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Fri Apr 18 02:36:37 2008 UTC (11 years, 4 months ago) by artak
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stopping criteriums are combined for GMRES,MINRES and TFQMR in stoppingcriterium2
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 artak 1489 if self.verbose: print "iter: #s: inner(rhat,r) = #e"#(len(self.history)-1, self.history[-1])
478 gross 1330 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 artak 1489 if self.verbose: print "iter: #s: norm(r) = #e"#(len(self.history)-1, self.history[-1])
495 gross 1469 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 1514 def GMRES(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100, iter_restart=20):
583 artak 1475 m=iter_restart
584 gross 1467 iter=0
585 artak 1475 while True:
586 artak 1488 if iter >= iter_max: raise MaxIterReached,"maximum number of %s steps reached"%iter_max
587 artak 1475 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 artak 1514 def GMRESm(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100, iter_restart=20):
593 artak 1475 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 artak 1488 x=0
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 artak 1488
614 gross 1467 v.append(r/rho)
615 artak 1465 g[0]=rho
616    
617 artak 1475 while not (stoppingcriterium(rho,norm_b) or iter==iter_restart-1):
618 artak 1465
619     if iter >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
620    
621    
622 gross 1467 p=Msolve(Aprod(v[iter]))
623 artak 1465
624     v.append(p)
625    
626     v_norm1=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
627    
628     # Modified Gram-Schmidt
629     for j in range(iter+1):
630     h[j][iter]=bilinearform(v[j],v[iter+1])
631     v[iter+1]+=(-1.)*h[j][iter]*v[j]
632    
633     h[iter+1][iter]=math.sqrt(bilinearform(v[iter+1],v[iter+1]))
634     v_norm2=h[iter+1][iter]
635    
636    
637     # Reorthogonalize if needed
638     if v_norm1 + 0.001*v_norm2 == v_norm1: #Brown/Hindmarsh condition (default)
639     for j in range(iter+1):
640     hr=bilinearform(v[j],v[iter+1])
641     h[j][iter]=h[j][iter]+hr #vhat
642     v[iter+1] +=(-1.)*hr*v[j]
643    
644     v_norm2=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
645     h[iter+1][iter]=v_norm2
646    
647     # watch out for happy breakdown
648     if v_norm2 != 0:
649     v[iter+1]=v[iter+1]/h[iter+1][iter]
650    
651     # Form and store the information for the new Givens rotation
652     if iter > 0 :
653     hhat=[]
654     for i in range(iter+1) : hhat.append(h[i][iter])
655 gross 1467 hhat=givapp(c[0:iter],s[0:iter],hhat);
656 artak 1465 for i in range(iter+1) : h[i][iter]=hhat[i]
657    
658     mu=math.sqrt(h[iter][iter]*h[iter][iter]+h[iter+1][iter]*h[iter+1][iter])
659     if mu!=0 :
660     c[iter]=h[iter][iter]/mu
661     s[iter]=-h[iter+1][iter]/mu
662     h[iter][iter]=c[iter]*h[iter][iter]-s[iter]*h[iter+1][iter]
663     h[iter+1][iter]=0.0
664 gross 1467 g[iter:iter+2]=givapp(c[iter],s[iter],g[iter:iter+2])
665 artak 1465
666     # Update the residual norm
667     rho=abs(g[iter+1])
668     iter+=1
669    
670     # At this point either iter > iter_max or rho < tol.
671     # It's time to compute x and leave.
672    
673     if iter > 0 :
674 gross 1467 y=numarray.zeros(iter,numarray.Float64)
675 artak 1465 y[iter-1] = g[iter-1] / h[iter-1][iter-1]
676     if iter > 1 :
677     i=iter-2
678     while i>=0 :
679 gross 1467 y[i] = ( g[i] - numarray.dot(h[i][i+1:iter], y[i+1:iter])) / h[i][i]
680 artak 1465 i=i-1
681     xhat=v[iter-1]*y[iter-1]
682     for i in range(iter-1):
683     xhat += v[i]*y[i]
684     else : xhat=v[0]
685    
686     x += xhat
687 artak 1488 if iter<iter_restart-1:
688 artak 1475 stopped=True
689     else:
690     stopped=False
691 artak 1465
692 artak 1475 return x,stopped
693 artak 1481
694     def MINRES(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100):
695    
696     #
697     # minres solves the system of linear equations Ax = b
698     # where A is a symmetric matrix (possibly indefinite or singular)
699     # and b is a given vector.
