/[escript]/trunk/escript/py_src/pdetools.py
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Revision 3675 - (hide annotations)
Thu Nov 17 00:53:38 2011 UTC (8 years, 3 months ago) by jfenwick
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pasowrap joins the trunk.

1 ksteube 1809
2     ########################################################
3 ksteube 1312 #
4 jfenwick 2881 # Copyright (c) 2003-2010 by University of Queensland
5 ksteube 1809 # Earth Systems Science Computational Center (ESSCC)
6     # http://www.uq.edu.au/esscc
7 ksteube 1312 #
8 ksteube 1809 # Primary Business: Queensland, Australia
9     # Licensed under the Open Software License version 3.0
10     # http://www.opensource.org/licenses/osl-3.0.php
11 ksteube 1312 #
12 ksteube 1809 ########################################################
13 jgs 121
14 jfenwick 2881 __copyright__="""Copyright (c) 2003-2010 by University of Queensland
15 ksteube 1809 Earth Systems Science Computational Center (ESSCC)
16     http://www.uq.edu.au/esscc
17     Primary Business: Queensland, Australia"""
18     __license__="""Licensed under the Open Software License version 3.0
19     http://www.opensource.org/licenses/osl-3.0.php"""
20 jfenwick 2344 __url__="https://launchpad.net/escript-finley"
21 ksteube 1809
22 jgs 121 """
23 caltinay 2169 Provides some tools related to PDEs.
24 jgs 121
25 jgs 149 Currently includes:
26 caltinay 2169 - Projector - to project a discontinuous function onto a continuous function
27 gross 351 - Locator - to trace values in data objects at a certain location
28 caltinay 2169 - TimeIntegrationManager - to handle extrapolation in time
29 gross 867 - SaddlePointProblem - solver for Saddle point problems using the inexact uszawa scheme
30 gross 637
31 jfenwick 2625 :var __author__: name of author
32     :var __copyright__: copyrights
33     :var __license__: licence agreement
34     :var __url__: url entry point on documentation
35     :var __version__: version
36     :var __date__: date of the version
37 jgs 121 """
38    
39 gross 637 __author__="Lutz Gross, l.gross@uq.edu.au"
40 elspeth 609
41 gross 637
42 jgs 149 import escript
43     import linearPDEs
44 jfenwick 2455 import numpy
45 jgs 149 import util
46 ksteube 1312 import math
47 jgs 121
48 gross 351 class TimeIntegrationManager:
49     """
50 caltinay 2169 A simple mechanism to manage time dependend values.
51 gross 351
52 caltinay 2169 Typical usage is::
53 gross 351
54 gross 720 dt=0.1 # time increment
55     tm=TimeIntegrationManager(inital_value,p=1)
56     while t<1.
57     v_guess=tm.extrapolate(dt) # extrapolate to t+dt
58     v=...
59     tm.checkin(dt,v)
60     t+=dt
61 gross 351
62 jfenwick 2625 :note: currently only p=1 is supported.
63 gross 351 """
64     def __init__(self,*inital_values,**kwargs):
65     """
66 jfenwick 2625 Sets up the value manager where ``inital_values`` are the initial values
67 caltinay 2169 and p is the order used for extrapolation.
68 gross 351 """
69     if kwargs.has_key("p"):
70     self.__p=kwargs["p"]
71     else:
72     self.__p=1
73     if kwargs.has_key("time"):
74     self.__t=kwargs["time"]
75     else:
76     self.__t=0.
77     self.__v_mem=[inital_values]
78     self.__order=0
79     self.__dt_mem=[]
80     self.__num_val=len(inital_values)
81    
82     def getTime(self):
83     return self.__t
84 caltinay 2169
85 gross 396 def getValue(self):
86 gross 409 out=self.__v_mem[0]
87     if len(out)==1:
88     return out[0]
89     else:
90     return out
91    
92 gross 351 def checkin(self,dt,*values):
93     """
94 caltinay 2169 Adds new values to the manager. The p+1 last values are lost.
95 gross 351 """
96     o=min(self.__order+1,self.__p)
97     self.__order=min(self.__order+1,self.__p)
98     v_mem_new=[values]
99     dt_mem_new=[dt]
100     for i in range(o-1):
101     v_mem_new.append(self.__v_mem[i])
102     dt_mem_new.append(self.__dt_mem[i])
103     v_mem_new.append(self.__v_mem[o-1])
104     self.__order=o
105     self.__v_mem=v_mem_new
106     self.__dt_mem=dt_mem_new
107     self.__t+=dt
108    
109     def extrapolate(self,dt):
110     """
111 jfenwick 2625 Extrapolates to ``dt`` forward in time.
112 gross 351 """
113     if self.__order==0:
114     out=self.__v_mem[0]
115     else:
116     out=[]
117     for i in range(self.__num_val):
118     out.append((1.+dt/self.__dt_mem[0])*self.__v_mem[0][i]-dt/self.__dt_mem[0]*self.__v_mem[1][i])
119    
120     if len(out)==0:
121     return None
122     elif len(out)==1:
123     return out[0]
124     else:
125     return out
126    
127 caltinay 2169
128 jgs 121 class Projector:
129 jgs 149 """
130 caltinay 2169 The Projector is a factory which projects a discontinuous function onto a
131     continuous function on a given domain.
132 jgs 149 """
133 caltinay 2169 def __init__(self, domain, reduce=True, fast=True):
134 jgs 121 """
135 caltinay 2169 Creates a continuous function space projector for a domain.
136 jgs 121
137 jfenwick 2625 :param domain: Domain of the projection.
138     :param reduce: Flag to reduce projection order
139     :param fast: Flag to use a fast method based on matrix lumping
140 jgs 121 """
141 jgs 149 self.__pde = linearPDEs.LinearPDE(domain)
142 jgs 148 if fast:
143 gross 2474 self.__pde.getSolverOptions().setSolverMethod(linearPDEs.SolverOptions.LUMPING)
144 jgs 121 self.__pde.setSymmetryOn()
145     self.__pde.setReducedOrderTo(reduce)
146     self.__pde.setValue(D = 1.)
147 ksteube 1312 return
148 gross 2474 def getSolverOptions(self):
149 artak 2693 """
150     Returns the solver options of the PDE solver.
151    
152     :rtype: `linearPDEs.SolverOptions`
153     """
154     return self.__pde.getSolverOptions()
155 jgs 121
156 gross 2867 def getValue(self, input_data):
157     """
158     Projects ``input_data`` onto a continuous function.
159    
160     :param input_data: the data to be projected
161     """
162     return self(input_data)
163    
164 jgs 121 def __call__(self, input_data):
165     """
166 jfenwick 2625 Projects ``input_data`` onto a continuous function.
167 jgs 121
168 jfenwick 2625 :param input_data: the data to be projected
169 jgs 121 """
170 gross 525 out=escript.Data(0.,input_data.getShape(),self.__pde.getFunctionSpaceForSolution())
171 gross 1122 self.__pde.setValue(Y = escript.Data(), Y_reduced = escript.Data())
172 jgs 121 if input_data.getRank()==0:
173     self.__pde.setValue(Y = input_data)
174     out=self.__pde.getSolution()
175     elif input_data.getRank()==1:
176     for i0 in range(input_data.getShape()[0]):
177     self.__pde.setValue(Y = input_data[i0])
178     out[i0]=self.__pde.getSolution()
179     elif input_data.getRank()==2:
180     for i0 in range(input_data.getShape()[0]):
181     for i1 in range(input_data.getShape()[1]):
182     self.__pde.setValue(Y = input_data[i0,i1])
183     out[i0,i1]=self.__pde.getSolution()
184     elif input_data.getRank()==3:
185     for i0 in range(input_data.getShape()[0]):
186     for i1 in range(input_data.getShape()[1]):
187     for i2 in range(input_data.getShape()[2]):
188     self.__pde.setValue(Y = input_data[i0,i1,i2])
189     out[i0,i1,i2]=self.__pde.getSolution()
190     else:
191     for i0 in range(input_data.getShape()[0]):
192     for i1 in range(input_data.getShape()[1]):
193     for i2 in range(input_data.getShape()[2]):
194     for i3 in range(input_data.getShape()[3]):
195     self.__pde.setValue(Y = input_data[i0,i1,i2,i3])
196     out[i0,i1,i2,i3]=self.__pde.getSolution()
197     return out
198    
199 gross 525 class NoPDE:
200     """
201 caltinay 2169 Solves the following problem for u:
202 jgs 121
203 jfenwick 2625 *kronecker[i,j]*D[j]*u[j]=Y[i]*
204 gross 525
205     with constraint
206    
207 jfenwick 2625 *u[j]=r[j]* where *q[j]>0*
208 gross 525
209 jfenwick 2625 where *D*, *Y*, *r* and *q* are given functions of rank 1.
210 gross 525
211     In the case of scalars this takes the form
212    
213 jfenwick 2625 *D*u=Y*
214 gross 525
215     with constraint
216    
217 jfenwick 2625 *u=r* where *q>0*
218 gross 525
219 jfenwick 2625 where *D*, *Y*, *r* and *q* are given scalar functions.
220 gross 525
221 caltinay 2169 The constraint overwrites any other condition.
222 gross 525
223 jfenwick 2625 :note: This class is similar to the `linearPDEs.LinearPDE` class with
224 caltinay 2169 A=B=C=X=0 but has the intention that all input parameters are given
225 jfenwick 2625 in `Solution` or `ReducedSolution`.
226 gross 525 """
227 caltinay 2169 # The whole thing is a bit strange and I blame Rob Woodcock (CSIRO) for
228     # this.
229 gross 525 def __init__(self,domain,D=None,Y=None,q=None,r=None):
230     """
231 caltinay 2169 Initializes the problem.
232 gross 525
233 jfenwick 2625 :param domain: domain of the PDE
234     :type domain: `Domain`
235     :param D: coefficient of the solution
236     :type D: ``float``, ``int``, ``numpy.ndarray``, `Data`
237     :param Y: right hand side
238     :type Y: ``float``, ``int``, ``numpy.ndarray``, `Data`
239     :param q: location of constraints
240     :type q: ``float``, ``int``, ``numpy.ndarray``, `Data`
241     :param r: value of solution at locations of constraints
242     :type r: ``float``, ``int``, ``numpy.ndarray``, `Data`
243 gross 525 """
244     self.__domain=domain
245     self.__D=D
246     self.__Y=Y
247     self.__q=q
248     self.__r=r
249     self.__u=None
250     self.__function_space=escript.Solution(self.__domain)
251 caltinay 2169
252 gross 525 def setReducedOn(self):
253     """
254 jfenwick 2625 Sets the `FunctionSpace` of the solution to `ReducedSolution`.
