983 lines
36 KiB
Python
983 lines
36 KiB
Python
|
"""
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Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG).
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References
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----------
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.. [1] A. V. Knyazev (2001),
|
||
|
|
Toward the Optimal Preconditioned Eigensolver: Locally Optimal
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||
|
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Block Preconditioned Conjugate Gradient Method.
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||
|
|
SIAM Journal on Scientific Computing 23, no. 2,
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pp. 517-541. :doi:`10.1137/S1064827500366124`
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|
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.. [2] A. V. Knyazev, I. Lashuk, M. E. Argentati, and E. Ovchinnikov (2007),
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||
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Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX)
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in hypre and PETSc. :arxiv:`0705.2626`
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|
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.. [3] A. V. Knyazev's C and MATLAB implementations:
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https://github.com/lobpcg/blopex
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"""
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import warnings
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import numpy as np
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from scipy.linalg import (inv, eigh, cho_factor, cho_solve,
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cholesky, LinAlgError)
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from scipy.sparse.linalg import LinearOperator
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from scipy.sparse import isspmatrix
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from numpy import block as bmat
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__all__ = ["lobpcg"]
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def _report_nonhermitian(M, name):
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"""
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Report if `M` is not a Hermitian matrix given its type.
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"""
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from scipy.linalg import norm
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md = M - M.T.conj()
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nmd = norm(md, 1)
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tol = 10 * np.finfo(M.dtype).eps
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tol = max(tol, tol * norm(M, 1))
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if nmd > tol:
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warnings.warn(
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f"Matrix {name} of the type {M.dtype} is not Hermitian: "
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f"condition: {nmd} < {tol} fails.",
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UserWarning, stacklevel=4
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)
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def _as2d(ar):
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"""
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If the input array is 2D return it, if it is 1D, append a dimension,
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making it a column vector.
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"""
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if ar.ndim == 2:
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return ar
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else: # Assume 1!
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aux = np.array(ar, copy=False)
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aux.shape = (ar.shape[0], 1)
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return aux
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def _makeMatMat(m):
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if m is None:
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return None
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elif callable(m):
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return lambda v: m(v)
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else:
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return lambda v: m @ v
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def _applyConstraints(blockVectorV, factYBY, blockVectorBY, blockVectorY):
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"""Changes blockVectorV in place."""
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YBV = np.dot(blockVectorBY.T.conj(), blockVectorV)
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tmp = cho_solve(factYBY, YBV)
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blockVectorV -= np.dot(blockVectorY, tmp)
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def _b_orthonormalize(B, blockVectorV, blockVectorBV=None,
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verbosityLevel=0):
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"""in-place B-orthonormalize the given block vector using Cholesky."""
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normalization = blockVectorV.max(axis=0) + np.finfo(blockVectorV.dtype).eps
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blockVectorV = blockVectorV / normalization
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if blockVectorBV is None:
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if B is not None:
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try:
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blockVectorBV = B(blockVectorV)
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except Exception as e:
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if verbosityLevel:
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warnings.warn(
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f"Secondary MatMul call failed with error\n"
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f"{e}\n",
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UserWarning, stacklevel=3
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)
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return None, None, None, normalization
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if blockVectorBV.shape != blockVectorV.shape:
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raise ValueError(
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f"The shape {blockVectorV.shape} "
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f"of the orthogonalized matrix not preserved\n"
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f"and changed to {blockVectorBV.shape} "
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f"after multiplying by the secondary matrix.\n"
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)
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else:
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blockVectorBV = blockVectorV # Shared data!!!
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else:
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blockVectorBV = blockVectorBV / normalization
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VBV = blockVectorV.T.conj() @ blockVectorBV
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try:
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# VBV is a Cholesky factor from now on...
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VBV = cholesky(VBV, overwrite_a=True)
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VBV = inv(VBV, overwrite_a=True)
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blockVectorV = blockVectorV @ VBV
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# blockVectorV = (cho_solve((VBV.T, True), blockVectorV.T)).T
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if B is not None:
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blockVectorBV = blockVectorBV @ VBV
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# blockVectorBV = (cho_solve((VBV.T, True), blockVectorBV.T)).T
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return blockVectorV, blockVectorBV, VBV, normalization
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except LinAlgError:
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if verbosityLevel:
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warnings.warn(
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"Cholesky has failed.",
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UserWarning, stacklevel=3
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)
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return None, None, None, normalization
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def _get_indx(_lambda, num, largest):
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"""Get `num` indices into `_lambda` depending on `largest` option."""
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ii = np.argsort(_lambda)
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if largest:
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ii = ii[:-num - 1:-1]
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else:
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ii = ii[:num]
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return ii
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def _handle_gramA_gramB_verbosity(gramA, gramB, verbosityLevel):
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if verbosityLevel:
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_report_nonhermitian(gramA, "gramA")
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_report_nonhermitian(gramB, "gramB")
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def lobpcg(
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A,
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X,
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B=None,
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M=None,
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Y=None,
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tol=None,
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maxiter=None,
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largest=True,
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verbosityLevel=0,
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retLambdaHistory=False,
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retResidualNormsHistory=False,
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restartControl=20,
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):
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"""Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG).