700     #
701     # "A" may be a dense or sparse matrix (preferably sparse!)
702     # or the name of a function such that
703     # y = A(x)
704     # returns the product y = Ax for any given vector x.
705     #
706     # "M" defines a positive-definite preconditioner M = C C'.
707     # "M" may be a dense or sparse matrix (preferably sparse!)
708     # or the name of a function such that
709     # solves the system My = x for any given vector x.
710     #
711     #
712 artak 1482
713 artak 1481 #------------------------------------------------------------------
714     # Set up y and v for the first Lanczos vector v1.
715     # y = beta1 P' v1, where P = C**(-1).
716     # v is really P' v1.
717     #------------------------------------------------------------------
718 artak 1482 if x==None:
719     x=0*b
720     else:
721     b += (-1)*Aprod(x)
722    
723 artak 1481 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 artak 1486 beta1 = math.sqrt(beta1)
734 artak 1481
735     #------------------------------------------------------------------
736 artak 1484 # Initialize quantities.
737 artak 1481 # ------------------------------------------------------------------
738 artak 1482 iter = 0
739     Anorm = 0
740     ynorm = 0
741 artak 1481 oldb = 0
742     beta = beta1
743     dbar = 0
744     epsln = 0
745     phibar = beta1
746     rhs1 = beta1
747     rhs2 = 0
748     rnorm = phibar
749     tnorm2 = 0
750     ynorm2 = 0
751     cs = -1
752     sn = 0
753     w = b*0.
754     w2 = b*0.
755     r2 = r1
756     eps = 0.0001
757    
758     #---------------------------------------------------------------------
759     # Main iteration loop.
760     # --------------------------------------------------------------------
761 artak 1517 while not stoppingcriterium(rnorm,Anorm*ynorm,'MINRES'): # checks ||r|| < (||A|| ||x||) * TOL
762 artak 1481
763     if iter >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
764     iter = iter + 1
765    
766     #-----------------------------------------------------------------
767     # Obtain quantities for the next Lanczos vector vk+1, k = 1, 2,...
768     # The general iteration is similar to the case k = 1 with v0 = 0:
769     #
770     # p1 = Operator * v1 - beta1 * v0,
771     # alpha1 = v1'p1,
772     # q2 = p2 - alpha1 * v1,
773     # beta2^2 = q2'q2,
774     # v2 = (1/beta2) q2.
775     #
776     # Again, y = betak P vk, where P = C**(-1).
777     #-----------------------------------------------------------------
778     s = 1/beta # Normalize previous vector (in y).
779     v = s*y # v = vk if P = I
780    
781     y = Aprod(v)
782 artak 1465
783 artak 1481 if iter >= 2:
784     y = y - (beta/oldb)*r1
785    
786     alfa = bilinearform(v,y) # alphak
787     y = (- alfa/beta)*r2 + y
788     r1 = r2
789     r2 = y
790     y = Msolve(r2)
791     oldb = beta # oldb = betak
792     beta = bilinearform(r2,y) # beta = betak+1^2
793     if beta < 0: raise NegativeNorm,"negative norm."
794    
795     beta = math.sqrt( beta )
796     tnorm2 = tnorm2 + alfa*alfa + oldb*oldb + beta*beta
797    
798     if iter==1: # Initialize a few things.
799     gmax = abs( alfa ) # alpha1
800     gmin = gmax # alpha1
801    
802     # Apply previous rotation Qk-1 to get
803     # [deltak epslnk+1] = [cs sn][dbark 0 ]
804     # [gbar k dbar k+1] [sn -cs][alfak betak+1].
805    
806     oldeps = epsln
807     delta = cs * dbar + sn * alfa # delta1 = 0 deltak
808     gbar = sn * dbar - cs * alfa # gbar 1 = alfa1 gbar k
809     epsln = sn * beta # epsln2 = 0 epslnk+1
810     dbar = - cs * beta # dbar 2 = beta2 dbar k+1
811    
812     # Compute the next plane rotation Qk
813    
814     gamma = math.sqrt(gbar*gbar+beta*beta) # gammak
815     gamma = max(gamma,eps)
816     cs = gbar / gamma # ck
817     sn = beta / gamma # sk
818     phi = cs * phibar # phik
819     phibar = sn * phibar # phibark+1
820    
821     # Update x.