255 gross 525 """
256     self.__function_space=escript.ReducedSolution(self.__domain)
257     self.__u=None
258    
259     def setReducedOff(self):
260     """
261 jfenwick 2625 Sets the `FunctionSpace` of the solution to `Solution`.
262 gross 525 """
263     self.__function_space=escript.Solution(self.__domain)
264     self.__u=None
265 caltinay 2169
266 gross 525 def setValue(self,D=None,Y=None,q=None,r=None):
267     """
268 caltinay 2169 Assigns values to the parameters.
269 gross 525
270 jfenwick 2625 :param D: coefficient of the solution
271     :type D: ``float``, ``int``, ``numpy.ndarray``, `Data`
272     :param Y: right hand side
273     :type Y: ``float``, ``int``, ``numpy.ndarray``, `Data`
274     :param q: location of constraints
275     :type q: ``float``, ``int``, ``numpy.ndarray``, `Data`
276     :param r: value of solution at locations of constraints
277     :type r: ``float``, ``int``, ``numpy.ndarray``, `Data`
278 gross 525 """
279     if not D==None:
280     self.__D=D
281     self.__u=None
282     if not Y==None:
283     self.__Y=Y
284     self.__u=None
285     if not q==None:
286     self.__q=q
287     self.__u=None
288     if not r==None:
289     self.__r=r
290     self.__u=None
291    
292     def getSolution(self):
293     """
294 caltinay 2169 Returns the solution.
295    
296 jfenwick 2625 :return: the solution of the problem
297     :rtype: `Data` object in the `FunctionSpace` `Solution` or
298     `ReducedSolution`
299 gross 525 """
300     if self.__u==None:
301     if self.__D==None:
302     raise ValueError,"coefficient D is undefined"
303     D=escript.Data(self.__D,self.__function_space)
304     if D.getRank()>1:
305     raise ValueError,"coefficient D must have rank 0 or 1"
306     if self.__Y==None:
307     self.__u=escript.Data(0.,D.getShape(),self.__function_space)
308     else:
309     self.__u=util.quotient(self.__Y,D)
310     if not self.__q==None:
311     q=util.wherePositive(escript.Data(self.__q,self.__function_space))
312     self.__u*=(1.-q)
313     if not self.__r==None: self.__u+=q*self.__r
314     return self.__u
315 caltinay 2169
316 jgs 147 class Locator:
317     """
318 caltinay 2169 Locator provides access to the values of data objects at a given spatial
319     coordinate x.
320    
321 jgs 149 In fact, a Locator object finds the sample in the set of samples of a
322 caltinay 2169 given function space or domain which is closest to the given point x.
323 jgs 147 """
324    
325 jfenwick 2455 def __init__(self,where,x=numpy.zeros((3,))):
326 jgs 149 """
327 caltinay 2169 Initializes a Locator to access values in Data objects on the Doamin
328     or FunctionSpace for the sample point which is closest to the given
329     point x.
330 gross 880
331 jfenwick 2625 :param where: function space
332     :type where: `escript.FunctionSpace`
333     :param x: location(s) of the Locator
334     :type x: ``numpy.ndarray`` or ``list`` of ``numpy.ndarray``
335 jgs 149 """
336     if isinstance(where,escript.FunctionSpace):
337 jgs 147 self.__function_space=where
338 jgs 121 else:
339 jgs 149 self.__function_space=escript.ContinuousFunction(where)
340 gross 2484 iterative=False
341 gross 880 if isinstance(x, list):
342 gross 2484 if len(x)==0:
343 gross 2948 raise ValueError, "At least one point must be given."
344 gross 2484 try:
345     iter(x[0])
346     iterative=True
347     except TypeError:
348     iterative=False
349 gross 2954 xxx=self.__function_space.getX()
350 gross 2484 if iterative:
351 gross 880 self.__id=[]
352     for p in x:
353 gross 2948 self.__id.append(util.length(xxx-p[:self.__function_space.getDim()]).minGlobalDataPoint())
354 gross 880 else:
355 gross 2948 self.__id=util.length(xxx-x[:self.__function_space.getDim()]).minGlobalDataPoint()
356 jgs 121
357 jgs 147 def __str__(self):
358 jgs 149 """
359     Returns the coordinates of the Locator as a string.
360     """
361 gross 880 x=self.getX()
362 gross 2676 if isinstance(x,list):
363 gross 880 out="["
364     first=True
365     for xx in x:
366     if not first:
367     out+=","
368     else:
369     first=False
370     out+=str(xx)
371     out+="]>"
372     else:
373     out=str(x)
374     return out
375 jgs 121
376 gross 880 def getX(self):
377     """
378 caltinay 2169 Returns the exact coordinates of the Locator.
379     """
380 gross 880 return self(self.getFunctionSpace().getX())
381    
382 jgs 147 def getFunctionSpace(self):
383 jgs 149 """
384 caltinay 2169 Returns the function space of the Locator.
385     """
386 jgs 147 return self.__function_space
387    
388 gross 880 def getId(self,item=None):
389 jgs 149 """
390 caltinay 2169 Returns the identifier of the location.
391     """
392 gross 880 if item == None:
393     return self.__id
394     else:
395     if isinstance(self.__id,list):
396     return self.__id[item]
397     else:
398     return self.__id
399 jgs 121
400    
401 jgs 147 def __call__(self,data):
402 jgs 149 """
403 caltinay 2169 Returns the value of data at the Locator of a Data object.
404     """
405 jgs 147 return self.getValue(data)
406 jgs 121
407 jgs 147 def getValue(self,data):
408 jgs 149 """
409 jfenwick 2625 Returns the value of ``data`` at the Locator if ``data`` is a `Data`
410 caltinay 2169 object otherwise the object is returned.
411     """
412 jfenwick 3675 print "Norris"
413 jgs 149 if isinstance(data,escript.Data):
414 jfenwick 2517 dat=util.interpolate(data,self.getFunctionSpace())
415 gross 880 id=self.getId()
416     r=data.getRank()
417     if isinstance(id,list):
418     out=[]
419 jfenwick 3675 print "Zorro"
420 gross 880 for i in id:
421 jfenwick 2517 o=numpy.array(dat.getTupleForGlobalDataPoint(*i))
422 gross 880 if data.getRank()==0:
423     out.append(o[0])
424     else:
425     out.append(o)
426     return out
427 jgs 147 else:
428 jfenwick 3675 print "Thom"
429 jfenwick 2517 out=numpy.array(dat.getTupleForGlobalDataPoint(*id))
430 gross 880 if data.getRank()==0:
431     return out[0]
432     else:
433     return out
434 jgs 147 else:
435     return data
436 jfenwick 3574
437 jfenwick 3584 # def setValue(self, data, v):
438     # """
439     # Sets the value of the ``data`` at the Locator.
440     # """
441     # data.expand() # Need to ensure that this is done globally
442     # if isinstance(data, escript.Data):
443     # id=self.getId()
444     # if isinstance(id, list):
445     # for i in id:
446     # data._setTupleForGlobalDataPoint(i[1], i[0], v)
447     # else:
448     # data._setTupleForGlobalDataPoint(id[1], id[0], v)
449     # else:
450     # raise TypeError, "setValue: Invalid argument type."
451 jgs 149
452 jfenwick 2745
453     def getInfLocator(arg):
454     """
455     Return a Locator for a point with the inf value over all arg.
456     """
457     if not isinstance(arg, escript.Data):
458     raise TypeError,"getInfLocator: Unknown argument type."
459     a_inf=util.inf(arg)
460     loc=util.length(arg-a_inf).minGlobalDataPoint() # This gives us the location but not coords
461     x=arg.getFunctionSpace().getX()
462     x_min=x.getTupleForGlobalDataPoint(*loc)
463     return Locator(arg.getFunctionSpace(),x_min)
464    
465     def getSupLocator(arg):
466     """
467     Return a Locator for a point with the sup value over all arg.
468     """
469     if not isinstance(arg, escript.Data):
470     raise TypeError,"getInfLocator: Unknown argument type."
471     a_inf=util.sup(arg)
472     loc=util.length(arg-a_inf).minGlobalDataPoint() # This gives us the location but not coords
473     x=arg.getFunctionSpace().getX()
474     x_min=x.getTupleForGlobalDataPoint(*loc)
475     return Locator(arg.getFunctionSpace(),x_min)
476    
477    
478 ksteube 1312 class SolverSchemeException(Exception):
479     """
480 caltinay 2169 This is a generic exception thrown by solvers.
481 ksteube 1312 """
482     pass
483    
484     class IndefinitePreconditioner(SolverSchemeException):
485     """
486 caltinay 2169 Exception thrown if the preconditioner is not positive definite.
487 ksteube 1312 """
488     pass
489 caltinay 2169
490 ksteube 1312 class MaxIterReached(SolverSchemeException):
491     """
492 caltinay 2169 Exception thrown if the maximum number of iteration steps is reached.
493 ksteube 1312 """
494     pass
495 caltinay 2169
496 gross 2100 class CorrectionFailed(SolverSchemeException):
497     """
498 caltinay 2169 Exception thrown if no convergence has been achieved in the solution
499     correction scheme.
500 gross 2100 """
501     pass
502 caltinay 2169
503 ksteube 1312 class IterationBreakDown(SolverSchemeException):
504     """
505 caltinay 2169 Exception thrown if the iteration scheme encountered an incurable breakdown.
506 ksteube 1312 """
507     pass
508 caltinay 2169
509 ksteube 1312 class NegativeNorm(SolverSchemeException):
510     """
511 caltinay 2169 Exception thrown if a norm calculation returns a negative norm.
512 ksteube 1312 """
513     pass
514    
515 gross 2156 def PCG(r, Aprod, x, Msolve, bilinearform, atol=0, rtol=1.e-8, iter_max=100, initial_guess=True, verbose=False):
516 ksteube 1312 """
517 caltinay 2169 Solver for
518 ksteube 1312
519 jfenwick 2625 *Ax=b*
520 ksteube 1312
521 caltinay 2169 with a symmetric and positive definite operator A (more details required!).
522     It uses the conjugate gradient method with preconditioner M providing an
523     approximation of A.
524 ksteube 1312
525 caltinay 2169 The iteration is terminated if
526 ksteube 1312
527 jfenwick 2625 *|r| <= atol+rtol*|r0|*
528 gross 2156
529 jfenwick 2625 where *r0* is the initial residual and *|.|* is the energy norm. In fact
530 gross 2156
531 jfenwick 2625 *|r| = sqrt( bilinearform(Msolve(r),r))*
532 gross 2156
533 ksteube 1312 For details on the preconditioned conjugate gradient method see the book:
534    
535 caltinay 2169 I{Templates for the Solution of Linear Systems by R. Barrett, M. Berry,
536     T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo,
537     C. Romine, and H. van der Vorst}.
538 ksteube 1312
539 jfenwick 2625 :param r: initial residual *r=b-Ax*. ``r`` is altered.