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LOBPCG is a preconditioned eigensolver for large symmetric positive
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definite (SPD) generalized eigenproblems.
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Parameters
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----------
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A : {sparse matrix, dense matrix, LinearOperator, callable object}
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The symmetric linear operator of the problem, usually a
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sparse matrix. Often called the "stiffness matrix".
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X : ndarray, float32 or float64
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Initial approximation to the ``k`` eigenvectors (non-sparse). If `A`
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has ``shape=(n,n)`` then `X` should have shape ``shape=(n,k)``.
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B : {dense matrix, sparse matrix, LinearOperator, callable object}
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Optional.
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The right hand side operator in a generalized eigenproblem.
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By default, ``B = Identity``. Often called the "mass matrix".
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M : {dense matrix, sparse matrix, LinearOperator, callable object}
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Optional.
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Preconditioner to `A`; by default ``M = Identity``.
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`M` should approximate the inverse of `A`.
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Y : ndarray, float32 or float64, optional.
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An n-by-sizeY matrix of constraints (non-sparse), sizeY < n.
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The iterations will be performed in the B-orthogonal complement
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of the column-space of Y. Y must be full rank.
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tol : scalar, optional.
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Solver tolerance (stopping criterion).
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The default is ``tol=n*sqrt(eps)``.
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maxiter : int, optional.
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Maximum number of iterations. The default is ``maxiter=20``.
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largest : bool, optional.
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When True, solve for the largest eigenvalues, otherwise the smallest.
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verbosityLevel : int, optional
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Controls solver output. The default is ``verbosityLevel=0``.
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retLambdaHistory : bool, optional.
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Whether to return eigenvalue history. Default is False.
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retResidualNormsHistory : bool, optional.
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Whether to return history of residual norms. Default is False.
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restartControl : int, optional.
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Iterations restart if the residuals jump up 2**restartControl times
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compared to the smallest ones recorded in retResidualNormsHistory.
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The default is ``restartControl=20``, making the restarts rare for
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backward compatibility.
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Returns
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-------
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w : ndarray
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Array of ``k`` eigenvalues.
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v : ndarray
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An array of ``k`` eigenvectors. `v` has the same shape as `X`.
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lambdas : ndarray, optional
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The eigenvalue history, if `retLambdaHistory` is True.
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rnorms : ndarray, optional
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The history of residual norms, if `retResidualNormsHistory` is True.
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Notes
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-----
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The iterative loop in lobpcg runs maxit=maxiter (or 20 if maxit=None)
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iterations at most and finishes earler if the tolerance is met.
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Breaking backward compatibility with the previous version, lobpcg
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now returns the block of iterative vectors with the best accuracy rather
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than the last one iterated, as a cure for possible divergence.
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The size of the iteration history output equals to the number of the best
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(limited by maxit) iterations plus 3 (initial, final, and postprocessing).
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If both ``retLambdaHistory`` and ``retResidualNormsHistory`` are True,
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the return tuple has the following format
|
||
|
|
``(lambda, V, lambda history, residual norms history)``.
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In the following ``n`` denotes the matrix size and ``k`` the number
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of required eigenvalues (smallest or largest).
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The LOBPCG code internally solves eigenproblems of the size ``3k`` on every
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iteration by calling the dense eigensolver `eigh`, so if ``k`` is not
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small enough compared to ``n``, it makes no sense to call the LOBPCG code.
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Moreover, if one calls the LOBPCG algorithm for ``5k > n``, it would likely
|
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break internally, so the code calls the standard function `eigh` instead.
|
||
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It is not that ``n`` should be large for the LOBPCG to work, but rather the
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ratio ``n / k`` should be large. It you call LOBPCG with ``k=1``
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and ``n=10``, it works though ``n`` is small. The method is intended
|
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for extremely large ``n / k``.
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The convergence speed depends basically on two factors:
|
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1. Relative separation of the seeking eigenvalues from the rest
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|
|
of the eigenvalues. One can vary ``k`` to improve the absolute
|
||
|
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separation and use proper preconditioning to shrink the spectral spread.
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||
|
|
For example, a rod vibration test problem (under tests
|
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|
|
directory) is ill-conditioned for large ``n``, so convergence will be
|
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slow, unless efficient preconditioning is used. For this specific
|
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problem, a good simple preconditioner function would be a linear solve
|
||
|
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for `A`, which is easy to code since `A` is tridiagonal.
