/
lobpcg.py
568 lines (445 loc) · 18.7 KB
/
lobpcg.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
"""
Pure SciPy implementation of Locally Optimal Block Preconditioned Conjugate
Gradient Method (LOBPCG), see
https://bitbucket.org/joseroman/blopex
License: BSD
Authors: Robert Cimrman, Andrew Knyazev
Examples in tests directory contributed by Nils Wagner.
"""
from __future__ import division, print_function, absolute_import
import sys
import numpy as np
from numpy.testing import assert_allclose
from scipy._lib.six import xrange
from scipy.linalg import inv, eigh, cho_factor, cho_solve, cholesky
from scipy.sparse.linalg import aslinearoperator, LinearOperator
__all__ = ['lobpcg']
def pause():
# Used only when verbosity level > 10.
input()
def save(ar, fileName):
# Used only when verbosity level > 10.
from numpy import savetxt
savetxt(fileName, ar, precision=8)
def _assert_symmetric(M, rtol=1e-5, atol=1e-8):
assert_allclose(M.T, M, rtol=rtol, atol=atol)
##
# 21.05.2007, c
def as2d(ar):
"""
If the input array is 2D return it, if it is 1D, append a dimension,
making it a column vector.
"""
if ar.ndim == 2:
return ar
else: # Assume 1!
aux = np.array(ar, copy=False)
aux.shape = (ar.shape[0], 1)
return aux
def _makeOperator(operatorInput, expectedShape):
"""Takes a dense numpy array or a sparse matrix or
a function and makes an operator performing matrix * blockvector
products.
Examples
--------
>>> A = _makeOperator( arrayA, (n, n) )
>>> vectorB = A( vectorX )
"""
if operatorInput is None:
def ident(x):
return x
operator = LinearOperator(expectedShape, ident, matmat=ident)
else:
operator = aslinearoperator(operatorInput)
if operator.shape != expectedShape:
raise ValueError('operator has invalid shape')
return operator
def _applyConstraints(blockVectorV, factYBY, blockVectorBY, blockVectorY):
"""Changes blockVectorV in place."""
gramYBV = np.dot(blockVectorBY.T, blockVectorV)
tmp = cho_solve(factYBY, gramYBV)
blockVectorV -= np.dot(blockVectorY, tmp)
def _b_orthonormalize(B, blockVectorV, blockVectorBV=None, retInvR=False):
if blockVectorBV is None:
if B is not None:
blockVectorBV = B(blockVectorV)
else:
blockVectorBV = blockVectorV # Shared data!!!
gramVBV = np.dot(blockVectorV.T, blockVectorBV)
gramVBV = cholesky(gramVBV)
gramVBV = inv(gramVBV, overwrite_a=True)
# gramVBV is now R^{-1}.
blockVectorV = np.dot(blockVectorV, gramVBV)
if B is not None:
blockVectorBV = np.dot(blockVectorBV, gramVBV)
if retInvR:
return blockVectorV, blockVectorBV, gramVBV
else:
return blockVectorV, blockVectorBV
def lobpcg(A, X,
B=None, M=None, Y=None,
tol=None, maxiter=20,
largest=True, verbosityLevel=0,
retLambdaHistory=False, retResidualNormsHistory=False):
"""Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG)
LOBPCG is a preconditioned eigensolver for large symmetric positive
definite (SPD) generalized eigenproblems.
Parameters
----------
A : {sparse matrix, dense matrix, LinearOperator}
The symmetric linear operator of the problem, usually a
sparse matrix. Often called the "stiffness matrix".
X : array_like
Initial approximation to the k eigenvectors. If A has
shape=(n,n) then X should have shape shape=(n,k).
B : {dense matrix, sparse matrix, LinearOperator}, optional
the right hand side operator in a generalized eigenproblem.
by default, B = Identity
often called the "mass matrix"
M : {dense matrix, sparse matrix, LinearOperator}, optional
preconditioner to A; by default M = Identity
M should approximate the inverse of A
Y : array_like, optional
n-by-sizeY matrix of constraints, sizeY < n
The iterations will be performed in the B-orthogonal complement
of the column-space of Y. Y must be full rank.
Returns
-------
w : array
Array of k eigenvalues
v : array
An array of k eigenvectors. V has the same shape as X.
Other Parameters
----------------
tol : scalar, optional
Solver tolerance (stopping criterion)
by default: tol=n*sqrt(eps)
maxiter : integer, optional
maximum number of iterations
by default: maxiter=min(n,20)
largest : bool, optional
when True, solve for the largest eigenvalues, otherwise the smallest
verbosityLevel : integer, optional
controls solver output. default: verbosityLevel = 0.
retLambdaHistory : boolean, optional
whether to return eigenvalue history
retResidualNormsHistory : boolean, optional
whether to return history of residual norms
Examples
--------
Solve A x = lambda B x with constraints and preconditioning.