540     :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
541     :param x: an initial guess for the solution
542     :type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
543     :param Aprod: returns the value Ax
544     :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
545     argument ``x``. The returned object needs to be of the same type
546     like argument ``r``.
547     :param Msolve: solves Mx=r
548     :type Msolve: function ``Msolve(r)`` where ``r`` is of the same type like
549     argument ``r``. The returned object needs to be of the same
550     type like argument ``x``.
551     :param bilinearform: inner product ``<x,r>``
552     :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
553     type like argument ``x`` and ``r`` is. The returned value
554     is a ``float``.
555     :param atol: absolute tolerance
556     :type atol: non-negative ``float``
557     :param rtol: relative tolerance
558     :type rtol: non-negative ``float``
559     :param iter_max: maximum number of iteration steps
560     :type iter_max: ``int``
561     :return: the solution approximation and the corresponding residual
562     :rtype: ``tuple``
563     :warning: ``r`` and ``x`` are altered.
564 ksteube 1312 """
565     iter=0
566     rhat=Msolve(r)
567 caltinay 2169 d = rhat
568 ksteube 1312 rhat_dot_r = bilinearform(rhat, r)
569 gross 1330 if rhat_dot_r<0: raise NegativeNorm,"negative norm."
570 gross 2156 norm_r0=math.sqrt(rhat_dot_r)
571     atol2=atol+rtol*norm_r0
572     if atol2<=0:
573     raise ValueError,"Non-positive tolarance."
574     atol2=max(atol2, 100. * util.EPSILON * norm_r0)
575 ksteube 1312
576 gross 2156 if verbose: print "PCG: initial residual norm = %e (absolute tolerance = %e)"%(norm_r0, atol2)
577    
578 caltinay 2169
579 gross 2156 while not math.sqrt(rhat_dot_r) <= atol2:
580 gross 1330 iter+=1
581 ksteube 1312 if iter >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
582    
583     q=Aprod(d)
584     alpha = rhat_dot_r / bilinearform(d, q)
585     x += alpha * d
586 jfenwick 2455 if isinstance(q,ArithmeticTuple):
587     r += q * (-alpha) # Doing it the other way calls the float64.__mul__ not AT.__rmul__
588     else:
589     r += (-alpha) * q
590 ksteube 1312 rhat=Msolve(r)
591     rhat_dot_r_new = bilinearform(rhat, r)
592     beta = rhat_dot_r_new / rhat_dot_r
593     rhat+=beta * d
594     d=rhat
595    
596     rhat_dot_r = rhat_dot_r_new
597 gross 1330 if rhat_dot_r<0: raise NegativeNorm,"negative norm."
598 gross 2100 if verbose: print "PCG: iteration step %s: residual norm = %e"%(iter, math.sqrt(rhat_dot_r))
599     if verbose: print "PCG: tolerance reached after %s steps."%iter
600 gross 2264 return x,r,math.sqrt(rhat_dot_r)
601 ksteube 1312
602 gross 1878 class Defect(object):
603     """
604 caltinay 2169 Defines a non-linear defect F(x) of a variable x.
605 gross 1878 """
606     def __init__(self):
607     """
608 caltinay 2169 Initializes defect.
609 gross 1878 """
610     self.setDerivativeIncrementLength()
611 artak 1465
612 gross 1878 def bilinearform(self, x0, x1):
613     """
614 caltinay 2169 Returns the inner product of x0 and x1
615    
616 jfenwick 2625 :param x0: value for x0
617     :param x1: value for x1
618     :return: the inner product of x0 and x1
619     :rtype: ``float``
620 gross 1878 """
621     return 0
622 caltinay 2169
623 gross 1878 def norm(self,x):
624     """
625 jfenwick 2625 Returns the norm of argument ``x``.
626 caltinay 2169
627 jfenwick 2625 :param x: a value
628     :return: norm of argument x
629     :rtype: ``float``
630     :note: by default ``sqrt(self.bilinearform(x,x)`` is returned.
631 gross 1878 """
632     s=self.bilinearform(x,x)
633     if s<0: raise NegativeNorm,"negative norm."
634     return math.sqrt(s)
635 artak 1465
636 gross 1878 def eval(self,x):
637     """
638 jfenwick 2625 Returns the value F of a given ``x``.
639 gross 1878
640 jfenwick 2625 :param x: value for which the defect ``F`` is evaluated
641     :return: value of the defect at ``x``
642 gross 1878 """
643     return 0
644    
645     def __call__(self,x):
646     return self.eval(x)
647    
648 gross 2683 def setDerivativeIncrementLength(self,inc=1000.*math.sqrt(util.EPSILON)):
649 gross 1878 """
650 caltinay 2169 Sets the relative length of the increment used to approximate the
651     derivative of the defect. The increment is inc*norm(x)/norm(v)*v in the
652     direction of v with x as a starting point.
653 gross 1878
654 jfenwick 2625 :param inc: relative increment length
655     :type inc: positive ``float``
656 gross 1878 """
657     if inc<=0: raise ValueError,"positive increment required."
658     self.__inc=inc
659    
660     def getDerivativeIncrementLength(self):
661     """
662 caltinay 2169 Returns the relative increment length used to approximate the
663     derivative of the defect.
664 jfenwick 2625 :return: value of the defect at ``x``
665     :rtype: positive ``float``
666 gross 1878 """
667     return self.__inc
668    
669     def derivative(self, F0, x0, v, v_is_normalised=True):
670     """
671 jfenwick 2625 Returns the directional derivative at ``x0`` in the direction of ``v``.
672 gross 1878
673 jfenwick 2625 :param F0: value of this defect at x0
674     :param x0: value at which derivative is calculated
675     :param v: direction
676     :param v_is_normalised: True to indicate that ``v`` is nomalized
677 caltinay 2169 (self.norm(v)=0)
678 jfenwick 2625 :return: derivative of this defect at x0 in the direction of ``v``
679     :note: by default numerical evaluation (self.eval(x0+eps*v)-F0)/eps is
680 caltinay 2169 used but this method maybe overwritten to use exact evaluation.
681 gross 1878 """
682     normx=self.norm(x0)
683     if normx>0:
684     epsnew = self.getDerivativeIncrementLength() * normx
685     else:
686     epsnew = self.getDerivativeIncrementLength()
687     if not v_is_normalised:
688     normv=self.norm(v)
689     if normv<=0:
690     return F0*0
691     else:
692     epsnew /= normv
693     F1=self.eval(x0 + epsnew * v)
694     return (F1-F0)/epsnew
695    
696 caltinay 2169 ######################################
697 gross 2719 def NewtonGMRES(defect, x, iter_max=100, sub_iter_max=20, atol=0,rtol=1.e-4, subtol_max=0.5, gamma=0.9, verbose=False):
698 gross 1878 """
699 jfenwick 2625 Solves a non-linear problem *F(x)=0* for unknown *x* using the stopping
700 caltinay 2169 criterion:
701 gross 1878
702 jfenwick 2625 *norm(F(x) <= atol + rtol * norm(F(x0)*
703 caltinay 2169
704 jfenwick 2625 where *x0* is the initial guess.
705 gross 1878
706 jfenwick 2625 :param defect: object defining the function *F*. ``defect.norm`` defines the
707     *norm* used in the stopping criterion.
708     :type defect: `Defect`
709     :param x: initial guess for the solution, ``x`` is altered.
710     :type x: any object type allowing basic operations such as
711     ``numpy.ndarray``, `Data`
712     :param iter_max: maximum number of iteration steps
713     :type iter_max: positive ``int``
714     :param sub_iter_max: maximum number of inner iteration steps
715     :type sub_iter_max: positive ``int``
716     :param atol: absolute tolerance for the solution
717     :type atol: positive ``float``
718     :param rtol: relative tolerance for the solution
719     :type rtol: positive ``float``
720     :param gamma: tolerance safety factor for inner iteration
721     :type gamma: positive ``float``, less than 1
722 gross 2719 :param subtol_max: upper bound for inner tolerance
723     :type subtol_max: positive ``float``, less than 1
724 jfenwick 2625 :return: an approximation of the solution with the desired accuracy
725     :rtype: same type as the initial guess
726 gross 1878 """
727     lmaxit=iter_max
728     if atol<0: raise ValueError,"atol needs to be non-negative."
729     if rtol<0: raise ValueError,"rtol needs to be non-negative."
730     if rtol+atol<=0: raise ValueError,"rtol or atol needs to be non-negative."
731     if gamma<=0 or gamma>=1: raise ValueError,"tolerance safety factor for inner iteration (gamma =%s) needs to be positive and less than 1."%gamma
732 gross 2719 if subtol_max<=0 or subtol_max>=1: raise ValueError,"upper bound for inner tolerance for inner iteration (subtol_max =%s) needs to be positive and less than 1."%subtol_max
733 gross 1878
734     F=defect(x)
735     fnrm=defect.norm(F)
736     stop_tol=atol + rtol*fnrm
737 gross 2719 subtol=subtol_max
738 gross 1878 if verbose: print "NewtonGMRES: initial residual = %e."%fnrm
739 gross 2719 if verbose: print " tolerance = %e."%subtol
740 gross 1878 iter=1
741     #
742     # main iteration loop
743     #
744     while not fnrm<=stop_tol:
745 caltinay 2169 if iter >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
746 gross 1878 #
747 gross 2719 # adjust subtol_
748 gross 1878 #
749     if iter > 1:
750     rat=fnrm/fnrmo
751 gross 2719 subtol_old=subtol
752     subtol=gamma*rat**2
753     if gamma*subtol_old**2 > .1: subtol=max(subtol,gamma*subtol_old**2)
754     subtol=max(min(subtol,subtol_max), .5*stop_tol/fnrm)
755 gross 1878 #
756     # calculate newton increment xc
757     # if iter_max in __FDGMRES is reached MaxIterReached is thrown
758     # if iter_restart -1 is returned as sub_iter
759     # if atol is reached sub_iter returns the numer of steps performed to get there
760 caltinay 2169 #
761     #
762 gross 2719 if verbose: print " subiteration (GMRES) is called with relative tolerance %e."%subtol
763 gross 1878 try:
764 gross 2719 xc, sub_iter=__FDGMRES(F, defect, x, subtol*fnrm, iter_max=iter_max-iter, iter_restart=sub_iter_max)
765 gross 1878 except MaxIterReached:
766     raise MaxIterReached,"maximum number of %s steps reached."%iter_max
767     if sub_iter<0:
768     iter+=sub_iter_max
769     else:
770     iter+=sub_iter
771     # ====
772     x+=xc
773     F=defect(x)
774     iter+=1
775     fnrmo, fnrm=fnrm, defect.norm(F)
776     if verbose: print " step %s: residual %e."%(iter,fnrm)
777     if verbose: print "NewtonGMRES: completed after %s steps."%iter
778     return x
779    
780     def __givapp(c,s,vin):
781     """
782 caltinay 2169 Applies a sequence of Givens rotations (c,s) recursively to the vector
783 jfenwick 2625 ``vin``
784 caltinay 2169
785 jfenwick 2625 :warning: ``vin`` is altered.