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2. Quality of the initial approximations `X` to the seeking eigenvectors.
|
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Randomly distributed around the origin vectors work well if no better
|
||
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|
choice is known.
|
||
|
|
|
||
|
|
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. :doi:`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. :arxiv:`0705.2626`
|
||
|
|
|
||
|
|
.. [3] A. V. Knyazev's C and MATLAB implementations:
|
||
|
|
https://github.com/lobpcg/blopex
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
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Solve ``A x = lambda x`` with constraints and preconditioning.
|
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|
|
|
||
|
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>>> import numpy as np
|
||
|
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>>> from scipy.sparse import spdiags, issparse
|
||
|
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>>> from scipy.sparse.linalg import lobpcg, LinearOperator
|
||
|
|
|
||
|
|
The square matrix size:
|
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|
|
|
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|
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>>> n = 100
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|
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>>> vals = np.arange(1, n + 1)
|
||
|
|
|
||
|
|
The first mandatory input parameter, in this test
|
||
|
|
a sparse 2D array representing the square matrix
|
||
|
|
of the eigenvalue problem to solve:
|
||
|
|
|
||
|
|
>>> A = spdiags(vals, 0, n, n)
|
||
|
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>>> A.toarray()
|
||
|
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array([[ 1, 0, 0, ..., 0, 0, 0],
|
||
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[ 0, 2, 0, ..., 0, 0, 0],
|
||
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[ 0, 0, 3, ..., 0, 0, 0],
|
||
|
|
...,
|
||
|
|
[ 0, 0, 0, ..., 98, 0, 0],
|
||
|
|
[ 0, 0, 0, ..., 0, 99, 0],
|
||
|
|
[ 0, 0, 0, ..., 0, 0, 100]])
|
||
|
|
|
||
|
|
Initial guess for eigenvectors, should have linearly independent
|
||
|
|
columns. The second mandatory input parameter, a 2D array with the
|
||
|
|
row dimension determining the number of requested eigenvalues.
|
||
|
|
If no initial approximations available, randomly oriented vectors
|
||
|
|
commonly work best, e.g., with components normally disrtibuted
|
||
|
|
around zero or uniformly distributed on the interval [-1 1].
|
||
|
|
|
||
|
|
>>> rng = np.random.default_rng()
|
||
|
|
>>> X = rng.normal(size=(n, 3))
|
||
|
|
|
||
|
|
Constraints - an optional input parameter is a 2D array comprising
|
||
|
|
of column vectors that the eigenvectors must be orthogonal to:
|
||
|
|
|
||
|
|
>>> Y = np.eye(n, 3)
|
||
|
|
|
||
|
|
Preconditioner in the inverse of A in this example:
|
||
|
|
|
||
|
|
>>> invA = spdiags([1./vals], 0, n, n)
|
||
|
|
|
||
|
|
The preconditiner must be defined by a function:
|
||
|
|
|
||
|
|
>>> def precond( x ):
|
||
|
|
... return invA @ x
|
||
|
|
|
||
|
|
The argument x of the preconditioner function is a matrix inside `lobpcg`,
|
||
|
|
thus the use of matrix-matrix product ``@``.
|
||
|
|
|
||
|
|
The preconditioner function is passed to lobpcg as a `LinearOperator`:
|
||
|
|
|
||
|
|
>>> M = LinearOperator(matvec=precond, matmat=precond,
|
||
|
|
... shape=(n, n), dtype=np.float64)
|
||
|
|
|
||
|
|
Let us now solve the eigenvalue problem for the matrix A:
|
||
|
|
|
||
|
|
>>> eigenvalues, _ = lobpcg(A, X, Y=Y, M=M, largest=False)
|
||
|
|
>>> eigenvalues
|
||
|
|
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.
|
||
|
|
"""
|
||
|
|
blockVectorX = X
|
||
|
|
bestblockVectorX = blockVectorX
|
||
|
|
blockVectorY = Y
|
||
|
|
residualTolerance = tol
|
||
|
|
if maxiter is None:
|
||
|
|
maxiter = 20
|
||
|
|
|
||
|
|
bestIterationNumber = maxiter
|
||
|
|
|
||
|
|
sizeY = 0
|
||
|
|
if blockVectorY is not None:
|
||
|
|
if len(blockVectorY.shape) != 2:
|
||
|
|
warnings.warn(
|
||
|
|
f"Expected rank-2 array for argument Y, instead got "
|
||
|
|
f"{len(blockVectorY.shape)}, "
|
||
|
|
f"so ignore it and use no constraints.",
|
||
|
|
UserWarning, stacklevel=2
|
||
|
|
)
|
||
|
|
blockVectorY = None
|
||
|
|
else:
|
||
|
|
sizeY = blockVectorY.shape[1]
|
||
|
|
|
||
|
|
# Block size.
|
||
|
|
if blockVectorX is None:
|
||
|
|
raise ValueError("The mandatory initial matrix X cannot be None")
|
||
|
|
if len(blockVectorX.shape) != 2:
|
||
|
|
raise ValueError("expected rank-2 array for argument X")
|
||
|
|
|
||
|
|
n, sizeX = blockVectorX.shape
|
||
|
|
|
||
|
|
# Data type of iterates, determined by X, must be inexact
|
||
|
|
if not np.issubdtype(blockVectorX.dtype, np.inexact):
|
||
|
|
warnings.warn(
|
||
|
|
f"Data type for argument X is {blockVectorX.dtype}, "
|
||
|
|
f"which is not inexact, so casted to np.float32.",
|
||
|
|
UserWarning, stacklevel=2
|
||
|
|
)
|
||
|
|
blockVectorX = np.asarray(blockVectorX, dtype=np.float32)
|
||
|
|
|
||
|
|
if retLambdaHistory:
|
||
|
|
lambdaHistory = np.zeros((maxiter + 3, sizeX),
|
||
|
|
dtype=blockVectorX.dtype)
|
||
|
|
if retResidualNormsHistory:
|
||
|
|
residualNormsHistory = np.zeros((maxiter + 3, sizeX),
|
||
|
|
dtype=blockVectorX.dtype)
|
||
|
|
|
||
|
|
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)
|
||
|
|
|
||
|
|
if (n - sizeY) < (5 * sizeX):
|
||
|
|
warnings.warn(
|
||
|
|
f"The problem size {n} minus the constraints size {sizeY} "
|
||
|
|
f"is too small relative to the block size {sizeX}. "
|
||
|
|
f"Using a dense eigensolver instead of LOBPCG iterations."