822    
823     denom = 1/gamma
824     w1 = w2
825     w2 = w
826     w = (v - oldeps*w1 - delta*w2) * denom
827     x = x + phi*w
828    
829     # Go round again.
830    
831     gmax = max(gmax,gamma)
832     gmin = min(gmin,gamma)
833     z = rhs1 / gamma
834     ynorm2 = z*z + ynorm2
835     rhs1 = rhs2 - delta*z
836     rhs2 = - epsln*z
837    
838     # Estimate various norms and test for convergence.
839    
840     Anorm = math.sqrt( tnorm2 )
841     ynorm = math.sqrt( ynorm2 )
842    
843     rnorm = phibar
844    
845     return x
846    
847 artak 1489
848     def TFQMR(b, Aprod, Msolve, bilinearform, stoppingcriterium, x=None, iter_max=100):
849    
850     # TFQMR solver for linear systems
851     #
852     #
853     # initialization
854     #
855     errtol = math.sqrt(bilinearform(b,b))
856     norm_b=errtol
857     kmax = iter_max
858     error = []
859    
860     if math.sqrt(bilinearform(x,x)) != 0.0:
861     r = b - Aprod(x)
862     else:
863     r = b
864    
865     r=Msolve(r)
866    
867     u1=0
868     u2=0
869     y1=0
870     y2=0
871    
872     w = r
873     y1 = r
874     iter = 0
875     d = 0
876    
877     v = Msolve(Aprod(y1))
878     u1 = v
879    
880     theta = 0.0;
881     eta = 0.0;
882     tau = math.sqrt(bilinearform(r,r))
883     error = [ error, tau ]
884     rho = tau * tau
885     m=1
886     #
887     # TFQMR iteration
888     #
889     # while ( iter < kmax-1 ):
890    
891 artak 1517 while not stoppingcriterium(tau*math.sqrt ( m + 1 ),norm_b,'TFQMR'):
892 artak 1489 if iter >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
893    
894     sigma = bilinearform(r,v)
895    
896     if ( sigma == 0.0 ):
897     raise 'TFQMR breakdown, sigma=0'
898    
899    
900     alpha = rho / sigma
901    
902     for j in range(2):
903     #
904     # Compute y2 and u2 only if you have to
905     #
906     if ( j == 1 ):
907     y2 = y1 - alpha * v
908     u2 = Msolve(Aprod(y2))
909    
910     m = 2 * (iter+1) - 2 + (j+1)
911     if j==0:
912     w = w - alpha * u1
913     d = y1 + ( theta * theta * eta / alpha ) * d
914     if j==1:
915     w = w - alpha * u2
916     d = y2 + ( theta * theta * eta / alpha ) * d
917    
918     theta = math.sqrt(bilinearform(w,w))/ tau
919     c = 1.0 / math.sqrt ( 1.0 + theta * theta )
920     tau = tau * theta * c
921     eta = c * c * alpha
922     x = x + eta * d
923     #
924     # Try to terminate the iteration at each pass through the loop
925     #
926     # if ( tau * math.sqrt ( m + 1 ) <= errtol ):
927     # error = [ error, tau ]
928     # total_iters = iter
929     # break
930    
931    
932     if ( rho == 0.0 ):
933     raise 'TFQMR breakdown, rho=0'
934    
935    
936     rhon = bilinearform(r,w)
937     beta = rhon / rho;
938     rho = rhon;
939     y1 = w + beta * y2;
940     u1 = Msolve(Aprod(y1))
941     v = u1 + beta * ( u2 + beta * v )
942     error = [ error, tau ]
943     total_iters = iter
944    
945     iter = iter + 1
946    
947     return x
948    
949    
950 artak 1465 #############################################
951    
952 gross 1331 class ArithmeticTuple(object):
953     """
954     tuple supporting inplace update x+=y and scaling x=a*y where x,y is an ArithmeticTuple and a is a float.
955    
956     example of usage:
957    
958     from esys.escript import Data
959     from numarray import array
960     a=Data(...)
961     b=array([1.,4.])