>>> from scipy.sparse import spdiags, issparse
>>> from scipy.sparse.linalg import lobpcg, LinearOperator
>>> n = 100
>>> vals = [np.arange(n, dtype=np.float64) + 1]
>>> A = spdiags(vals, 0, n, n)
>>> A.toarray()
array([[ 1., 0., 0., ..., 0., 0., 0.],
[ 0., 2., 0., ..., 0., 0., 0.],
[ 0., 0., 3., ..., 0., 0., 0.],
...,
[ 0., 0., 0., ..., 98., 0., 0.],
[ 0., 0., 0., ..., 0., 99., 0.],
[ 0., 0., 0., ..., 0., 0., 100.]])
Constraints.
>>> Y = np.eye(n, 3)
Initial guess for eigenvectors, should have linearly independent
columns. Column dimension = number of requested eigenvalues.
>>> X = np.random.rand(n, 3)
Preconditioner -- inverse of A (as an abstract linear operator).
>>> invA = spdiags([1./vals[0]], 0, n, n)
>>> def precond( x ):
... return invA * x
>>> M = LinearOperator(matvec=precond, shape=(n, n), dtype=float)
Here, ``invA`` could of course have been used directly as a preconditioner.
Let us then solve the problem:
>>> eigs, vecs = lobpcg(A, X, Y=Y, M=M, tol=1e-4, maxiter=40, largest=False)
>>> eigs
array([ 4., 5., 6.])
Note that the vectors passed in Y are the eigenvectors of the 3 smallest
eigenvalues. The results returned are orthogonal to those.
Notes
-----
If both retLambdaHistory and retResidualNormsHistory are True,
the return tuple has the following format
(lambda, V, lambda history, residual norms history).
In the following ``n`` denotes the matrix size and ``m`` the number
of required eigenvalues (smallest or largest).
The LOBPCG code internally solves eigenproblems of the size 3``m`` on every
iteration by calling the "standard" dense eigensolver, so if ``m`` is not
small enough compared to ``n``, it does not make sense to call the LOBPCG
code, but rather one should use the "standard" eigensolver,
e.g. numpy or scipy function in this case.
If one calls the LOBPCG algorithm for 5``m``>``n``,
it will most likely break internally, so the code tries to call the standard
function instead.
It is not that n should be large for the LOBPCG to work, but rather the
ratio ``n``/``m`` should be large. It you call the LOBPCG code with ``m``=1
and ``n``=10, it should work, though ``n`` is small. The method is intended
for extremely large ``n``/``m``, see e.g., reference [28] in
http://arxiv.org/abs/0705.2626
The convergence speed depends basically on two factors:
1. How well relatively separated the seeking eigenvalues are
from the rest of the eigenvalues.
One can try to vary ``m`` to make this better.
2. How well conditioned the problem is. This can be changed by using proper
preconditioning. For example, a rod vibration test problem (under tests
directory) is ill-conditioned for large ``n``, so convergence will be
slow, unless efficient preconditioning is used.
For this specific problem, a good simple preconditioner function would
be a linear solve for A, which is easy to code since A is tridiagonal.
*Acknowledgements*
lobpcg.py code was written by Robert Cimrman.
Many thanks belong to Andrew Knyazev, the author of the algorithm,
for lots of advice and support.
References
----------
.. [1] A. V. Knyazev (2001),
Toward the Optimal Preconditioned Eigensolver: Locally Optimal
Block Preconditioned Conjugate Gradient Method.
SIAM Journal on Scientific Computing 23, no. 2,
pp. 517-541. http://dx.doi.org/10.1137/S1064827500366124
.. [2] A. V. Knyazev, I. Lashuk, M. E. Argentati, and E. Ovchinnikov (2007),
Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX)
in hypre and PETSc. http://arxiv.org/abs/0705.2626
.. [3] A. V. Knyazev's C and MATLAB implementations:
https://bitbucket.org/joseroman/blopex
"""
blockVectorX = X
blockVectorY = Y
residualTolerance = tol
maxIterations = maxiter
if blockVectorY is not None:
sizeY = blockVectorY.shape[1]
else:
sizeY = 0
# Block size.
if len(blockVectorX.shape) != 2:
raise ValueError('expected rank-2 array for argument X')
n, sizeX = blockVectorX.shape
if sizeX > n:
raise ValueError('X column dimension exceeds the row dimension')
A = _makeOperator(A, (n,n))
B = _makeOperator(B, (n,n))
M = _makeOperator(M, (n,n))
if (n - sizeY) < (5 * sizeX):
# warn('The problem size is small compared to the block size.' \
# ' Using dense eigensolver instead of LOBPCG.')
if blockVectorY is not None:
raise NotImplementedError('The dense eigensolver '
'does not support constraints.')