|
||
|
|
f"No output of the history of the iterations.",
|
||
|
|
UserWarning, stacklevel=2
|
||
|
|
)
|
||
|
|
|
||
|
|
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)
|
||
|
|
|
||
|
|
try:
|
||
|
|
if isinstance(A, LinearOperator):
|
||
|
|
A = A(np.eye(n, dtype=int))
|
||
|
|
elif callable(A):
|
||
|
|
A = A(np.eye(n, dtype=int))
|
||
|
|
if A.shape != (n, n):
|
||
|
|
raise ValueError(
|
||
|
|
f"The shape {A.shape} of the primary matrix\n"
|
||
|
|
f"defined by a callable object is wrong.\n"
|
||
|
|
)
|
||
|
|
elif isspmatrix(A):
|
||
|
|
A = A.toarray()
|
||
|
|
else:
|
||
|
|
A = np.asarray(A)
|
||
|
|
except Exception as e:
|
||
|
|
raise Exception(
|
||
|
|
f"Primary MatMul call failed with error\n"
|
||
|
|
f"{e}\n")
|
||
|
|
|
||
|
|
if B is not None:
|
||
|
|
try:
|
||
|
|
if isinstance(B, LinearOperator):
|
||
|
|
B = B(np.eye(n, dtype=int))
|
||
|
|
elif callable(B):
|
||
|
|
B = B(np.eye(n, dtype=int))
|
||
|
|
if B.shape != (n, n):
|
||
|
|
raise ValueError(
|
||
|
|
f"The shape {B.shape} of the secondary matrix\n"
|
||
|
|
f"defined by a callable object is wrong.\n"
|
||
|
|
)
|
||
|
|
elif isspmatrix(B):
|
||
|
|
B = B.toarray()
|
||
|
|
else:
|
||
|
|
B = np.asarray(B)
|
||
|
|
except Exception as e:
|
||
|
|
raise Exception(
|
||
|
|
f"Secondary MatMul call failed with error\n"
|
||
|
|
f"{e}\n")
|
||
|
|
|
||
|
|
try:
|
||
|
|
vals, vecs = eigh(A,
|
||
|
|
B,
|
||
|
|
subset_by_index=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
|
||
|
|
except Exception as e:
|
||
|
|
raise Exception(
|
||
|
|
f"Dense eigensolver failed with error\n"
|
||
|
|
f"{e}\n"
|
||
|
|
)
|
||
|
|
|
||
|
|
if (residualTolerance is None) or (residualTolerance <= 0.0):
|
||
|
|
residualTolerance = np.sqrt(np.finfo(blockVectorX.dtype).eps) * n
|
||
|
|
|
||
|
|
A = _makeMatMat(A)
|
||
|
|
B = _makeMatMat(B)
|
||
|
|
M = _makeMatMat(M)
|
||
|
|
|
||
|
|
# Apply constraints to X.
|
||
|
|
if blockVectorY is not None:
|
||
|
|
|
||
|
|
if B is not None:
|
||
|
|
blockVectorBY = B(blockVectorY)
|
||
|
|
if blockVectorBY.shape != blockVectorY.shape:
|
||
|
|
raise ValueError(
|
||
|
|
f"The shape {blockVectorY.shape} "
|
||
|
|
f"of the constraint not preserved\n"
|
||
|
|
f"and changed to {blockVectorBY.shape} "
|
||
|
|
f"after multiplying by the secondary matrix.\n"
|
||
|
|
)
|
||
|
|
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 as e:
|
||
|
|
raise ValueError("Linearly dependent constraints") from e
|
||
|
|
|
||
|
|
_applyConstraints(blockVectorX, gramYBY, blockVectorBY, blockVectorY)
|
||
|
|
|
||
|
|
##
|
||
|
|
# B-orthonormalize X.
|
||
|
|
blockVectorX, blockVectorBX, _, _ = _b_orthonormalize(
|
||
|
|
B, blockVectorX, verbosityLevel=verbosityLevel)
|
||
|
|
if blockVectorX is None:
|
||
|
|
raise ValueError("Linearly dependent initial approximations")