962     x=ArithmeticTuple(a,b)
963     y=5.*x
964    
965     """
966     def __init__(self,*args):
967     """
968     initialize object with elements args.
969    
970     @param args: tuple of object that support implace add (x+=y) and scaling (x=a*y)
971     """
972     self.__items=list(args)
973    
974     def __len__(self):
975     """
976     number of items
977    
978     @return: number of items
979     @rtype: C{int}
980     """
981     return len(self.__items)
982    
983     def __getitem__(self,index):
984     """
985     get an item
986    
987     @param index: item to be returned
988     @type index: C{int}
989     @return: item with index C{index}
990     """
991     return self.__items.__getitem__(index)
992    
993     def __mul__(self,other):
994     """
995     scaling from the right
996    
997     @param other: scaling factor
998     @type other: C{float}
999     @return: itemwise self*other
1000     @rtype: L{ArithmeticTuple}
1001     """
1002     out=[]
1003     for i in range(len(self)):
1004     out.append(self[i]*other)
1005     return ArithmeticTuple(*tuple(out))
1006    
1007     def __rmul__(self,other):
1008     """
1009     scaling from the left
1010    
1011     @param other: scaling factor
1012     @type other: C{float}
1013     @return: itemwise other*self
1014     @rtype: L{ArithmeticTuple}
1015     """
1016     out=[]
1017     for i in range(len(self)):
1018     out.append(other*self[i])
1019     return ArithmeticTuple(*tuple(out))
1020    
1021 artak 1465 #########################
1022     # Added by Artak
1023     #########################
1024     def __div__(self,other):
1025     """
1026     dividing from the right
1027    
1028     @param other: scaling factor
1029     @type other: C{float}
1030     @return: itemwise self/other
1031     @rtype: L{ArithmeticTuple}
1032     """
1033     out=[]
1034     for i in range(len(self)):
1035     out.append(self[i]/other)
1036     return ArithmeticTuple(*tuple(out))
1037    
1038     def __rdiv__(self,other):
1039     """
1040     dividing from the left
1041    
1042     @param other: scaling factor
1043     @type other: C{float}
1044     @return: itemwise other/self
1045     @rtype: L{ArithmeticTuple}
1046     """
1047     out=[]
1048     for i in range(len(self)):
1049     out.append(other/self[i])
1050     return ArithmeticTuple(*tuple(out))
1051    
1052     ##########################################33
1053    
1054 gross 1331 def __iadd__(self,other):
1055     """
1056     in-place add of other to self
1057    
1058     @param other: increment
1059     @type other: C{ArithmeticTuple}
1060     """
1061     if len(self) != len(other):
1062     raise ValueError,"tuple length must match."
1063     for i in range(len(self)):
1064     self.__items[i]+=other[i]
1065     return self
1066    
1067 gross 1414 class HomogeneousSaddlePointProblem(object):
1068     """
1069     This provides a framwork for solving homogeneous saddle point problem of the form
1070    
1071     Av+B^*p=f
1072     Bv =0
1073    
1074     for the unknowns v and p and given operators A and B and given right hand side f.
1075     B^* is the adjoint operator of B is the given inner product.
1076    
1077     """
1078     def __init__(self,**kwargs):
1079     self.setTolerance()
1080     self.setToleranceReductionFactor()
1081    
1082     def initialize(self):
1083     """
1084     initialize the problem (overwrite)
1085     """
1086     pass
1087     def B(self,v):
1088     """
1089     returns Bv (overwrite)
1090     @rtype: equal to the type of p
1091    
1092     @note: boundary conditions on p should be zero!
1093     """
1094     pass
1095    
1096     def inner(self,p0,p1):
1097     """
1098     returns inner product of two element p0 and p1 (overwrite)
1099    
1100     @type p0: equal to the type of p
1101     @type p1: equal to the type of p
1102     @rtype: C{float}
1103    
1104     @rtype: equal to the type of p
1105     """
1106     pass
1107    
1108     def solve_A(self,u,p):
1109     """
1110     solves Av=f-Au-B^*p with accuracy self.getReducedTolerance() (overwrite)
1111    
1112     @rtype: equal to the type of v
1113     @note: boundary conditions on v should be zero!
