol: C{float}
1030 gross 867 @return: increment dp
1031     @rtype: L{escript.Data}
1032     @note: this method has to be overwritten by a particular saddle point problem
1033     """
1034     pass
1035    
1036     def inner(self,p0,p1):
1037     """
1038     inner product of p0 and p1 approximating p. Typically this returns integrate(p0*p1)
1039     @return: inner product of p0 and p1
1040     @rtype: C{float}
1041     """
1042     pass
1043    
1044 gross 877 subiter_max=3
1045     def solve(self,u0,p0,tolerance=1.e-6,tolerance_u=None,iter_max=100,accepted_reduction=0.995,relaxation=None):
1046     """
1047     runs the solver
1048 gross 873
1049 gross 877 @param u0: initial guess for C{u}
1050     @type u0: L{esys.escript.Data}
1051     @param p0: initial guess for C{p}
1052     @type p0: L{esys.escript.Data}
1053     @param tolerance: tolerance for relative error in C{u} and C{p}
1054     @type tolerance: positive C{float}
1055     @param tolerance_u: tolerance for relative error in C{u} if different from C{tolerance}
1056     @type tolerance_u: positive C{float}
1057     @param iter_max: maximum number of iteration steps.
1058     @type iter_max: C{int}
1059     @param accepted_reduction: if the norm g cannot be reduced by C{accepted_reduction} backtracking to adapt the
1060     relaxation factor. If C{accepted_reduction=None} no backtracking is used.
1061     @type accepted_reduction: positive C{float} or C{None}
1062     @param relaxation: initial relaxation factor. If C{relaxation==None}, the last relaxation factor is used.
1063     @type relaxation: C{float} or C{None}
1064     """
1065     tol=1.e-2
1066     if tolerance_u==None: tolerance_u=tolerance
1067     if not relaxation==None: self.relaxation=relaxation
1068     if accepted_reduction ==None:
1069     angle_limit=0.
1070     elif accepted_reduction>=1.:
1071     angle_limit=0.
1072     else:
1073     angle_limit=util.sqrt(1-accepted_reduction**2)
1074     self.iter=0
1075     u=u0
1076     p=p0
1077     #
1078     # initialize things:
1079     #
1080     converged=False
1081     #
1082     # start loop:
1083     #
1084     # initial search direction is g
1085     #
1086     while not converged :
1087     if self.iter>iter_max:
1088     raise ArithmeticError("no convergence after %s steps."%self.iter)
1089     f_new=self.solve_f(u,p,tol)
1090     norm_f_new = util.Lsup(f_new)
1091     u_new=u-f_new
1092     g_new=self.solve_g(u_new,tol)
1093     self.iter+=1
1094     norm_g_new = util.sqrt(self.inner(g_new,g_new))
1095     if norm_f_new==0. and norm_g_new==0.: return u, p
1096     if self.iter>1 and not accepted_reduction==None:
1097     #
1098     # did we manage to reduce the norm of G? I
1099     # if not we start a backtracking procedure
1100     #
1101     # print "new/old norm = ",norm_g_new, norm_g, norm_g_new/norm_g
1102     if norm_g_new > accepted_reduction * norm_g:
1103     sub_iter=0
1104     s=self.relaxation
1105     d=g
1106     g_last=g
1107     self.trace(" start substepping: f = %s, g = %s, relaxation = %s."%(norm_f_new, norm_g_new, s))
1108     while sub_iter < self.subiter_max and norm_g_new > accepted_reduction * norm_g:
1109     dg= g_new-g_last
1110     norm_dg=abs(util.sqrt(self.inner(dg,dg))/self.relaxation)
1111     rad=self.inner(g_new,dg)/self.relaxation
1112     # print " ",sub_iter,": rad, norm_dg:",abs(rad), norm_dg*norm_g_new * angle_limit
1113     # print " ",sub_iter,": rad, norm_dg:",rad, norm_dg, norm_g_new, norm_g
1114     if abs(rad) < norm_dg*norm_g_new * angle_limit:
1115     if sub_iter>0: self.trace(" no further improvements expected from backtracking.")
1116     break
1117     r=self.relaxation
1118     self.relaxation= - rad/norm_dg**2
1119     s+=self.relaxation
1120     #####
1121     # a=g_new+self.relaxation*dg/r