|
||
|
|
|
||
|
|
##
|
||
|
|
# Compute the initial Ritz vectors: solve the eigenproblem.
|
||
|
|
blockVectorAX = A(blockVectorX)
|
||
|
|
if blockVectorAX.shape != blockVectorX.shape:
|
||
|
|
raise ValueError(
|
||
|
|
f"The shape {blockVectorX.shape} "
|
||
|
|
f"of the initial approximations not preserved\n"
|
||
|
|
f"and changed to {blockVectorAX.shape} "
|
||
|
|
f"after multiplying by the primary matrix.\n"
|
||
|
|
)
|
||
|
|
|
||
|
|
gramXAX = np.dot(blockVectorX.T.conj(), blockVectorAX)
|
||
|
|
|
||
|
|
_lambda, eigBlockVector = eigh(gramXAX, check_finite=False)
|
||
|
|
ii = _get_indx(_lambda, sizeX, largest)
|
||
|
|
_lambda = _lambda[ii]
|
||
|
|
if retLambdaHistory:
|
||
|
|
lambdaHistory[0, :] = _lambda
|
||
|
|
|
||
|
|
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)
|
||
|
|
|
||
|
|
##
|
||
|
|
# Main iteration loop.
|
||
|
|
|
||
|
|
blockVectorP = None # set during iteration
|
||
|
|
blockVectorAP = None
|
||
|
|
blockVectorBP = None
|
||
|
|
|
||
|
|
smallestResidualNorm = np.abs(np.finfo(blockVectorX.dtype).max)
|
||
|
|
|
||
|
|
iterationNumber = -1
|
||
|
|
restart = True
|
||
|
|
forcedRestart = False
|
||
|
|
explicitGramFlag = False
|
||
|
|
while iterationNumber < maxiter:
|
||
|
|
iterationNumber += 1
|
||
|
|
|
||
|
|
if B is not None:
|
||
|
|
aux = blockVectorBX * _lambda[np.newaxis, :]
|
||
|
|
else:
|
||
|
|
aux = blockVectorX * _lambda[np.newaxis, :]
|
||
|
|
|
||
|
|
blockVectorR = blockVectorAX - aux
|
||
|
|
|
||
|
|
aux = np.sum(blockVectorR.conj() * blockVectorR, 0)
|
||
|
|
residualNorms = np.sqrt(np.abs(aux))
|
||
|
|
if retResidualNormsHistory:
|
||
|
|
residualNormsHistory[iterationNumber, :] = residualNorms
|
||
|
|
residualNorm = np.sum(np.abs(residualNorms)) / sizeX
|
||
|
|
|
||
|
|
if residualNorm < smallestResidualNorm:
|
||
|
|
smallestResidualNorm = residualNorm
|
||
|
|
bestIterationNumber = iterationNumber
|
||
|
|
bestblockVectorX = blockVectorX
|
||
|
|
elif residualNorm > 2**restartControl * smallestResidualNorm:
|
||
|
|
forcedRestart = True
|
||
|
|
blockVectorAX = A(blockVectorX)
|
||
|
|
if blockVectorAX.shape != blockVectorX.shape:
|
||
|
|
raise ValueError(
|
||
|
|
f"The shape {blockVectorX.shape} "
|
||
|
|
f"of the restarted iterate not preserved\n"
|
||
|
|
f"and changed to {blockVectorAX.shape} "
|
||
|
|
f"after multiplying by the primary matrix.\n"
|
||
|
|
)
|
||
|
|
if B is not None:
|
||
|
|
blockVectorBX = B(blockVectorX)
|
||
|
|
if blockVectorBX.shape != blockVectorX.shape:
|
||
|
|
raise ValueError(
|
||
|
|
f"The shape {blockVectorX.shape} "
|
||
|
|
f"of the restarted iterate not preserved\n"
|
||
|
|
f"and changed to {blockVectorBX.shape} "
|
||
|
|
f"after multiplying by the secondary matrix.\n"
|
||
|
|
)
|
||
|
|
|
||
|
|
ii = np.where(residualNorms > residualTolerance, True, False)
|
||
|
|
activeMask = activeMask & ii
|
||
|
|
currentBlockSize = activeMask.sum()
|
||
|
|
|
||
|
|
if verbosityLevel:
|
||
|
|
print(f"iteration {iterationNumber}")
|
||
|
|
print(f"current block size: {currentBlockSize}")
|
||
|
|
print(f"eigenvalue(s):\n{_lambda}")
|
||
|
|
print(f"residual norm(s):\n{residualNorms}")
|
||
|
|
|
||
|
|
if currentBlockSize == 0:
|
||
|
|
break
|
||
|
|
|
||
|
|
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-orthogonalize the preconditioned residuals to X.