786 gross 1878 """
787 caltinay 2169 vrot=vin
788 gross 1467 if isinstance(c,float):
789 artak 2303 vrot=[c*vrot[0]-s*vrot[1],s*vrot[0]+c*vrot[1]]
790 gross 1467 else:
791     for i in range(len(c)):
792     w1=c[i]*vrot[i]-s[i]*vrot[i+1]
793     w2=s[i]*vrot[i]+c[i]*vrot[i+1]
794 gross 2284 vrot[i]=w1
795     vrot[i+1]=w2
796 artak 1465 return vrot
797    
798 gross 1878 def __FDGMRES(F0, defect, x0, atol, iter_max=100, iter_restart=20):
799 jfenwick 2455 h=numpy.zeros((iter_restart,iter_restart),numpy.float64)
800     c=numpy.zeros(iter_restart,numpy.float64)
801     s=numpy.zeros(iter_restart,numpy.float64)
802     g=numpy.zeros(iter_restart,numpy.float64)
803 artak 1465 v=[]
804    
805 gross 1878 rho=defect.norm(F0)
806     if rho<=0.: return x0*0
807 caltinay 2169
808 gross 1878 v.append(-F0/rho)
809 artak 1465 g[0]=rho
810 gross 1878 iter=0
811     while rho > atol and iter<iter_restart-1:
812 caltinay 2169 if iter >= iter_max:
813     raise MaxIterReached,"maximum number of %s steps reached."%iter_max
814 artak 1557
815 gross 1878 p=defect.derivative(F0,x0,v[iter], v_is_normalised=True)
816 caltinay 2169 v.append(p)
817 artak 1465
818 caltinay 2169 v_norm1=defect.norm(v[iter+1])
819 artak 1465
820 caltinay 2169 # Modified Gram-Schmidt
821     for j in range(iter+1):
822     h[j,iter]=defect.bilinearform(v[j],v[iter+1])
823     v[iter+1]-=h[j,iter]*v[j]
824 artak 1465
825 caltinay 2169 h[iter+1,iter]=defect.norm(v[iter+1])
826     v_norm2=h[iter+1,iter]
827    
828 gross 1878 # Reorthogonalize if needed
829 caltinay 2169 if v_norm1 + 0.001*v_norm2 == v_norm1: #Brown/Hindmarsh condition (default)
830     for j in range(iter+1):
831     hr=defect.bilinearform(v[j],v[iter+1])
832     h[j,iter]=h[j,iter]+hr
833     v[iter+1] -= hr*v[j]
834 artak 1465
835 caltinay 2169 v_norm2=defect.norm(v[iter+1])
836     h[iter+1,iter]=v_norm2
837     # watch out for happy breakdown
838 artak 1550 if not v_norm2 == 0:
839 caltinay 2169 v[iter+1]=v[iter+1]/h[iter+1,iter]
840 artak 1465
841 gross 1878 # Form and store the information for the new Givens rotation
842 caltinay 2169 if iter > 0 :
843 jfenwick 2455 hhat=numpy.zeros(iter+1,numpy.float64)
844 caltinay 2169 for i in range(iter+1) : hhat[i]=h[i,iter]
845     hhat=__givapp(c[0:iter],s[0:iter],hhat);
846     for i in range(iter+1) : h[i,iter]=hhat[i]
847 artak 1465
848 caltinay 2169 mu=math.sqrt(h[iter,iter]*h[iter,iter]+h[iter+1,iter]*h[iter+1,iter])
849 artak 1557
850 caltinay 2169 if mu!=0 :
851     c[iter]=h[iter,iter]/mu
852     s[iter]=-h[iter+1,iter]/mu
853     h[iter,iter]=c[iter]*h[iter,iter]-s[iter]*h[iter+1,iter]
854     h[iter+1,iter]=0.0
855 artak 2303 gg=__givapp(c[iter],s[iter],[g[iter],g[iter+1]])
856 gross 2284 g[iter]=gg[0]
857     g[iter+1]=gg[1]
858 artak 1465
859 gross 1878 # Update the residual norm
860 artak 1465 rho=abs(g[iter+1])
861 caltinay 2169 iter+=1
862 artak 1465
863 gross 1878 # At this point either iter > iter_max or rho < tol.
864 caltinay 2169 # It's time to compute x and leave.
865     if iter > 0 :
866 jfenwick 2455 y=numpy.zeros(iter,numpy.float64)
867 gross 2100 y[iter-1] = g[iter-1] / h[iter-1,iter-1]
868 caltinay 2169 if iter > 1 :
869     i=iter-2
870 artak 1465 while i>=0 :
871 jfenwick 2455 y[i] = ( g[i] - numpy.dot(h[i,i+1:iter], y[i+1:iter])) / h[i,i]
872 artak 1465 i=i-1
873     xhat=v[iter-1]*y[iter-1]
874     for i in range(iter-1):
875     xhat += v[i]*y[i]
876 caltinay 2169 else :
877 gross 1878 xhat=v[0] * 0
878 artak 1557
879 caltinay 2169 if iter<iter_restart-1:
880 gross 1878 stopped=iter
881 caltinay 2169 else:
882 gross 1878 stopped=-1
883 artak 1465
884 gross 1878 return xhat,stopped
885 artak 1481
886 gross 2261 def GMRES(r, Aprod, x, bilinearform, atol=0, rtol=1.e-8, iter_max=100, iter_restart=20, verbose=False,P_R=None):
887 gross 2156 """
888 caltinay 2169 Solver for
889 gross 2156
890 jfenwick 2625 *Ax=b*
891 gross 2156
892 caltinay 2169 with a general operator A (more details required!).
893 gross 2156 It uses the generalized minimum residual method (GMRES).
894    
895 caltinay 2169 The iteration is terminated if
896 gross 2156
897 jfenwick 2625 *|r| <= atol+rtol*|r0|*
898 gross 2156
899 jfenwick 2625 where *r0* is the initial residual and *|.|* is the energy norm. In fact
900 gross 2156
901 jfenwick 2625 *|r| = sqrt( bilinearform(r,r))*
902 gross 2156
903 jfenwick 2625 :param r: initial residual *r=b-Ax*. ``r`` is altered.
904     :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
905     :param x: an initial guess for the solution
906     :type x: same like ``r``
907     :param Aprod: returns the value Ax
908     :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
909     argument ``x``. The returned object needs to be of the same
910     type like argument ``r``.
911     :param bilinearform: inner product ``<x,r>``
912     :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
913     type like argument ``x`` and ``r``. The returned value is
914     a ``float``.
915     :param atol: absolute tolerance
916     :type atol: non-negative ``float``
917     :param rtol: relative tolerance
918     :type rtol: non-negative ``float``
919     :param iter_max: maximum number of iteration steps
920     :type iter_max: ``int``
921     :param iter_restart: in order to save memory the orthogonalization process
922     is terminated after ``iter_restart`` steps and the
923 caltinay 2169 iteration is restarted.
924 jfenwick 2625 :type iter_restart: ``int``
925     :return: the solution approximation and the corresponding residual
926     :rtype: ``tuple``
927     :warning: ``r`` and ``x`` are altered.
928 gross 2156 """
929 gross 1878 m=iter_restart
930 gross 2156 restarted=False
931 gross 1878 iter=0
932 gross 2100 if rtol>0:
933 gross 2156 r_dot_r = bilinearform(r, r)
934 gross 2100 if r_dot_r<0: raise NegativeNorm,"negative norm."
935     atol2=atol+rtol*math.sqrt(r_dot_r)
936     if verbose: print "GMRES: norm of right hand side = %e (absolute tolerance = %e)"%(math.sqrt(r_dot_r), atol2)
937     else:
938     atol2=atol
939     if verbose: print "GMRES: absolute tolerance = %e"%atol2
940 gross 2156 if atol2<=0:
941     raise ValueError,"Non-positive tolarance."
942 caltinay 2169
943 gross 1878 while True:
944     if iter >= iter_max: raise MaxIterReached,"maximum number of %s steps reached"%iter_max
945 caltinay 2169 if restarted:
946 gross 2156 r2 = r-Aprod(x-x2)
947     else:
948     r2=1*r
949     x2=x*1.
950 gross 2261 x,stopped=_GMRESm(r2, Aprod, x, bilinearform, atol2, iter_max=iter_max-iter, iter_restart=m, verbose=verbose,P_R=P_R)
951 caltinay 2169 iter+=iter_restart
952 gross 1878 if stopped: break
953 gross 2100 if verbose: print "GMRES: restart."
954 gross 2156 restarted=True
955 gross 2251 if verbose: print "GMRES: tolerance has been reached."
956 gross 2156 return x
957 artak 1550
958 gross 2261 def _GMRESm(r, Aprod, x, bilinearform, atol, iter_max=100, iter_restart=20, verbose=False, P_R=None):
959 gross 1878 iter=0
960 caltinay 2169
961 jfenwick 2455 h=numpy.zeros((iter_restart+1,iter_restart),numpy.float64)
962     c=numpy.zeros(iter_restart,numpy.float64)
963     s=numpy.zeros(iter_restart,numpy.float64)
964     g=numpy.zeros(iter_restart+1,numpy.float64)
965 artak 1519 v=[]
966    
967 gross 2100 r_dot_r = bilinearform(r, r)
968     if r_dot_r<0: raise NegativeNorm,"negative norm."