# Define the closed range of indices of eigenvalues to return.
if largest:
eigvals = (n - sizeX, n-1)
else:
eigvals = (0, sizeX-1)
A_dense = A(np.eye(n))
B_dense = None if B is None else B(np.eye(n))
return eigh(A_dense, B_dense, eigvals=eigvals, check_finite=False)
if residualTolerance is None:
residualTolerance = np.sqrt(1e-15) * n
maxIterations = min(n, maxIterations)
if verbosityLevel:
aux = "Solving "
if B is None:
aux += "standard"
else:
aux += "generalized"
aux += " eigenvalue problem with"
if M is None:
aux += "out"
aux += " preconditioning\n\n"
aux += "matrix size %d\n" % n
aux += "block size %d\n\n" % sizeX
if blockVectorY is None:
aux += "No constraints\n\n"
else:
if sizeY > 1:
aux += "%d constraints\n\n" % sizeY
else:
aux += "%d constraint\n\n" % sizeY
print(aux)
##
# Apply constraints to X.
if blockVectorY is not None:
if B is not None:
blockVectorBY = B(blockVectorY)
else:
blockVectorBY = blockVectorY
# gramYBY is a dense array.
gramYBY = np.dot(blockVectorY.T, blockVectorBY)
try:
# gramYBY is a Cholesky factor from now on...
gramYBY = cho_factor(gramYBY)
except:
raise ValueError('cannot handle linearly dependent constraints')
_applyConstraints(blockVectorX, gramYBY, blockVectorBY, blockVectorY)
##
# B-orthonormalize X.
blockVectorX, blockVectorBX = _b_orthonormalize(B, blockVectorX)
##
# Compute the initial Ritz vectors: solve the eigenproblem.
blockVectorAX = A(blockVectorX)
gramXAX = np.dot(blockVectorX.T, blockVectorAX)
_lambda, eigBlockVector = eigh(gramXAX, check_finite=False)
ii = np.argsort(_lambda)[:sizeX]
if largest:
ii = ii[::-1]
_lambda = _lambda[ii]
eigBlockVector = np.asarray(eigBlockVector[:,ii])
blockVectorX = np.dot(blockVectorX, eigBlockVector)
blockVectorAX = np.dot(blockVectorAX, eigBlockVector)
if B is not None:
blockVectorBX = np.dot(blockVectorBX, eigBlockVector)
##
# Active index set.
activeMask = np.ones((sizeX,), dtype=bool)
lambdaHistory = [_lambda]
residualNormsHistory = []
previousBlockSize = sizeX
ident = np.eye(sizeX, dtype=A.dtype)
ident0 = np.eye(sizeX, dtype=A.dtype)
##
# Main iteration loop.
blockVectorP = None # set during iteration
blockVectorAP = None
blockVectorBP = None
for iterationNumber in xrange(maxIterations):
if verbosityLevel > 0:
print('iteration %d' % iterationNumber)
aux = blockVectorBX * _lambda[np.newaxis,:]
blockVectorR = blockVectorAX - aux
aux = np.sum(blockVectorR.conjugate() * blockVectorR, 0)
residualNorms = np.sqrt(aux)
residualNormsHistory.append(residualNorms)
ii = np.where(residualNorms > residualTolerance, True, False)
activeMask = activeMask & ii
if verbosityLevel > 2:
print(activeMask)
currentBlockSize = activeMask.sum()
if currentBlockSize != previousBlockSize:
previousBlockSize = currentBlockSize
ident = np.eye(currentBlockSize, dtype=A.dtype)
if currentBlockSize == 0:
break
if verbosityLevel > 0:
print('current block size:', currentBlockSize)
print('eigenvalue:', _lambda)
print('residual norms:', residualNorms)
if verbosityLevel > 10:
print(eigBlockVector)
activeBlockVectorR = as2d(blockVectorR[:,activeMask])
if iterationNumber > 0:
activeBlockVectorP = as2d(blockVectorP[:,activeMask])
activeBlockVectorAP = as2d(blockVectorAP[:,activeMask])
activeBlockVectorBP = as2d(blockVectorBP[:,activeMask])
if M is not None:
# Apply preconditioner T to the active residuals.
activeBlockVectorR = M(activeBlockVectorR)
##
# Apply constraints to the preconditioned residuals.
if blockVectorY is not None:
_applyConstraints(activeBlockVectorR,
gramYBY, blockVectorBY, blockVectorY)