|
||
|
|
if B is not None:
|
||
|
|
activeBlockVectorR = activeBlockVectorR - (
|
||
|
|
blockVectorX @
|
||
|
|
(blockVectorBX.T.conj() @ activeBlockVectorR)
|
||
|
|
)
|
||
|
|
else:
|
||
|
|
activeBlockVectorR = activeBlockVectorR - (
|
||
|
|
blockVectorX @
|
||
|
|
(blockVectorX.T.conj() @ activeBlockVectorR)
|
||
|
|
)
|
||
|
|
|
||
|
|
##
|
||
|
|
# B-orthonormalize the preconditioned residuals.
|
||
|
|
aux = _b_orthonormalize(
|
||
|
|
B, activeBlockVectorR, verbosityLevel=verbosityLevel)
|
||
|
|
activeBlockVectorR, activeBlockVectorBR, _, _ = aux
|
||
|
|
|
||
|
|
if activeBlockVectorR is None:
|
||
|
|
warnings.warn(
|
||
|
|
f"Failed at iteration {iterationNumber} with accuracies "
|
||
|
|
f"{residualNorms}\n not reaching the requested "
|
||
|
|
f"tolerance {residualTolerance}.",
|
||
|
|
UserWarning, stacklevel=2
|
||
|
|
)
|
||
|
|
break
|
||
|
|
activeBlockVectorAR = A(activeBlockVectorR)
|
||
|
|
|
||
|
|
if iterationNumber > 0:
|
||
|
|
if B is not None:
|
||
|
|
aux = _b_orthonormalize(
|
||
|
|
B, activeBlockVectorP, activeBlockVectorBP,
|
||
|
|
verbosityLevel=verbosityLevel
|
||
|
|
)
|
||
|
|
activeBlockVectorP, activeBlockVectorBP, invR, normal = aux
|
||
|
|
else:
|
||
|
|
aux = _b_orthonormalize(B, activeBlockVectorP,
|
||
|
|
verbosityLevel=verbosityLevel)
|
||
|
|
activeBlockVectorP, _, invR, normal = aux
|
||
|
|
# Function _b_orthonormalize returns None if Cholesky fails
|
||
|
|
if activeBlockVectorP is not None:
|
||
|
|
activeBlockVectorAP = activeBlockVectorAP / normal
|
||
|
|
activeBlockVectorAP = np.dot(activeBlockVectorAP, invR)
|
||
|
|
restart = forcedRestart
|
||
|
|
else:
|
||
|
|
restart = True
|
||
|
|
|
||
|
|
##
|
||
|
|
# Perform the Rayleigh Ritz Procedure:
|
||
|
|
# Compute symmetric Gram matrices:
|
||
|
|
|
||
|
|
if activeBlockVectorAR.dtype == "float32":
|
||
|
|
myeps = 1
|
||
|
|
else:
|
||
|
|
myeps = np.sqrt(np.finfo(activeBlockVectorR.dtype).eps)
|
||
|
|
|
||
|
|
if residualNorms.max() > myeps and not explicitGramFlag:
|
||
|
|
explicitGramFlag = False
|
||
|
|
else:
|
||
|
|
# Once explicitGramFlag, forever explicitGramFlag.
|
||
|
|
explicitGramFlag = True
|
||
|
|
|
||
|
|
# Shared memory assingments to simplify the code
|
||
|
|
if B is None:
|
||
|
|
blockVectorBX = blockVectorX
|
||
|
|
activeBlockVectorBR = activeBlockVectorR
|
||
|
|
if not restart:
|
||
|
|
activeBlockVectorBP = activeBlockVectorP
|
||
|
|
|
||
|
|
# Common submatrices:
|
||
|
|
gramXAR = np.dot(blockVectorX.T.conj(), activeBlockVectorAR)
|
||
|
|
gramRAR = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAR)
|
||
|
|
|
||
|
|
gramDtype = activeBlockVectorAR.dtype
|
||
|
|
if explicitGramFlag:
|
||
|
|
gramRAR = (gramRAR + gramRAR.T.conj()) / 2
|
||
|
|
gramXAX = np.dot(blockVectorX.T.conj(), blockVectorAX)
|
||
|
|
gramXAX = (gramXAX + gramXAX.T.conj()) / 2
|
||
|
|
gramXBX = np.dot(blockVectorX.T.conj(), blockVectorBX)
|
||
|
|
gramRBR = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorBR)
|
||
|
|
gramXBR = np.dot(blockVectorX.T.conj(), activeBlockVectorBR)
|
||
|
|
else:
|
||
|
|
gramXAX = np.diag(_lambda).astype(gramDtype)
|
||
|
|