969 artak 1519 rho=math.sqrt(r_dot_r)
970 caltinay 2169
971 artak 1519 v.append(r/rho)
972     g[0]=rho
973    
974 gross 2100 if verbose: print "GMRES: initial residual %e (absolute tolerance = %e)"%(rho,atol)
975     while not (rho<=atol or iter==iter_restart):
976 artak 1519
977     if iter >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
978    
979 gross 2261 if P_R!=None:
980     p=Aprod(P_R(v[iter]))
981     else:
982     p=Aprod(v[iter])
983 artak 1519 v.append(p)
984    
985 caltinay 2169 v_norm1=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
986 artak 1519
987 caltinay 2169 # Modified Gram-Schmidt
988     for j in range(iter+1):
989     h[j,iter]=bilinearform(v[j],v[iter+1])
990 gross 2100 v[iter+1]-=h[j,iter]*v[j]
991 caltinay 2169
992     h[iter+1,iter]=math.sqrt(bilinearform(v[iter+1],v[iter+1]))
993 gross 2100 v_norm2=h[iter+1,iter]
994 artak 1519
995     # Reorthogonalize if needed
996     if v_norm1 + 0.001*v_norm2 == v_norm1: #Brown/Hindmarsh condition (default)
997 caltinay 2169 for j in range(iter+1):
998 artak 1519 hr=bilinearform(v[j],v[iter+1])
999 caltinay 2169 h[j,iter]=h[j,iter]+hr
1000 gross 1878 v[iter+1] -= hr*v[j]
1001 artak 1519
1002 caltinay 2169 v_norm2=math.sqrt(bilinearform(v[iter+1], v[iter+1]))
1003 gross 2100 h[iter+1,iter]=v_norm2
1004 artak 1519
1005 caltinay 2169 # watch out for happy breakdown
1006 gross 1878 if not v_norm2 == 0:
1007 gross 2100 v[iter+1]=v[iter+1]/h[iter+1,iter]
1008 artak 1519
1009     # Form and store the information for the new Givens rotation
1010 gross 2100 if iter > 0: h[:iter+1,iter]=__givapp(c[:iter],s[:iter],h[:iter+1,iter])
1011     mu=math.sqrt(h[iter,iter]*h[iter,iter]+h[iter+1,iter]*h[iter+1,iter])
1012 artak 1519
1013     if mu!=0 :
1014 gross 2100 c[iter]=h[iter,iter]/mu
1015     s[iter]=-h[iter+1,iter]/mu
1016     h[iter,iter]=c[iter]*h[iter,iter]-s[iter]*h[iter+1,iter]
1017     h[iter+1,iter]=0.0
1018 artak 2303 gg=__givapp(c[iter],s[iter],[g[iter],g[iter+1]])
1019 gross 2284 g[iter]=gg[0]
1020     g[iter+1]=gg[1]
1021 artak 1519 # Update the residual norm
1022 caltinay 2169
1023 artak 1519 rho=abs(g[iter+1])
1024 gross 2100 if verbose: print "GMRES: iteration step %s: residual %e"%(iter,rho)
1025 artak 1519 iter+=1
1026    
1027 gross 1878 # At this point either iter > iter_max or rho < tol.
1028 caltinay 2169 # It's time to compute x and leave.
1029 gross 1878
1030 gross 2100 if verbose: print "GMRES: iteration stopped after %s step."%iter
1031 caltinay 2169 if iter > 0 :
1032 jfenwick 2455 y=numpy.zeros(iter,numpy.float64)
1033 gross 2100 y[iter-1] = g[iter-1] / h[iter-1,iter-1]
1034 caltinay 2169 if iter > 1 :
1035     i=iter-2
1036 artak 1519 while i>=0 :
1037 jfenwick 2455 y[i] = ( g[i] - numpy.dot(h[i,i+1:iter], y[i+1:iter])) / h[i,i]
1038 artak 1519 i=i-1
1039     xhat=v[iter-1]*y[iter-1]
1040     for i in range(iter-1):
1041     xhat += v[i]*y[i]
1042 caltinay 2169 else:
1043 gross 2100 xhat=v[0] * 0
1044 gross 2261 if P_R!=None:
1045     x += P_R(xhat)
1046     else:
1047     x += xhat
1048 caltinay 2169 if iter<iter_restart-1:
1049     stopped=True
1050     else:
1051 artak 1519 stopped=False
1052    
1053     return x,stopped
1054    
1055 gross 2156 def MINRES(r, Aprod, x, Msolve, bilinearform, atol=0, rtol=1.e-8, iter_max=100):
1056     """
1057 caltinay 2169 Solver for
1058    
1059 jfenwick 2625 *Ax=b*
1060 caltinay 2169
1061     with a symmetric and positive definite operator A (more details required!).
1062     It uses the minimum residual method (MINRES) with preconditioner M
1063     providing an approximation of A.
1064    
1065     The iteration is terminated if
1066    
1067 jfenwick 2625 *|r| <= atol+rtol*|r0|*
1068 caltinay 2169
1069 jfenwick 2625 where *r0* is the initial residual and *|.|* is the energy norm. In fact
1070 gross 2156
1071 jfenwick 2625 *|r| = sqrt( bilinearform(Msolve(r),r))*
1072 caltinay 2169
1073 gross 2156 For details on the preconditioned conjugate gradient method see the book:
1074    
1075 caltinay 2169 I{Templates for the Solution of Linear Systems by R. Barrett, M. Berry,
1076     T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo,
1077     C. Romine, and H. van der Vorst}.
1078    
1079 jfenwick 2625 :param r: initial residual *r=b-Ax*. ``r`` is altered.
1080     :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1081     :param x: an initial guess for the solution
1082     :type x: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1083     :param Aprod: returns the value Ax
1084     :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
1085     argument ``x``. The returned object needs to be of the same
1086     type like argument ``r``.
1087     :param Msolve: solves Mx=r
1088     :type Msolve: function ``Msolve(r)`` where ``r`` is of the same type like
1089     argument ``r``. The returned object needs to be of the same
1090     type like argument ``x``.
1091     :param bilinearform: inner product ``<x,r>``
1092     :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
1093     type like argument ``x`` and ``r`` is. The returned value
1094     is a ``float``.
1095     :param atol: absolute tolerance
1096     :type atol: non-negative ``float``
1097     :param rtol: relative tolerance
1098     :type rtol: non-negative ``float``
1099     :param iter_max: maximum number of iteration steps
1100     :type iter_max: ``int``
1101     :return: the solution approximation and the corresponding residual
1102     :rtype: ``tuple``
1103     :warning: ``r`` and ``x`` are altered.
1104 gross 2156 """
1105 artak 1481 #------------------------------------------------------------------
1106     # Set up y and v for the first Lanczos vector v1.
1107     # y = beta1 P' v1, where P = C**(-1).
1108     # v is really P' v1.
1109     #------------------------------------------------------------------
1110 gross 2156 r1 = r
1111     y = Msolve(r)
1112     beta1 = bilinearform(y,r)
1113 caltinay 2169
1114 artak 1481 if beta1< 0: raise NegativeNorm,"negative norm."
1115    
1116 gross 2156 # If r = 0 exactly, stop with x
1117     if beta1==0: return x
1118 artak 1481
1119 caltinay 2169 if beta1> 0: beta1 = math.sqrt(beta1)
1120 artak 1481
1121     #------------------------------------------------------------------
1122 artak 1484 # Initialize quantities.
1123 artak 1481 # ------------------------------------------------------------------
1124 artak 1482 iter = 0
1125     Anorm = 0
1126     ynorm = 0
1127 artak 1481 oldb = 0
1128     beta = beta1
1129     dbar = 0
1130     epsln = 0
1131     phibar = beta1
1132     rhs1 = beta1
1133     rhs2 = 0
1134     rnorm = phibar
1135     tnorm2 = 0
1136     ynorm2 = 0
1137     cs = -1
1138     sn = 0
1139 gross 2156 w = r*0.
1140     w2 = r*0.
1141 artak 1481 r2 = r1
1142     eps = 0.0001
1143    
1144     #---------------------------------------------------------------------
1145     # Main iteration loop.
1146     # --------------------------------------------------------------------
1147 gross 2100 while not rnorm<=atol+rtol*Anorm*ynorm: # checks ||r|| < (||A|| ||x||) * TOL
1148 artak 1481
1149     if iter >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
1150     iter = iter + 1
1151    
1152     #-----------------------------------------------------------------
1153     # Obtain quantities for the next Lanczos vector vk+1, k = 1, 2,...
1154     # The general iteration is similar to the case k = 1 with v0 = 0:
1155     #
1156     # p1 = Operator * v1 - beta1 * v0,
1157     # alpha1 = v1'p1,
1158     # q2 = p2 - alpha1 * v1,
1159     # beta2^2 = q2'q2,
1160     # v2 = (1/beta2) q2.
1161     #
1162     # Again, y = betak P vk, where P = C**(-1).
1163     #-----------------------------------------------------------------
1164     s = 1/beta # Normalize previous vector (in y).
1165     v = s*y # v = vk if P = I
1166 caltinay 2169
1167 artak 1481 y = Aprod(v)
1168 caltinay 2169
1169 artak 1481 if iter >= 2:
1170     y = y - (beta/oldb)*r1
1171    
1172     alfa = bilinearform(v,y) # alphak
1173 caltinay 2169 y += (- alfa/beta)*r2
1174 artak 1481 r1 = r2
1175     r2 = y
1176     y = Msolve(r2)
1177     oldb = beta # oldb = betak
1178 artak 1550 beta = bilinearform(y,r2) # beta = betak+1^2
1179 artak 1481 if beta < 0: raise NegativeNorm,"negative norm."
1180    
1181     beta = math.sqrt( beta )
1182     tnorm2 = tnorm2 + alfa*alfa + oldb*oldb + beta*beta
1183 caltinay 2169
1184 artak 1481 if iter==1: # Initialize a few things.
1185     gmax = abs( alfa ) # alpha1
1186     gmin = gmax # alpha1
1187    
1188     # Apply previous rotation Qk-1 to get
1189     # [deltak epslnk+1] = [cs sn][dbark 0 ]
1190     # [gbar k dbar k+1] [sn -cs][alfak betak+1].
1191 caltinay 2169
1192 artak 1481 oldeps = epsln
1193     delta = cs * dbar + sn * alfa # delta1 = 0 deltak
1194     gbar = sn * dbar - cs * alfa # gbar 1 = alfa1 gbar k
1195     epsln = sn * beta # epsln2 = 0 epslnk+1
1196     dbar = - cs * beta # dbar 2 = beta2 dbar k+1
1197    
1198     # Compute the next plane rotation Qk
1199    
1200     gamma = math.sqrt(gbar*gbar+beta*beta) # gammak
1201 caltinay 2169 gamma = max(gamma,eps)
1202 artak 1481 cs = gbar / gamma # ck
1203     sn = beta / gamma # sk
1204     phi = cs * phibar # phik
1205     phibar = sn * phibar # phibark+1
1206    
1207     # Update x.
1208    
1209 caltinay 2169 denom = 1/gamma
1210     w1 = w2
1211     w2 = w
1212 artak 1481 w = (v - oldeps*w1 - delta*w2) * denom
1213 artak 1550 x += phi*w
1214 artak 1481
1215     # Go round again.
1216    
1217     gmax = max(gmax,gamma)
1218     gmin = min(gmin,gamma)
1219     z = rhs1 / gamma
1220     ynorm2 = z*z + ynorm2
1221     rhs1 = rhs2 - delta*z
1222     rhs2 = - epsln*z
1223    
1224     # Estimate various norms and test for convergence.