1114     """
1115     pass
1116    
1117     def solve_prec(self,p):
1118     """
1119     provides a preconditioner for BA^{-1}B^* with accuracy self.getReducedTolerance() (overwrite)
1120    
1121     @rtype: equal to the type of p
1122     """
1123     pass
1124    
1125     def stoppingcriterium(self,Bv,v,p):
1126     """
1127     returns a True if iteration is terminated. (overwrite)
1128    
1129     @rtype: C{bool}
1130     """
1131     pass
1132    
1133     def __inner(self,p,r):
1134     return self.inner(p,r[1])
1135    
1136 artak 1465 def __inner_p(self,p1,p2):
1137     return self.inner(p1,p2)
1138    
1139 gross 1414 def __stoppingcriterium(self,norm_r,r,p):
1140     return self.stoppingcriterium(r[1],r[0],p)
1141    
1142 artak 1517 def __stoppingcriterium2(self,norm_r,norm_b,solver='GMRES'):
1143     return self.stoppingcriterium2(norm_r,norm_b,solver)
1144 artak 1465
1145 gross 1414 def setTolerance(self,tolerance=1.e-8):
1146     self.__tol=tolerance
1147     def getTolerance(self):
1148     return self.__tol
1149     def setToleranceReductionFactor(self,reduction=0.01):
1150     self.__reduction=reduction
1151     def getSubProblemTolerance(self):
1152     return self.__reduction*self.getTolerance()
1153    
1154 artak 1517 def solve(self,v,p,max_iter=20, verbose=False, show_details=False, solver='PCG',iter_restart=20):
1155 gross 1414 """
1156     solves the saddle point problem using initial guesses v and p.
1157    
1158     @param max_iter: maximum number of iteration steps.
1159     """
1160     self.verbose=verbose
1161     self.show_details=show_details and self.verbose
1162    
1163 gross 1469 # assume p is known: then v=A^-1(f-B^*p)
1164     # which leads to BA^-1B^*p = BA^-1f
1165    
1166 gross 1414 # Az=f is solved as A(z-v)=f-Av (z-v = 0 on fixed_u_mask)
1167    
1168 artak 1465
1169 gross 1414 self.__z=v+self.solve_A(v,p*0)
1170 artak 1465
1171 gross 1414 Bz=self.B(self.__z)
1172     #
1173     # solve BA^-1B^*p = Bz
1174     #
1175     # note that the residual r=Bz-BA^-1B^*p = B(z-A^-1B^*p) = Bv
1176     #
1177     # with Av=Az-B^*p = f - B^*p (v=z on fixed_u_mask)
1178     # A(v-z)=Az-B^*p-Az = f -Az - B^*p (v-z=0 on fixed_u_mask)
1179     #
1180     self.iter=0
1181 artak 1465 if solver=='GMRES':
1182     if self.verbose: print "enter GMRES method (iter_max=%s)"%max_iter
1183 artak 1517 p=GMRES(Bz,self.__Aprod2,self.__Msolve2,self.__inner_p,self.__stoppingcriterium2,iter_max=max_iter, x=p*1.,iter_restart=iter_restart)
1184 gross 1467 # solve Au=f-B^*p
1185     # A(u-v)=f-B^*p-Av
1186     # u=v+(u-v)
1187 artak 1465 u=v+self.solve_A(v,p)
1188 artak 1481
1189 artak 1489 if solver=='TFQMR':
1190 artak 1517 if self.verbose: print "enter TFQMR method (iter_max=%s)"%max_iter
1191     p=TFQMR(Bz,self.__Aprod2,self.__Msolve2,self.__inner_p,self.__stoppingcriterium2,iter_max=max_iter, x=p*1.)
1192 artak 1489 # solve Au=f-B^*p
1193     # A(u-v)=f-B^*p-Av
1194     # u=v+(u-v)
1195     u=v+self.solve_A(v,p)
1196    
1197 artak 1481 if solver=='MINRES':
1198     if self.verbose: print "enter MINRES method (iter_max=%s)"%max_iter
1199 artak 1517 p=MINRES(Bz,self.__Aprod2,self.__Msolve2,self.__inner_p,self.__stoppingcriterium2,iter_max=max_iter, x=p*1.)