gramXBX = np.eye(sizeX, dtype=gramDtype)
|
||
|
|
gramRBR = np.eye(currentBlockSize, dtype=gramDtype)
|
||
|
|
gramXBR = np.zeros((sizeX, currentBlockSize), dtype=gramDtype)
|
||
|
|
|
||
|
|
if not restart:
|
||
|
|
gramXAP = np.dot(blockVectorX.T.conj(), activeBlockVectorAP)
|
||
|
|
gramRAP = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAP)
|
||
|
|
gramPAP = np.dot(activeBlockVectorP.T.conj(), activeBlockVectorAP)
|
||
|
|
gramXBP = np.dot(blockVectorX.T.conj(), activeBlockVectorBP)
|
||
|
|
gramRBP = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorBP)
|
||
|
|
if explicitGramFlag:
|
||
|
|
gramPAP = (gramPAP + gramPAP.T.conj()) / 2
|
||
|
|
gramPBP = np.dot(activeBlockVectorP.T.conj(),
|
||
|
|
activeBlockVectorBP)
|
||
|
|
else:
|
||
|
|
gramPBP = np.eye(currentBlockSize, dtype=gramDtype)
|
||
|
|
|
||
|
|
gramA = bmat(
|
||
|
|
[
|
||
|
|
[gramXAX, gramXAR, gramXAP],
|
||
|
|
[gramXAR.T.conj(), gramRAR, gramRAP],
|
||
|
|
[gramXAP.T.conj(), gramRAP.T.conj(), gramPAP],
|
||
|
|
]
|
||
|
|
)
|
||
|
|
gramB = bmat(
|
||
|
|
[
|
||
|
|
[gramXBX, gramXBR, gramXBP],
|
||
|
|
[gramXBR.T.conj(), gramRBR, gramRBP],
|
||
|
|
[gramXBP.T.conj(), gramRBP.T.conj(), gramPBP],
|
||
|
|
]
|
||
|
|
)
|
||
|
|
|
||
|
|
_handle_gramA_gramB_verbosity(gramA, gramB, verbosityLevel)
|
||
|
|
|
||
|
|
try:
|
||
|
|
_lambda, eigBlockVector = eigh(gramA,
|
||
|
|
gramB,
|
||
|
|
check_finite=False)
|
||
|
|
except LinAlgError as e:
|
||
|
|
# raise ValueError("eigh failed in lobpcg iterations") from e
|
||
|
|
if verbosityLevel:
|
||
|
|
warnings.warn(
|
||
|
|
f"eigh failed at iteration {iterationNumber} \n"
|
||
|
|
f"with error {e} causing a restart.\n",
|
||
|
|
UserWarning, stacklevel=2
|
||
|
|
)
|
||
|
|
# try again after dropping the direction vectors P from RR
|
||
|
|
restart = True
|
||
|
|
|
||
|
|
if restart:
|
||
|
|
gramA = bmat([[gramXAX, gramXAR], [gramXAR.T.conj(), gramRAR]])
|
||
|
|
gramB = bmat([[gramXBX, gramXBR], [gramXBR.T.conj(), gramRBR]])
|
||
|
|
|
||
|
|
_handle_gramA_gramB_verbosity(gramA, gramB, verbosityLevel)
|
||
|
|
|
||
|
|
try:
|
||
|
|
_lambda, eigBlockVector = eigh(gramA,
|
||
|
|
gramB,
|
||
|
|
check_finite=False)
|
||
|
|
except LinAlgError as e:
|
||
|
|
# raise ValueError("eigh failed in lobpcg iterations") from e
|
||
|
|
warnings.warn(
|
||
|
|
f"eigh failed at iteration {iterationNumber} with error\n"
|
||
|
|
f"{e}\n",
|
||
|
|
UserWarning, stacklevel=2
|
||
|
|
)
|
||
|
|
break
|
||
|
|
|
||
|
|
ii = _get_indx(_lambda, sizeX, largest)
|
||
|
|
_lambda = _lambda[ii]
|
||
|
|
eigBlockVector = eigBlockVector[:, ii]
|
||
|
|
if retLambdaHistory:
|
||
|
|
lambdaHistory[iterationNumber + 1, :] = _lambda
|
||
|
|
|
||
|
|
# Compute Ritz vectors.
|
||
|
|
if B is not None:
|
||
|
|
if not restart:
|
||
|
|
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)
|
||
|
|
|
||
|
|
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 not restart:
|
||
|
|
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)
|
||
|
|
|
||
|
|
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.conj() * blockVectorR, 0)
|
||
|
|
residualNorms = np.sqrt(np.abs(aux))