1225    
1226 caltinay 2169 Anorm = math.sqrt( tnorm2 )
1227     ynorm = math.sqrt( ynorm2 )
1228 artak 1481
1229     rnorm = phibar
1230    
1231     return x
1232 artak 1489
1233 gross 2156 def TFQMR(r, Aprod, x, bilinearform, atol=0, rtol=1.e-8, iter_max=100):
1234     """
1235 caltinay 2169 Solver for
1236 artak 1489
1237 jfenwick 2625 *Ax=b*
1238 artak 1489
1239 caltinay 2169 with a general operator A (more details required!).
1240     It uses the Transpose-Free Quasi-Minimal Residual method (TFQMR).
1241 artak 1489
1242 caltinay 2169 The iteration is terminated if
1243 artak 1489
1244 jfenwick 2625 *|r| <= atol+rtol*|r0|*
1245 gross 2156
1246 jfenwick 2625 where *r0* is the initial residual and *|.|* is the energy norm. In fact
1247 gross 2156
1248 jfenwick 2625 *|r| = sqrt( bilinearform(r,r))*
1249 gross 2156
1250 jfenwick 2625 :param r: initial residual *r=b-Ax*. ``r`` is altered.
1251     :type r: any object supporting inplace add (x+=y) and scaling (x=scalar*y)
1252     :param x: an initial guess for the solution
1253     :type x: same like ``r``
1254     :param Aprod: returns the value Ax
1255     :type Aprod: function ``Aprod(x)`` where ``x`` is of the same object like
1256     argument ``x``. The returned object needs to be of the same type
1257     like argument ``r``.
1258     :param bilinearform: inner product ``<x,r>``
1259     :type bilinearform: function ``bilinearform(x,r)`` where ``x`` is of the same
1260     type like argument ``x`` and ``r``. The returned value is
1261     a ``float``.
1262     :param atol: absolute tolerance
1263     :type atol: non-negative ``float``
1264     :param rtol: relative tolerance
1265     :type rtol: non-negative ``float``
1266     :param iter_max: maximum number of iteration steps
1267     :type iter_max: ``int``
1268     :rtype: ``tuple``
1269     :warning: ``r`` and ``x`` are altered.
1270 gross 2156 """
1271 artak 1489 u1=0
1272     u2=0
1273     y1=0
1274     y2=0
1275    
1276     w = r
1277 caltinay 2169 y1 = r
1278     iter = 0
1279 artak 1489 d = 0
1280 gross 2156 v = Aprod(y1)
1281 artak 1489 u1 = v
1282 caltinay 2169
1283 artak 1489 theta = 0.0;
1284     eta = 0.0;
1285 gross 2156 rho=bilinearform(r,r)
1286     if rho < 0: raise NegativeNorm,"negative norm."
1287     tau = math.sqrt(rho)
1288     norm_r0=tau
1289     while tau>atol+rtol*norm_r0:
1290 artak 1489 if iter >= iter_max: raise MaxIterReached,"maximum number of %s steps reached."%iter_max
1291    
1292     sigma = bilinearform(r,v)
1293 gross 2156 if sigma == 0.0: raise IterationBreakDown,'TFQMR breakdown, sigma=0'
1294 caltinay 2169
1295 artak 1489 alpha = rho / sigma
1296    
1297     for j in range(2):
1298     #
1299     # Compute y2 and u2 only if you have to
1300     #
1301     if ( j == 1 ):
1302     y2 = y1 - alpha * v
1303 gross 2156 u2 = Aprod(y2)
1304 caltinay 2169
1305 artak 1489 m = 2 * (iter+1) - 2 + (j+1)
1306 caltinay 2169 if j==0:
1307 artak 1489 w = w - alpha * u1
1308     d = y1 + ( theta * theta * eta / alpha ) * d
1309     if j==1:
1310     w = w - alpha * u2
1311     d = y2 + ( theta * theta * eta / alpha ) * d
1312    
1313     theta = math.sqrt(bilinearform(w,w))/ tau
1314     c = 1.0 / math.sqrt ( 1.0 + theta * theta )
1315     tau = tau * theta * c
1316     eta = c * c * alpha
1317     x = x + eta * d
1318     #
1319     # Try to terminate the iteration at each pass through the loop
1320     #
1321 gross 2156 if rho == 0.0: raise IterationBreakDown,'TFQMR breakdown, rho=0'
1322 artak 1489
1323     rhon = bilinearform(r,w)
1324     beta = rhon / rho;
1325     rho = rhon;
1326     y1 = w + beta * y2;
1327 gross 2156 u1 = Aprod(y1)
1328 artak 1489 v = u1 + beta * ( u2 + beta * v )
1329 caltinay 2169
1330 gross 2156 iter += 1
1331 artak 1489
1332     return x
1333    
1334    
1335 artak 1465 #############################################
1336    
1337 gross 1331 class ArithmeticTuple(object):
1338     """
1339 jfenwick 2625 Tuple supporting inplace update x+=y and scaling x=a*y where ``x,y`` is an
1340     ArithmeticTuple and ``a`` is a float.
1341 gross 1331
1342 caltinay 2169 Example of usage::
1343 gross 1331
1344 caltinay 2169 from esys.escript import Data
1345 jfenwick 2455 from numpy import array
1346 caltinay 2169 a=Data(...)
1347     b=array([1.,4.])
1348     x=ArithmeticTuple(a,b)
1349     y=5.*x
1350 gross 1331
1351     """
1352     def __init__(self,*args):
1353     """
1354 jfenwick 2625 Initializes object with elements ``args``.
1355 gross 1331
1356 jfenwick 2625 :param args: tuple of objects that support inplace add (x+=y) and
1357 caltinay 2169 scaling (x=a*y)
1358 gross 1331 """
1359     self.__items=list(args)
1360    
1361     def __len__(self):
1362     """
1363 caltinay 2169 Returns the number of items.
1364 gross 1331
1365 jfenwick 2625 :return: number of items
1366     :rtype: ``int``
1367 gross 1331 """
1368     return len(self.__items)
1369    
1370     def __getitem__(self,index):
1371     """
1372 caltinay 2169 Returns item at specified position.
1373 gross 1331
1374 jfenwick 2625 :param index: index of item to be returned
1375     :type index: ``int``
1376     :return: item with index ``index``
1377 gross 1331 """
1378     return self.__items.__getitem__(index)
1379    
1380     def __mul__(self,other):
1381     """
1382 jfenwick 2625 Scales by ``other`` from the right.
1383 gross 1331
1384 jfenwick 2625 :param other: scaling factor
1385     :type other: ``float``
1386     :return: itemwise self*other
1387     :rtype: `ArithmeticTuple`
1388 gross 1331 """
1389     out=[]
1390 caltinay 2169 try:
1391 gross 1896 l=len(other)
1392     if l!=len(self):
1393 caltinay 2169 raise ValueError,"length of arguments don't match."
1394 gross 1896 for i in range(l): out.append(self[i]*other[i])
1395     except TypeError:
1396 caltinay 2169 for i in range(len(self)): out.append(self[i]*other)
1397 gross 1331 return ArithmeticTuple(*tuple(out))
1398    
1399     def __rmul__(self,other):
1400     """
1401 jfenwick 2625 Scales by ``other`` from the left.
1402 gross 1331
1403 jfenwick 2625 :param other: scaling factor
1404     :type other: ``float``
1405     :return: itemwise other*self
1406     :rtype: `ArithmeticTuple`
1407 gross 1331 """
1408     out=[]
1409 caltinay 2169 try:
1410 gross 1896 l=len(other)
1411     if l!=len(self):
1412 caltinay 2169 raise ValueError,"length of arguments don't match."
1413 gross 1896 for i in range(l): out.append(other[i]*self[i])
1414     except TypeError:
1415 caltinay 2169 for i in range(len(self)): out.append(other*self[i])
1416 gross 1331 return ArithmeticTuple(*tuple(out))
1417    
1418 artak 1465 def __div__(self,other):
1419     """
1420 jfenwick 2625 Scales by (1/``other``) from the right.
1421 artak 1465
1422 jfenwick 2625 :param other: scaling factor
1423     :type other: ``float``
1424     :return: itemwise self/other
1425     :rtype: `ArithmeticTuple`
1426 artak 1465 """
1427 gross 1896 return self*(1/other)
1428 artak 1465
1429     def __rdiv__(self,other):
1430     """
1431 jfenwick 2625 Scales by (1/``other``) from the left.
1432 artak 1465
1433 jfenwick 2625 :param other: scaling factor
1434     :type other: ``float``
1435     :return: itemwise other/self
1436     :rtype: `ArithmeticTuple`
1437 artak 1465 """
1438     out=[]
1439 caltinay 2169 try:
1440 gross 1896 l=len(other)
1441     if l!=len(self):
1442 caltinay 2169 raise ValueError,"length of arguments don't match."
1443 gross 1896 for i in range(l): out.append(other[i]/self[i])
1444     except TypeError:
1445 caltinay 2169 for i in range(len(self)): out.append(other/self[i])
1446 artak 1465 return ArithmeticTuple(*tuple(out))
1447 caltinay 2169
1448 gross 1331 def __iadd__(self,other):
1449     """
1450 jfenwick 2625 Inplace addition of ``other`` to self.
1451 gross 1331
1452 jfenwick 2625 :param other: increment
1453     :type other: ``ArithmeticTuple``
1454 gross 1331 """
1455     if len(self) != len(other):
1456 caltinay 2169 raise ValueError,"tuple lengths must match."
1457 gross 1331 for i in range(len(self)):
1458     self.__items[i]+=other[i]
1459     return self
1460    
1461 artak 1550 def __add__(self,other):
1462     """
1463 jfenwick 2625 Adds ``other`` to self.
1464 artak 1550
1465 jfenwick 2625 :param other: increment
1466     :type other: ``ArithmeticTuple``
1467 artak 1550 """
1468 gross 1896 out=[]
1469 caltinay 2169 try:
1470 gross 1896 l=len(other)
1471     if l!=len(self):
1472 caltinay 2169 raise ValueError,"length of arguments don't match."
1473 gross 1896 for i in range(l): out.append(self[i]+other[i])
1474     except TypeError:
1475 caltinay 2169 for i in range(len(self)): out.append(self[i]+other)
1476 gross 1896 return ArithmeticTuple(*tuple(out))
1477 artak 1550
1478     def __sub__(self,other):
1479     """
1480 jfenwick 2625 Subtracts ``other`` from self.
1481 artak 1550
1482 jfenwick 2625 :param other: decrement
1483     :type other: ``ArithmeticTuple``
1484 artak 1550 """
1485 gross 1896 out=[]
1486 caltinay 2169 try:
1487 gross 1896 l=len(other)
1488     if l!=len(self):
1489 caltinay 2169 raise ValueError,"length of arguments don't match."
