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""" Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG).
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. https://arxiv.org/abs/0705.2626
.. [3] A. V. Knyazev's C and MATLAB implementations: https://bitbucket.org/joseroman/blopex """
from __future__ import division, print_function, absolute_import import numpy as np from scipy.linalg import (inv, eigh, cho_factor, cho_solve, cholesky, LinAlgError) from scipy.sparse.linalg import aslinearoperator from scipy.sparse.sputils import bmat
__all__ = ['lobpcg']
def _save(ar, fileName): # Used only when verbosity level > 10. np.savetxt(fileName, ar)
def _report_nonhermitian(M, a, b, name): """ Report if `M` is not a hermitian matrix given the tolerances `a`, `b`. """ from scipy.linalg import norm
md = M - M.T.conj()
nmd = norm(md, 1) tol = np.spacing(max(10**a, (10**b)*norm(M, 1))) if nmd > tol: print('matrix %s is not sufficiently Hermitian for a=%d, b=%d:' % (name, a, b)) print('condition: %.e < %e' % (nmd, tol))
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.""" if operatorInput is None: return None 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.conj(), 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.conj(), 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) else: blockVectorBV = None
if retInvR: return blockVectorV, blockVectorBV, gramVBV else: return blockVectorV, blockVectorBV
def _get_indx(_lambda, num, largest): """Get `num` indices into `_lambda` depending on `largest` option.""" ii = np.argsort(_lambda) if largest: ii = ii[:-num-1:-1] else: ii = ii[:num]
return ii
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. 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
Returns ------- w : array Array of k eigenvalues v : array An array of k eigenvectors. V has the same shape as X. lambdas : list of arrays, optional The eigenvalue history, if `retLambdaHistory` is True. rnorms : list of arrays, optional The history of residual norms, if `retResidualNormsHistory` is True.
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, 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 LOBPCG with ``m``=1 and ``n``=10, it works though ``n`` is small. The method is intended for extremely large ``n``/``m``, see e.g., reference [28] in https://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. https://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 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)
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.')
sizeX = min(sizeX, n)
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, dtype=A.dtype)) B_dense = None if B is None else B(np.eye(n, dtype=B.dtype))
vals, vecs = eigh(A_dense, B_dense, eigvals=eigvals, check_finite=False) if largest: # Reverse order to be compatible with eigs() in 'LM' mode. vals = vals[::-1] vecs = vecs[:, ::-1]
return vals, vecs
if (residualTolerance is None) or (residualTolerance <= 0.0): residualTolerance = np.sqrt(1e-15) * n
# 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.conj(), blockVectorBY) try: # gramYBY is a Cholesky factor from now on... gramYBY = cho_factor(gramYBY) except LinAlgError: 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.conj(), blockVectorAX)
_lambda, eigBlockVector = eigh(gramXAX, check_finite=False) ii = _get_indx(_lambda, sizeX, largest) _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
iterationNumber = -1 while iterationNumber < maxIterations: iterationNumber += 1 if verbosityLevel > 0: print('iteration %d' % iterationNumber)
if B is not None: aux = blockVectorBX * _lambda[np.newaxis, :]
else: aux = blockVectorX * _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]) if B is not None: 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: if B is not None: aux = _b_orthonormalize(B, activeBlockVectorP, activeBlockVectorBP, retInvR=True) activeBlockVectorP, activeBlockVectorBP, invR = aux activeBlockVectorAP = np.dot(activeBlockVectorAP, invR)
else: aux = _b_orthonormalize(B, activeBlockVectorP, retInvR=True) activeBlockVectorP, _, invR = aux activeBlockVectorAP = np.dot(activeBlockVectorAP, invR)
## # Perform the Rayleigh Ritz Procedure: # Compute symmetric Gram matrices:
if B is not None: xaw = np.dot(blockVectorX.T.conj(), activeBlockVectorAR) waw = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAR) xbw = np.dot(blockVectorX.T.conj(), activeBlockVectorBR)
if iterationNumber > 0: xap = np.dot(blockVectorX.T.conj(), activeBlockVectorAP) wap = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAP) pap = np.dot(activeBlockVectorP.T.conj(), activeBlockVectorAP) xbp = np.dot(blockVectorX.T.conj(), activeBlockVectorBP) wbp = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorBP)
gramA = bmat([[np.diag(_lambda), xaw, xap], [xaw.T.conj(), waw, wap], [xap.T.conj(), wap.T.conj(), pap]])
gramB = bmat([[ident0, xbw, xbp], [xbw.T.conj(), ident, wbp], [xbp.T.conj(), wbp.T.conj(), ident]]) else: gramA = bmat([[np.diag(_lambda), xaw], [xaw.T.conj(), waw]]) gramB = bmat([[ident0, xbw], [xbw.T.conj(), ident]])
else: xaw = np.dot(blockVectorX.T.conj(), activeBlockVectorAR) waw = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAR) xbw = np.dot(blockVectorX.T.conj(), activeBlockVectorR)
if iterationNumber > 0: xap = np.dot(blockVectorX.T.conj(), activeBlockVectorAP) wap = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAP) pap = np.dot(activeBlockVectorP.T.conj(), activeBlockVectorAP) xbp = np.dot(blockVectorX.T.conj(), activeBlockVectorP) wbp = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorP)
gramA = bmat([[np.diag(_lambda), xaw, xap], [xaw.T.conj(), waw, wap], [xap.T.conj(), wap.T.conj(), pap]])
gramB = bmat([[ident0, xbw, xbp], [xbw.T.conj(), ident, wbp], [xbp.T.conj(), wbp.T.conj(), ident]]) else: gramA = bmat([[np.diag(_lambda), xaw], [xaw.T.conj(), waw]]) gramB = bmat([[ident0, xbw], [xbw.T.conj(), ident]])
if verbosityLevel > 0: _report_nonhermitian(gramA, 3, -1, 'gramA') _report_nonhermitian(gramB, 3, -1, 'gramB')
if verbosityLevel > 10: _save(gramA, 'gramA') _save(gramB, 'gramB')
# Solve the generalized eigenvalue problem. _lambda, eigBlockVector = eigh(gramA, gramB, check_finite=False) ii = _get_indx(_lambda, sizeX, largest)
if verbosityLevel > 10: print(ii) print(_lambda)
_lambda = _lambda[ii] eigBlockVector = eigBlockVector[:, ii]
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)
# Compute Ritz vectors. if B is not None: 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)
blockVectorX = np.dot(blockVectorX, eigBlockVectorX) + pp blockVectorAX = np.dot(blockVectorAX, eigBlockVectorX) + app blockVectorBX = np.dot(blockVectorBX, eigBlockVectorX) + bpp
blockVectorP, blockVectorAP, blockVectorBP = pp, app, bpp
else: 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) else: eigBlockVectorX = eigBlockVector[:sizeX] eigBlockVectorR = eigBlockVector[sizeX:]
pp = np.dot(activeBlockVectorR, eigBlockVectorR) app = np.dot(activeBlockVectorAR, eigBlockVectorR)
if verbosityLevel > 10: print(pp) print(app)
blockVectorX = np.dot(blockVectorX, eigBlockVectorX) + pp blockVectorAX = np.dot(blockVectorAX, eigBlockVectorX) + app
blockVectorP, blockVectorAP = pp, app
if B is not None: aux = blockVectorBX * _lambda[np.newaxis, :]
else: aux = blockVectorX * _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 |