|
||
|
|
# Use old lambda in case of early loop exit.
|
||
|
|
if retLambdaHistory:
|
||
|
|
lambdaHistory[iterationNumber + 1, :] = _lambda
|
||
|
|
if retResidualNormsHistory:
|
||
|
|
residualNormsHistory[iterationNumber + 1, :] = residualNorms
|
||
|
|
residualNorm = np.sum(np.abs(residualNorms)) / sizeX
|
||
|
|
if residualNorm < smallestResidualNorm:
|
||
|
|
smallestResidualNorm = residualNorm
|
||
|
|
bestIterationNumber = iterationNumber + 1
|
||
|
|
bestblockVectorX = blockVectorX
|
||
|
|
|
||
|
|
if np.max(np.abs(residualNorms)) > residualTolerance:
|
||
|
|
warnings.warn(
|
||
|
|
f"Exited at iteration {iterationNumber} with accuracies \n"
|
||
|
|
f"{residualNorms}\n"
|
||
|
|
f"not reaching the requested tolerance {residualTolerance}.\n"
|
||
|
|
f"Use iteration {bestIterationNumber} instead with accuracy \n"
|
||
|
|
f"{smallestResidualNorm}.\n",
|
||
|
|
UserWarning, stacklevel=2
|
||
|
|
)
|
||
|
|
|
||
|
|
if verbosityLevel:
|
||
|
|
print(f"Final iterative eigenvalue(s):\n{_lambda}")
|
||
|
|
print(f"Final iterative residual norm(s):\n{residualNorms}")
|
||
|
|
|
||
|
|
blockVectorX = bestblockVectorX
|
||
|
|
# Making eigenvectors "exactly" satisfy the blockVectorY constrains
|
||
|
|
if blockVectorY is not None:
|
||
|
|
_applyConstraints(blockVectorX,
|
||
|
|
gramYBY,
|
||
|
|
blockVectorBY,
|
||
|
|
blockVectorY)
|
||
|
|
|
||
|
|
# Making eigenvectors "exactly" othonormalized by final "exact" RR
|
||
|
|
blockVectorAX = A(blockVectorX)
|
||
|
|
if blockVectorAX.shape != blockVectorX.shape:
|
||
|
|
raise ValueError(
|
||
|
|
f"The shape {blockVectorX.shape} "
|
||
|
|
f"of the postprocessing iterate not preserved\n"
|
||
|
|
f"and changed to {blockVectorAX.shape} "
|
||
|
|
f"after multiplying by the primary matrix.\n"
|
||
|
|
)
|
||
|
|
gramXAX = np.dot(blockVectorX.T.conj(), blockVectorAX)
|
||
|
|
|
||
|
|
blockVectorBX = blockVectorX
|
||
|
|
if B is not None:
|
||
|
|
blockVectorBX = B(blockVectorX)
|
||
|
|
if blockVectorBX.shape != blockVectorX.shape:
|
||
|
|
raise ValueError(
|
||
|
|
f"The shape {blockVectorX.shape} "
|
||
|
|
f"of the postprocessing iterate not preserved\n"
|
||
|
|
f"and changed to {blockVectorBX.shape} "
|
||
|
|
f"after multiplying by the secondary matrix.\n"
|
||
|
|
)
|
||
|
|
|
||
|
|
gramXBX = np.dot(blockVectorX.T.conj(), blockVectorBX)
|
||
|
|
_handle_gramA_gramB_verbosity(gramXAX, gramXBX, verbosityLevel)
|
||
|
|
gramXAX = (gramXAX + gramXAX.T.conj()) / 2
|
||
|
|
gramXBX = (gramXBX + gramXBX.T.conj()) / 2
|
||
|
|
try:
|
||
|
|
_lambda, eigBlockVector = eigh(gramXAX,
|
||
|
|
gramXBX,
|
||
|
|
check_finite=False)
|
||
|
|
except LinAlgError as e:
|
||
|
|
raise ValueError("eigh has failed in lobpcg postprocessing") from e
|
||
|
|
|
||
|
|
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)
|
||
|
|
aux = blockVectorBX * _lambda[np.newaxis, :]
|
||
|
|
else:
|
||
|
|
aux = blockVectorX * _lambda[np.newaxis, :]
|
||
|
|
|
||
|
|
blockVectorR = blockVectorAX - aux
|
||
|
|
|
||
|
|
aux = np.sum(blockVectorR.conj() * blockVectorR, 0)
|
||
|
|
residualNorms = np.sqrt(np.abs(aux))
|
||
|
|
|
||
|
|
if retLambdaHistory:
|
||
|
|
lambdaHistory[bestIterationNumber + 1, :] = _lambda
|
||
|
|
if retResidualNormsHistory:
|
||
|
|
residualNormsHistory[bestIterationNumber + 1, :] = residualNorms
|
||
|
|
|
||
|
|
if retLambdaHistory:
|
||
|
|
lambdaHistory = lambdaHistory[
|
||
|
|
: bestIterationNumber + 2, :]
|
||
|
|
if retResidualNormsHistory:
|
||
|
|
residualNormsHistory = residualNormsHistory[
|
||
|
|
: bestIterationNumber + 2, :]
|
||
|
|
|
||
|
|
if np.max(np.abs(residualNorms)) > residualTolerance:
|
||
|
|
warnings.warn(
|
||
|
|
f"Exited postprocessing with accuracies \n"
|
||
|
|
f"{residualNorms}\n"
|
||
|
|
f"not reaching the requested tolerance {residualTolerance}.",
|
||
|
|
UserWarning, stacklevel=2
|
||
|
|
)
|
||
|
|
|
||
|
|
if verbosityLevel:
|
||
|
|
print(f"Final postprocessing eigenvalue(s):\n{_lambda}")
|
||
|
|
print(f"Final residual norm(s):\n{residualNorms}")
|
||
|
|
|
||
|
|
if retLambdaHistory:
|
||
|
|
lambdaHistory = np.vsplit(lambdaHistory, np.shape(lambdaHistory)[0])
|
||
|
|
lambdaHistory = [np.squeeze(i) for i in lambdaHistory]
|
||
|
|
if retResidualNormsHistory:
|
||
|
|
residualNormsHistory = np.vsplit(residualNormsHistory,
|
||
|
|
np.shape(residualNormsHistory)[0])
|
||
|
|
residualNormsHistory = [np.squeeze(i) for i in residualNormsHistory]
|
||
|
|
|
||
|
|
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
|