1122     # print "predicted new norm = ",util.sqrt(self.inner(a,a)),util.sqrt(self.inner(g_new,g_new)), self.relaxation
1123     #####
1124     g_last=g_new
1125     p+=self.relaxation*d
1126     f_new=self.solve_f(u,p,tol)
1127     u_new=u-f_new
1128     g_new=self.solve_g(u_new,tol)
1129     self.iter+=1
1130     norm_f_new = util.Lsup(f_new)
1131     norm_g_new = util.sqrt(self.inner(g_new,g_new))
1132     # print " ",sub_iter," new g norm",norm_g_new
1133     self.trace(" %s th sub-step: f = %s, g = %s, relaxation = %s."%(sub_iter, norm_f_new, norm_g_new, s))
1134     #
1135     # can we expect reduction of g?
1136     #
1137     # u_last=u_new
1138     sub_iter+=1
1139     self.relaxation=s
1140     #
1141     # check for convergence:
1142     #
1143     norm_u_new = util.Lsup(u_new)
1144     p_new=p+self.relaxation*g_new
1145     norm_p_new = util.sqrt(self.inner(p_new,p_new))
1146 ksteube 1125 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))
1147 gross 873
1148 gross 877 if self.iter>1:
1149     dg2=g_new-g
1150     df2=f_new-f
1151     norm_dg2=util.sqrt(self.inner(dg2,dg2))
1152     norm_df2=util.Lsup(df2)
1153     # print norm_g_new, norm_g, norm_dg, norm_p, tolerance
1154     tol_eq_g=tolerance*norm_dg2/(norm_g*abs(self.relaxation))*norm_p_new
1155     tol_eq_f=tolerance_u*norm_df2/norm_f*norm_u_new
1156     if norm_g_new <= tol_eq_g and norm_f_new <= tol_eq_f:
1157     converged=True
1158     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
1159     self.trace("convergence after %s steps."%self.iter)
1160     return u,p
1161     # def solve(self,u0,p0,tolerance=1.e-6,iter_max=10,self.relaxation=1.):
1162     # tol=1.e-2
1163     # iter=0
1164     # converged=False
1165     # u=u0*1.
1166     # p=p0*1.
1167     # while not converged and iter<iter_max:
1168     # du=self.solve_f(u,p,tol)
1169     # u-=du
1170     # norm_du=util.Lsup(du)
1171     # norm_u=util.Lsup(u)
1172     #
1173     # dp=self.relaxation*self.solve_g(u,tol)
1174     # p+=dp
1175     # norm_dp=util.sqrt(self.inner(dp,dp))
1176     # norm_p=util.sqrt(self.inner(p,p))
1177     # 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)
1178     # iter+=1
1179     #
1180     # converged = (norm_du <= tolerance*norm_u) and (norm_dp <= tolerance*norm_p)
1181     # if converged:
1182     # print "convergence after %s steps."%iter
1183     # else:
1184     # raise ArithmeticError("no convergence after %s steps."%iter)
1185     #
1186     # return u,p
1187 gross 873
1188 ksteube 1312 def MaskFromBoundaryTag(function_space,*tags):
1189     """
1190     create a mask on the given function space which one for samples
1191     that touch the boundary tagged by tags.
1192    
1193     usage: m=MaskFromBoundaryTag(Solution(domain),"left", "right")
1194    
1195     @param function_space: a given function space
1196     @type function_space: L{escript.FunctionSpace}
1197     @param tags: boundray tags
1198     @type tags: C{str}
1199     @return: a mask which marks samples used by C{function_space} that are touching the
1200     boundary tagged by any of the given tags.
1201     @rtype: L{escript.Data} of rank 0
1202     """
1203     pde=linearPDEs.LinearPDE(function_space.getDomain(),numEquations=1, numSolutions=1)
1204     d=escript.Scalar(0.,escript.FunctionOnBoundary(function_space.getDomain()))
1205     for t in tags: d.setTaggedValue(t,1.)
1206     pde.setValue(y=d)
1207     out=util.whereNonZero(pde.getRightHandSide())
1208     if out.getFunctionSpace() == function_space:
1209     return out
1210     else:
1211     return util.whereNonZero(util.interpolate(out,function_space))
1212    
1213 gross 1414
1214 artak 1465

Properties

Name Value
svn:eol-style native
svn:keywords Author Date Id Revision

  ViewVC Help
Powered by ViewVC 1.1.26