1200 artak 1481 # solve Au=f-B^*p
1201     # A(u-v)=f-B^*p-Av
1202     # u=v+(u-v)
1203     u=v+self.solve_A(v,p)
1204 artak 1488
1205     if solver=='PCG':
1206 artak 1465 if self.verbose: print "enter PCG method (iter_max=%s)"%max_iter
1207 gross 1467 p,r=PCG(ArithmeticTuple(self.__z*1.,Bz),self.__Aprod,self.__Msolve,self.__inner,self.__stoppingcriterium,iter_max=max_iter, x=p)
1208 artak 1465 u=r[0]
1209 artak 1481
1210 artak 1475 print "RESULT div(u)=",util.Lsup(self.B(u)),util.Lsup(u)
1211 gross 1414
1212 artak 1465 return u,p
1213    
1214 gross 1414 def __Msolve(self,r):
1215     return self.solve_prec(r[1])
1216    
1217 artak 1517 def __Msolve2(self,r):
1218 artak 1465 return self.solve_prec(r)
1219    
1220    
1221 gross 1414 def __Aprod(self,p):
1222     # return BA^-1B*p
1223     #solve Av =-B^*p as Av =f-Az-B^*p
1224 gross 1469 v=self.solve_A(self.__z,-p)
1225     return ArithmeticTuple(v, self.B(v))
1226 gross 1414
1227 artak 1517 def __Aprod2(self,p):
1228 artak 1465 # return BA^-1B*p
1229     #solve Av =-B^*p as Av =f-Az-B^*p
1230 gross 1469 v=self.solve_A(self.__z,-p)
1231     return self.B(v)
1232 gross 1414
1233 gross 867 class SaddlePointProblem(object):
1234     """
1235     This implements a solver for a saddlepoint problem
1236    
1237 gross 877 M{f(u,p)=0}
1238     M{g(u)=0}
1239 gross 867
1240     for u and p. The problem is solved with an inexact Uszawa scheme for p:
1241    
1242 ksteube 990 M{Q_f (u^{k+1}-u^{k}) = - f(u^{k},p^{k})}
1243 gross 877 M{Q_g (p^{k+1}-p^{k}) = g(u^{k+1})}
1244 gross 867
1245     where Q_f is an approximation of the Jacobiean A_f of f with respect to u and Q_f is an approximation of
1246     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'
1247     Q_g can be difficult, non-linear conjugate gradient method is applied to solve for p, so Q_g plays
1248     in fact the role of a preconditioner.
1249     """
1250     def __init__(self,verbose=False,*args):
1251     """
1252     initializes the problem
1253    
1254 ksteube 990 @param verbose: switches on the printing out some information
1255 gross 867 @type verbose: C{bool}
1256     @note: this method may be overwritten by a particular saddle point problem
1257     """
1258 gross 1107 if not isinstance(verbose,bool):
1259     raise TypeError("verbose needs to be of type bool.")
1260 gross 1106 self.__verbose=verbose
1261 gross 877 self.relaxation=1.
1262 gross 867
1263     def trace(self,text):
1264     """
1265     prints text if verbose has been set
1266    
1267 ksteube 990 @param text: a text message
1268 gross 867 @type text: C{str}
1269     """
1270 artak 1489 if self.__verbose: print "#s: #s"%(str(self),text)
1271 gross 867
1272 gross 873 def solve_f(self,u,p,tol=1.e-8):
1273 gross 867 """
1274     solves
1275    
1276     A_f du = f(u,p)
1277    
1278     with tolerance C{tol} and return du. A_f is Jacobiean of f with respect to u.
1279    
1280     @param u: current approximation of u
1281     @type u: L{escript.Data}
1282     @param p: current approximation of p
1283     @type p: L{escript.Data}
1284 gross 873 @param tol: tolerance expected for du
1285 gross 867 @type tol: C{float}
1286     @return: increment du
1287     @rtype: L{escript.Data}
1288     @note: this method has to be overwritten by a particular saddle point problem
1289     """
1290     pass
1291    
1292 gross 873 def solve_g(self,u,tol=1.e-8):
1293 gross 867 """
1294     solves
1295    
1296     Q_g dp = g(u)
1297    
1298     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.