1490 gross 1896 for i in range(l): out.append(self[i]-other[i])
1491     except TypeError:
1492 caltinay 2169 for i in range(len(self)): out.append(self[i]-other)
1493 gross 1896 return ArithmeticTuple(*tuple(out))
1494 caltinay 2169
1495 artak 1557 def __isub__(self,other):
1496     """
1497 jfenwick 2625 Inplace subtraction of ``other`` from self.
1498 artak 1550
1499 jfenwick 2625 :param other: decrement
1500     :type other: ``ArithmeticTuple``
1501 artak 1557 """
1502     if len(self) != len(other):
1503     raise ValueError,"tuple length must match."
1504     for i in range(len(self)):
1505     self.__items[i]-=other[i]
1506     return self
1507    
1508 artak 1550 def __neg__(self):
1509     """
1510 caltinay 2169 Negates values.
1511 artak 1550 """
1512 gross 1896 out=[]
1513 artak 1550 for i in range(len(self)):
1514 gross 1896 out.append(-self[i])
1515     return ArithmeticTuple(*tuple(out))
1516 artak 1550
1517    
1518 gross 1414 class HomogeneousSaddlePointProblem(object):
1519     """
1520 caltinay 2169 This class provides a framework for solving linear homogeneous saddle
1521     point problems of the form::
1522 gross 1414
1523 jfenwick 2625 *Av+B^*p=f*
1524     *Bv =0*
1525 gross 1414
1526 jfenwick 2625 for the unknowns *v* and *p* and given operators *A* and *B* and
1527     given right hand side *f*. *B^** is the adjoint operator of *B*.
1528 gross 2719 *A* may depend weakly on *v* and *p*.
1529 gross 1414 """
1530 gross 2719 def __init__(self, **kwargs):
1531 gross 2474 """
1532     initializes the saddle point problem
1533     """
1534 gross 2719 self.resetControlParameters()
1535 gross 1414 self.setTolerance()
1536 gross 2100 self.setAbsoluteTolerance()
1537 gross 2862 def resetControlParameters(self, K_p=1., K_v=1., rtol_max=0.01, rtol_min = 1.e-7, chi_max=0.5, reduction_factor=0.3, theta = 0.1):
1538 gross 2719 """
1539     sets a control parameter
1540    
1541 gross 2843 :param K_p: initial value for constant to adjust pressure tolerance
1542     :type K_p: ``float``
1543     :param K_v: initial value for constant to adjust velocity tolerance
1544     :type K_v: ``float``
1545     :param rtol_max: maximuim relative tolerance used to calculate presssure and velocity increment.
1546     :type rtol_max: ``float``
1547 gross 2719 :param chi_max: maximum tolerable converegence rate.
1548     :type chi_max: ``float``
1549 gross 2843 :param reduction_factor: reduction factor for adjustment factors.
1550     :type reduction_factor: ``float``
1551 gross 2719 """
1552 gross 2843 self.setControlParameter(K_p, K_v, rtol_max, rtol_min, chi_max, reduction_factor, theta)
1553 gross 2719
1554 gross 2843 def setControlParameter(self,K_p=None, K_v=None, rtol_max=None, rtol_min=None, chi_max=None, reduction_factor=None, theta=None):
1555 gross 2719 """
1556     sets a control parameter
1557    
1558 gross 2843
1559     :param K_p: initial value for constant to adjust pressure tolerance
1560     :type K_p: ``float``
1561     :param K_v: initial value for constant to adjust velocity tolerance
1562     :type K_v: ``float``
1563     :param rtol_max: maximuim relative tolerance used to calculate presssure and velocity increment.
1564     :type rtol_max: ``float``
1565 gross 2719 :param chi_max: maximum tolerable converegence rate.
1566     :type chi_max: ``float``
1567 gross 2843 :type reduction_factor: ``float``
1568 gross 2719 """
1569 gross 2843 if not K_p == None:
1570     if K_p<1:
1571     raise ValueError,"K_p need to be greater or equal to 1."
1572 gross 2719 else:
1573 gross 2843 K_p=self.__K_p
1574 gross 2719
1575 gross 2843 if not K_v == None:
1576     if K_v<1:
1577     raise ValueError,"K_v need to be greater or equal to 1."
1578 gross 2719 else:
1579 gross 2843 K_v=self.__K_v
1580 gross 2719
1581 gross 2843 if not rtol_max == None:
1582     if rtol_max<=0 or rtol_max>=1:
1583     raise ValueError,"rtol_max needs to be positive and less than 1."
1584 gross 2719 else:
1585 gross 2843 rtol_max=self.__rtol_max
1586 gross 2719
1587     if not rtol_min == None:
1588     if rtol_min<=0 or rtol_min>=1:
1589     raise ValueError,"rtol_min needs to be positive and less than 1."
1590     else:
1591     rtol_min=self.__rtol_min
1592    
1593 gross 2843 if not chi_max == None:
1594     if chi_max<=0 or chi_max>=1:
1595     raise ValueError,"chi_max needs to be positive and less than 1."
1596 gross 2719 else:
1597 gross 2843 chi_max = self.__chi_max
1598 gross 2719
1599 gross 2843 if not reduction_factor == None:
1600     if reduction_factor<=0 or reduction_factor>1:
1601     raise ValueError,"reduction_factor need to be between zero and one."
1602 gross 2719 else:
1603 gross 2843 reduction_factor=self.__reduction_factor
1604 gross 2719
1605 gross 2843 if not theta == None:
1606     if theta<=0 or theta>1:
1607     raise ValueError,"theta need to be between zero and one."
1608 gross 2719 else:
1609 gross 2843 theta=self.__theta
1610 gross 2719
1611 gross 2843 if rtol_min>=rtol_max:
1612     raise ValueError,"rtol_max = %e needs to be greater than rtol_min = %e"%(rtol_max,rtol_min)
1613 gross 2719 self.__chi_max = chi_max
1614     self.__rtol_max = rtol_max
1615 gross 2843 self.__K_p = K_p
1616     self.__K_v = K_v
1617     self.__reduction_factor = reduction_factor
1618     self.__theta = theta
1619     self.__rtol_min=rtol_min
1620 gross 2719
1621 gross 2100 #=============================================================
1622 gross 2445 def inner_pBv(self,p,Bv):
1623 gross 1414 """
1624 caltinay 2169 Returns inner product of element p and Bv (overwrite).
1625    
1626 jfenwick 2625 :param p: a pressure increment
1627     :param Bv: a residual
1628     :return: inner product of element p and Bv
1629     :rtype: ``float``
1630     :note: used if PCG is applied.
1631 gross 1414 """
1632 gross 2445 raise NotImplementedError,"no inner product for p and Bv implemented."
1633 gross 1414
1634 gross 2100 def inner_p(self,p0,p1):
1635 gross 1414 """
1636 gross 2251 Returns inner product of p0 and p1 (overwrite).
1637 caltinay 2169
1638 jfenwick 2625 :param p0: a pressure
1639     :param p1: a pressure
1640     :return: inner product of p0 and p1
1641     :rtype: ``float``
1642 gross 1414 """
1643 gross 2100 raise NotImplementedError,"no inner product for p implemented."
1644 gross 2251
1645     def norm_v(self,v):
1646     """
1647     Returns the norm of v (overwrite).
1648 gross 2100
1649 jfenwick 2625 :param v: a velovity
1650     :return: norm of v
1651     :rtype: non-negative ``float``
1652 gross 2100 """
1653 gross 2251 raise NotImplementedError,"no norm of v implemented."
1654 gross 2719 def getDV(self, p, v, tol):
1655 gross 2100 """
1656 gross 2719 return a correction to the value for a given v and a given p with accuracy `tol` (overwrite)
1657 gross 1414
1658 gross 2719 :param p: pressure
1659     :param v: pressure
1660     :return: dv given as *dv= A^{-1} (f-A v-B^*p)*
1661     :note: Only *A* may depend on *v* and *p*
1662 gross 1414 """
1663 gross 2719 raise NotImplementedError,"no dv calculation implemented."
1664 gross 2445
1665 gross 2251
1666 gross 2719 def Bv(self,v, tol):
1667 gross 2251 """
1668 gross 2719 Returns Bv with accuracy `tol` (overwrite)
1669 gross 2251
1670 jfenwick 2625 :rtype: equal to the type of p
1671     :note: boundary conditions on p should be zero!
1672 gross 2251 """
1673     raise NotImplementedError, "no operator B implemented."
1674    
1675 gross 2445 def norm_Bv(self,Bv):
1676     """
1677     Returns the norm of Bv (overwrite).
1678    
1679 jfenwick 2625 :rtype: equal to the type of p
1680     :note: boundary conditions on p should be zero!
1681 gross 2445 """
1682     raise NotImplementedError, "no norm of Bv implemented."
1683    
1684 gross 2719 def solve_AinvBt(self,dp, tol):
1685 gross 2251 """
1686 gross 2719 Solves *A dv=B^*dp* with accuracy `tol`
1687 gross 1414
1688 gross 2719 :param dp: a pressure increment
1689     :return: the solution of *A dv=B^*dp*
1690     :note: boundary conditions on dv should be zero! *A* is the operator used in ``getDV`` and must not be altered.
1691 gross 1414 """
1692 gross 2100 raise NotImplementedError,"no operator A implemented."
1693 gross 1414
1694 gross 2719 def solve_prec(self,Bv, tol):
1695 gross 1414 """
1696 gross 2719 Provides a preconditioner for *(BA^{-1}B^ * )* applied to Bv with accuracy `tol`
1697 gross 1414
1698 jfenwick 2625 :rtype: equal to the type of p
1699     :note: boundary conditions on p should be zero!
1700 gross 1414 """
1701 gross 2100 raise NotImplementedError,"no preconditioner for Schur complement implemented."
1702     #=============================================================
1703 gross 2719 def __Aprod_PCG(self,dp):
1704     dv=self.solve_AinvBt(dp, self.__subtol)
1705     return ArithmeticTuple(dv,self.Bv(dv, self.__subtol))
1706 gross 1414
1707 gross 2445 def __inner_PCG(self,p,r):
1708     return self.inner_pBv(p,r[1])
1709 gross 1414
1710 gross 2445 def __Msolve_PCG(self,r):
1711 gross 2719 return self.solve_prec(r[1], self.__subtol)
1712 gross 2100 #=============================================================
1713 gross 2251 def __Aprod_GMRES(self,p):
1714 gross 2719 return self.solve_prec(self.Bv(self.solve_AinvBt(p, self.__subtol), self.__subtol), self.__subtol)
1715 gross 2251 def __inner_GMRES(self,p0,p1):
1716     return self.inner_p(p0,p1)
1717 gross 2474
1718 gross 2100 #=============================================================
1719 gross 2251 def norm_p(self,p):
1720     """
1721 jfenwick 2625 calculates the norm of ``p``
1722 gross 2251
1723 jfenwick 2625 :param p: a pressure
1724     :return: the norm of ``p`` using the inner product for pressure
1725     :rtype: ``float``
1726 gross 2251 """
1727     f=self.inner_p(p,p)
1728     if f<0: raise ValueError,"negative pressure norm."