##
# B-orthonormalize the preconditioned residuals.
aux = _b_orthonormalize(B, activeBlockVectorR)
activeBlockVectorR, activeBlockVectorBR = aux
activeBlockVectorAR = A(activeBlockVectorR)
if iterationNumber > 0:
aux = _b_orthonormalize(B, activeBlockVectorP,
activeBlockVectorBP, retInvR=True)
activeBlockVectorP, activeBlockVectorBP, invR = aux
activeBlockVectorAP = np.dot(activeBlockVectorAP, invR)
##
# Perform the Rayleigh Ritz Procedure:
# Compute symmetric Gram matrices:
xaw = np.dot(blockVectorX.T, activeBlockVectorAR)
waw = np.dot(activeBlockVectorR.T, activeBlockVectorAR)
xbw = np.dot(blockVectorX.T, activeBlockVectorBR)
if iterationNumber > 0:
xap = np.dot(blockVectorX.T, activeBlockVectorAP)
wap = np.dot(activeBlockVectorR.T, activeBlockVectorAP)
pap = np.dot(activeBlockVectorP.T, activeBlockVectorAP)
xbp = np.dot(blockVectorX.T, activeBlockVectorBP)
wbp = np.dot(activeBlockVectorR.T, activeBlockVectorBP)
gramA = np.bmat([[np.diag(_lambda), xaw, xap],
[xaw.T, waw, wap],
[xap.T, wap.T, pap]])
gramB = np.bmat([[ident0, xbw, xbp],
[xbw.T, ident, wbp],
[xbp.T, wbp.T, ident]])
else:
gramA = np.bmat([[np.diag(_lambda), xaw],
[xaw.T, waw]])
gramB = np.bmat([[ident0, xbw],
[xbw.T, ident]])
_assert_symmetric(gramA)
_assert_symmetric(gramB)
if verbosityLevel > 10:
save(gramA, 'gramA')
save(gramB, 'gramB')
# Solve the generalized eigenvalue problem.
_lambda, eigBlockVector = eigh(gramA, gramB, check_finite=False)
ii = np.argsort(_lambda)[:sizeX]
if largest:
ii = ii[::-1]
if verbosityLevel > 10:
print(ii)
_lambda = _lambda[ii].astype(np.float64)
eigBlockVector = np.asarray(eigBlockVector[:,ii].astype(np.float64))
lambdaHistory.append(_lambda)
if verbosityLevel > 10:
print('lambda:', _lambda)
## # Normalize eigenvectors!
## aux = np.sum( eigBlockVector.conjugate() * eigBlockVector, 0 )
## eigVecNorms = np.sqrt( aux )
## eigBlockVector = eigBlockVector / eigVecNorms[np.newaxis,:]
# eigBlockVector, aux = _b_orthonormalize( B, eigBlockVector )
if verbosityLevel > 10:
print(eigBlockVector)
pause()
##
# Compute Ritz vectors.
if iterationNumber > 0:
eigBlockVectorX = eigBlockVector[:sizeX]
eigBlockVectorR = eigBlockVector[sizeX:sizeX+currentBlockSize]
eigBlockVectorP = eigBlockVector[sizeX+currentBlockSize:]
pp = np.dot(activeBlockVectorR, eigBlockVectorR)
pp += np.dot(activeBlockVectorP, eigBlockVectorP)
app = np.dot(activeBlockVectorAR, eigBlockVectorR)
app += np.dot(activeBlockVectorAP, eigBlockVectorP)
bpp = np.dot(activeBlockVectorBR, eigBlockVectorR)
bpp += np.dot(activeBlockVectorBP, eigBlockVectorP)
else:
eigBlockVectorX = eigBlockVector[:sizeX]
eigBlockVectorR = eigBlockVector[sizeX:]
pp = np.dot(activeBlockVectorR, eigBlockVectorR)
app = np.dot(activeBlockVectorAR, eigBlockVectorR)
bpp = np.dot(activeBlockVectorBR, eigBlockVectorR)
if verbosityLevel > 10:
print(pp)
print(app)
print(bpp)
pause()
blockVectorX = np.dot(blockVectorX, eigBlockVectorX) + pp
blockVectorAX = np.dot(blockVectorAX, eigBlockVectorX) + app
blockVectorBX = np.dot(blockVectorBX, eigBlockVectorX) + bpp
blockVectorP, blockVectorAP, blockVectorBP = pp, app, bpp
aux = blockVectorBX * _lambda[np.newaxis,:]
blockVectorR = blockVectorAX - aux
aux = np.sum(blockVectorR.conjugate() * blockVectorR, 0)
residualNorms = np.sqrt(aux)
if verbosityLevel > 0:
print('final eigenvalue:', _lambda)
print('final residual norms:', residualNorms)
if retLambdaHistory:
if retResidualNormsHistory:
return _lambda, blockVectorX, lambdaHistory, residualNormsHistory
else:
return _lambda, blockVectorX, lambdaHistory
else:
if retResidualNormsHistory:
return _lambda, blockVectorX, residualNormsHistory
else:
return _lambda, blockVectorX