1299    
1300     @param u: current approximation of u
1301     @type u: L{escript.Data}
1302 gross 873 @param tol: tolerance expected for dp
1303     @type tol: C{float}
1304 gross 867 @return: increment dp
1305     @rtype: L{escript.Data}
1306     @note: this method has to be overwritten by a particular saddle point problem
1307     """
1308     pass
1309    
1310     def inner(self,p0,p1):
1311     """
1312     inner product of p0 and p1 approximating p. Typically this returns integrate(p0*p1)
1313     @return: inner product of p0 and p1
1314     @rtype: C{float}
1315     """
1316     pass
1317    
1318 gross 877 subiter_max=3
1319     def solve(self,u0,p0,tolerance=1.e-6,tolerance_u=None,iter_max=100,accepted_reduction=0.995,relaxation=None):
1320     """
1321     runs the solver
1322 gross 873
1323 gross 877 @param u0: initial guess for C{u}
1324     @type u0: L{esys.escript.Data}
1325     @param p0: initial guess for C{p}
1326     @type p0: L{esys.escript.Data}
1327     @param tolerance: tolerance for relative error in C{u} and C{p}
1328     @type tolerance: positive C{float}
1329     @param tolerance_u: tolerance for relative error in C{u} if different from C{tolerance}
1330     @type tolerance_u: positive C{float}
1331     @param iter_max: maximum number of iteration steps.
1332     @type iter_max: C{int}
1333     @param accepted_reduction: if the norm g cannot be reduced by C{accepted_reduction} backtracking to adapt the
1334     relaxation factor. If C{accepted_reduction=None} no backtracking is used.
1335     @type accepted_reduction: positive C{float} or C{None}
1336     @param relaxation: initial relaxation factor. If C{relaxation==None}, the last relaxation factor is used.
1337     @type relaxation: C{float} or C{None}
1338     """
1339     tol=1.e-2
1340     if tolerance_u==None: tolerance_u=tolerance
1341     if not relaxation==None: self.relaxation=relaxation
1342     if accepted_reduction ==None:
1343     angle_limit=0.
1344     elif accepted_reduction>=1.:
1345     angle_limit=0.
1346     else:
1347     angle_limit=util.sqrt(1-accepted_reduction**2)
1348     self.iter=0
1349     u=u0
1350     p=p0
1351     #
1352     # initialize things:
1353     #
1354     converged=False
1355     #
1356     # start loop:
1357     #
1358     # initial search direction is g
1359     #
1360     while not converged :
1361     if self.iter>iter_max:
1362     raise ArithmeticError("no convergence after %s steps."%self.iter)
1363     f_new=self.solve_f(u,p,tol)
1364     norm_f_new = util.Lsup(f_new)
1365     u_new=u-f_new
1366     g_new=self.solve_g(u_new,tol)
1367     self.iter+=1
1368     norm_g_new = util.sqrt(self.inner(g_new,g_new))
1369     if norm_f_new==0. and norm_g_new==0.: return u, p
1370     if self.iter>1 and not accepted_reduction==None:
1371     #
1372     # did we manage to reduce the norm of G? I
1373     # if not we start a backtracking procedure
1374     #
1375     # print "new/old norm = ",norm_g_new, norm_g, norm_g_new/norm_g
1376     if norm_g_new > accepted_reduction * norm_g:
1377     sub_iter=0
1378     s=self.relaxation
1379     d=g
1380     g_last=g
1381     self.trace(" start substepping: f = %s, g = %s, relaxation = %s."%(norm_f_new, norm_g_new, s))
1382     while sub_iter < self.subiter_max and norm_g_new > accepted_reduction * norm_g:
1383     dg= g_new-g_last
1384     norm_dg=abs(util.sqrt(self.inner(dg,dg))/self.relaxation)
1385     rad=self.inner(g_new,dg)/self.relaxation
1386     # print " ",sub_iter,": rad, norm_dg:",abs(rad), norm_dg*norm_g_new * angle_limit
1387     # print " ",sub_iter,": rad, norm_dg:",rad, norm_dg, norm_g_new, norm_g
1388     if abs(rad) < norm_dg*norm_g_new * angle_limit:
1389     if sub_iter>0: self.trace(" no further improvements expected from backtracking.")