1729     return math.sqrt(f)
1730 gross 2474
1731     def solve(self,v,p,max_iter=20, verbose=False, usePCG=True, iter_restart=20, max_correction_steps=10):
1732 gross 2100 """
1733 caltinay 2169 Solves the saddle point problem using initial guesses v and p.
1734 gross 1414
1735 jfenwick 2625 :param v: initial guess for velocity
1736     :param p: initial guess for pressure
1737     :type v: `Data`
1738     :type p: `Data`
1739     :param usePCG: indicates the usage of the PCG rather than GMRES scheme.
1740     :param max_iter: maximum number of iteration steps per correction
1741 caltinay 2169 attempt
1742 jfenwick 2625 :param verbose: if True, shows information on the progress of the
1743 caltinay 2169 saddlepoint problem solver.
1744 jfenwick 2625 :param iter_restart: restart the iteration after ``iter_restart`` steps
1745 caltinay 2169 (only used if useUzaw=False)
1746 jfenwick 2625 :type usePCG: ``bool``
1747     :type max_iter: ``int``
1748     :type verbose: ``bool``
1749     :type iter_restart: ``int``
1750     :rtype: ``tuple`` of `Data` objects
1751 gross 2793 :note: typically this method is overwritten by a subclass. It provides a wrapper for the ``_solve`` method.
1752 gross 2100 """
1753 gross 2793 return self._solve(v=v,p=p,max_iter=max_iter,verbose=verbose, usePCG=usePCG, iter_restart=iter_restart, max_correction_steps=max_correction_steps)
1754    
1755     def _solve(self,v,p,max_iter=20, verbose=False, usePCG=True, iter_restart=20, max_correction_steps=10):
1756     """
1757     see `_solve` method.
1758     """
1759 gross 2100 self.verbose=verbose
1760     rtol=self.getTolerance()
1761     atol=self.getAbsoluteTolerance()
1762 gross 2843
1763     K_p=self.__K_p
1764     K_v=self.__K_v
1765 gross 2100 correction_step=0
1766     converged=False
1767 gross 2719 chi=None
1768 gross 2843 eps=None
1769    
1770     if self.verbose: print "HomogeneousSaddlePointProblem: start iteration: rtol= %e, atol=%e"%(rtol, atol)
1771 gross 2251 while not converged:
1772 gross 2843
1773     # get tolerance for velecity increment:
1774     if chi == None:
1775     rtol_v=self.__rtol_max
1776     else:
1777     rtol_v=min(chi/K_v,self.__rtol_max)
1778     rtol_v=max(rtol_v, self.__rtol_min)
1779     if self.verbose: print "HomogeneousSaddlePointProblem: step %s: rtol_v= %e"%(correction_step,rtol_v)
1780     # get velocity increment:
1781     dv1=self.getDV(p,v,rtol_v)
1782     v1=v+dv1
1783     Bv1=self.Bv(v1, rtol_v)
1784     norm_Bv1=self.norm_Bv(Bv1)
1785     norm_dv1=self.norm_v(dv1)
1786     if self.verbose: print "HomogeneousSaddlePointProblem: step %s: norm_Bv1 = %e, norm_dv1 = %e"%(correction_step, norm_Bv1, norm_dv1)
1787     if norm_dv1*self.__theta < norm_Bv1:
1788     # get tolerance for pressure increment:
1789     large_Bv1=True
1790     if chi == None or eps == None:
1791     rtol_p=self.__rtol_max
1792     else:
1793     rtol_p=min(chi**2*eps/K_p/norm_Bv1, self.__rtol_max)
1794     self.__subtol=max(rtol_p**2, self.__rtol_min)
1795     if self.verbose: print "HomogeneousSaddlePointProblem: step %s: rtol_p= %e"%(correction_step,rtol_p)
1796     # now we solve for the pressure increment dp from B*A^{-1}B^* dp = Bv1
1797     if usePCG:
1798 gross 2719 dp,r,a_norm=PCG(ArithmeticTuple(v1,Bv1),self.__Aprod_PCG,0*p,self.__Msolve_PCG,self.__inner_PCG,atol=0, rtol=rtol_p,iter_max=max_iter, verbose=self.verbose)
1799     v2=r[0]
1800     Bv2=r[1]
1801 gross 2843 else:
1802     # don't use!!!!
1803 gross 2719 dp=GMRES(self.solve_prec(Bv1,self.__subtol),self.__Aprod_GMRES, 0*p, self.__inner_GMRES,atol=0, rtol=rtol_p,iter_max=max_iter, iter_restart=iter_restart, verbose=self.verbose)
1804     dv2=self.solve_AinvBt(dp, self.__subtol)
1805     v2=v1-dv2
1806     Bv2=self.Bv(v2, self.__subtol)
1807 gross 2843 p2=p+dp
1808     else:
1809     large_Bv1=False
1810     v2=v1
1811     p2=p
1812     # update business:
1813     norm_dv2=self.norm_v(v2-v)
1814     norm_v2=self.norm_v(v2)
1815     if self.verbose: print "HomogeneousSaddlePointProblem: step %s: v2 = %e, norm_dv2 = %e"%(correction_step, norm_v2, self.norm_v(v2-v))
1816     eps, eps_old = max(norm_Bv1, norm_dv2), eps
1817     if eps_old == None:
1818     chi, chi_old = None, chi
1819     else:
1820     chi, chi_old = min(eps/ eps_old, self.__chi_max), chi
1821     if eps != None:
1822     if chi !=None:
1823     if self.verbose: print "HomogeneousSaddlePointProblem: step %s: convergence rate = %e, correction = %e"%(correction_step,chi, eps)
1824     else:
1825     if self.verbose: print "HomogeneousSaddlePointProblem: step %s: correction = %e"%(correction_step, eps)
1826     if eps <= rtol*norm_v2+atol :
1827     converged = True
1828     else:
1829     if correction_step>=max_correction_steps:
1830     raise CorrectionFailed,"Given up after %d correction steps."%correction_step
1831     if chi_old!=None:
1832     K_p=max(1,self.__reduction_factor*K_p,(chi-chi_old)/chi_old**2*K_p)
1833     K_v=max(1,self.__reduction_factor*K_v,(chi-chi_old)/chi_old**2*K_p)
1834     if self.verbose: print "HomogeneousSaddlePointProblem: step %s: new adjustment factor K = %e"%(correction_step,K_p)
1835     correction_step+=1
1836     v,p =v2, p2
1837 gross 2719 if self.verbose: print "HomogeneousSaddlePointProblem: tolerance reached after %s steps."%correction_step
1838 gross 2251 return v,p
1839 caltinay 2169 #========================================================================
1840 gross 2100 def setTolerance(self,tolerance=1.e-4):
1841     """
1842 caltinay 2169 Sets the relative tolerance for (v,p).
1843 gross 2100
1844 jfenwick 2625 :param tolerance: tolerance to be used
1845     :type tolerance: non-negative ``float``
1846 gross 2100 """
1847     if tolerance<0:
1848     raise ValueError,"tolerance must be positive."
1849     self.__rtol=tolerance
1850 gross 2156
1851 gross 1414 def getTolerance(self):
1852 gross 2100 """
1853 caltinay 2169 Returns the relative tolerance.
1854 gross 1414
1855 jfenwick 2625 :return: relative tolerance
1856     :rtype: ``float``
1857 gross 2100 """
1858     return self.__rtol
1859 caltinay 2169
1860 gross 2100 def setAbsoluteTolerance(self,tolerance=0.):
1861     """
1862 caltinay 2169 Sets the absolute tolerance.
1863 gross 1414
1864 jfenwick 2625 :param tolerance: tolerance to be used
1865     :type tolerance: non-negative ``float``
1866 gross 2100 """
1867     if tolerance<0:
1868     raise ValueError,"tolerance must be non-negative."
1869     self.__atol=tolerance
1870 caltinay 2169
1871 gross 2100 def getAbsoluteTolerance(self):
1872     """
1873 caltinay 2169 Returns the absolute tolerance.
1874 gross 1414
1875 jfenwick 2625 :return: absolute tolerance
1876     :rtype: ``float``
1877 gross 2100 """
1878     return self.__atol
1879 gross 1469
1880 caltinay 2169
1881 gross 1878 def MaskFromBoundaryTag(domain,*tags):
1882     """
1883 caltinay 2169 Creates a mask on the Solution(domain) function space where the value is
1884     one for samples that touch the boundary tagged by tags.
1885 gross 1878
1886 caltinay 2169 Usage: m=MaskFromBoundaryTag(domain, "left", "right")
1887 gross 1878
1888 jfenwick 2625 :param domain: domain to be used
1889     :type domain: `escript.Domain`
1890     :param tags: boundary tags
1891     :type tags: ``str``
1892     :return: a mask which marks samples that are touching the boundary tagged
1893 caltinay 2169 by any of the given tags
1894 jfenwick 2625 :rtype: `escript.Data` of rank 0
1895 gross 1878 """
1896     pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)
1897 gross 1956 d=escript.Scalar(0.,escript.FunctionOnBoundary(domain))
1898 gross 1878 for t in tags: d.setTaggedValue(t,1.)
1899     pde.setValue(y=d)
1900     return util.whereNonZero(pde.getRightHandSide())
1901 caltinay 2169
1902 gross 2498 def MaskFromTag(domain,*tags):
1903     """
1904     Creates a mask on the Solution(domain) function space where the value is
1905     one for samples that touch regions tagged by tags.
1906 gross 867
1907 gross 2498 Usage: m=MaskFromTag(domain, "ham")
1908    
1909 jfenwick 2625 :param domain: domain to be used
1910     :type domain: `escript.Domain`
1911     :param tags: boundary tags
1912     :type tags: ``str``
1913     :return: a mask which marks samples that are touching the boundary tagged
1914 gross 2498 by any of the given tags
1915 jfenwick 2625 :rtype: `escript.Data` of rank 0
1916 gross 2498 """
1917     pde=linearPDEs.LinearPDE(domain,numEquations=1, numSolutions=1)
1918     d=escript.Scalar(0.,escript.Function(domain))
1919     for t in tags: d.setTaggedValue(t,1.)
1920     pde.setValue(Y=d)
1921     return util.whereNonZero(pde.getRightHandSide())
1922    
1923    

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