1390     break
1391     r=self.relaxation
1392     self.relaxation= - rad/norm_dg**2
1393     s+=self.relaxation
1394     #####
1395     # a=g_new+self.relaxation*dg/r
1396     # print "predicted new norm = ",util.sqrt(self.inner(a,a)),util.sqrt(self.inner(g_new,g_new)), self.relaxation
1397     #####
1398     g_last=g_new
1399     p+=self.relaxation*d
1400     f_new=self.solve_f(u,p,tol)
1401     u_new=u-f_new
1402     g_new=self.solve_g(u_new,tol)
1403     self.iter+=1
1404     norm_f_new = util.Lsup(f_new)
1405     norm_g_new = util.sqrt(self.inner(g_new,g_new))
1406     # print " ",sub_iter," new g norm",norm_g_new
1407     self.trace(" %s th sub-step: f = %s, g = %s, relaxation = %s."%(sub_iter, norm_f_new, norm_g_new, s))
1408     #
1409     # can we expect reduction of g?
1410     #
1411     # u_last=u_new
1412     sub_iter+=1
1413     self.relaxation=s
1414     #
1415     # check for convergence:
1416     #
1417     norm_u_new = util.Lsup(u_new)
1418     p_new=p+self.relaxation*g_new
1419     norm_p_new = util.sqrt(self.inner(p_new,p_new))
1420 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))
1421 gross 873
1422 gross 877 if self.iter>1:
1423     dg2=g_new-g
1424     df2=f_new-f
1425     norm_dg2=util.sqrt(self.inner(dg2,dg2))
1426     norm_df2=util.Lsup(df2)
1427     # print norm_g_new, norm_g, norm_dg, norm_p, tolerance
1428     tol_eq_g=tolerance*norm_dg2/(norm_g*abs(self.relaxation))*norm_p_new
1429     tol_eq_f=tolerance_u*norm_df2/norm_f*norm_u_new
1430     if norm_g_new <= tol_eq_g and norm_f_new <= tol_eq_f:
1431     converged=True
1432     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
1433     self.trace("convergence after %s steps."%self.iter)
1434     return u,p
1435     # def solve(self,u0,p0,tolerance=1.e-6,iter_max=10,self.relaxation=1.):
1436     # tol=1.e-2
1437     # iter=0
1438     # converged=False
1439     # u=u0*1.
1440     # p=p0*1.
1441     # while not converged and iter<iter_max:
1442     # du=self.solve_f(u,p,tol)
1443     # u-=du
1444     # norm_du=util.Lsup(du)
1445     # norm_u=util.Lsup(u)
1446     #
1447     # dp=self.relaxation*self.solve_g(u,tol)
1448     # p+=dp
1449     # norm_dp=util.sqrt(self.inner(dp,dp))
1450     # norm_p=util.sqrt(self.inner(p,p))
1451     # 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)
1452     # iter+=1
1453     #
1454     # converged = (norm_du <= tolerance*norm_u) and (norm_dp <= tolerance*norm_p)
1455     # if converged:
1456     # print "convergence after %s steps."%iter
1457     # else:
1458     # raise ArithmeticError("no convergence after %s steps."%iter)
1459     #
1460     # return u,p
1461 gross 873
1462 ksteube 1312 def MaskFromBoundaryTag(function_space,*tags):
1463     """
1464     create a mask on the given function space which one for samples
1465     that touch the boundary tagged by tags.
1466    
1467     usage: m=MaskFromBoundaryTag(Solution(domain),"left", "right")
1468    
1469     @param function_space: a given function space
1470     @type function_space: L{escript.FunctionSpace}
1471     @param tags: boundray tags
1472     @type tags: C{str}
1473     @return: a mask which marks samples used by C{function_space} that are touching the
1474     boundary tagged by any of the given tags.
1475     @rtype: L{escript.Data} of rank 0
1476     """
1477     pde=linearPDEs.LinearPDE(function_space.getDomain(),numEquations=1, numSolutions=1)
1478     d=escript.Scalar(0.,escript.FunctionOnBoundary(function_space.getDomain()))
1479     for t in tags: d.setTaggedValue(t,1.)
1480     pde.setValue(y=d)
1481     out=util.whereNonZero(pde.getRightHandSide())
1482     if out.getFunctionSpace() == function_space:
1483     return out
1484     else:
1485     return util.whereNonZero(util.interpolate(out,function_space))
1486    
1487 gross 1414
1488 artak 1465

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