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488 lines
15 KiB
Python
488 lines
15 KiB
Python
# Copyright (C) 2015, Pauli Virtanen <pav@iki.fi>
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# Distributed under the same license as SciPy.
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import warnings
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import numpy as np
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from numpy.linalg import LinAlgError
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from scipy.linalg import (get_blas_funcs, qr, solve, svd, qr_insert, lstsq)
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from scipy.sparse.linalg.isolve.utils import make_system
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__all__ = ['gcrotmk']
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def _fgmres(matvec, v0, m, atol, lpsolve=None, rpsolve=None, cs=(), outer_v=(),
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prepend_outer_v=False):
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"""
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FGMRES Arnoldi process, with optional projection or augmentation
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Parameters
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----------
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matvec : callable
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Operation A*x
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v0 : ndarray
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Initial vector, normalized to nrm2(v0) == 1
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m : int
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Number of GMRES rounds
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atol : float
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Absolute tolerance for early exit
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lpsolve : callable
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Left preconditioner L
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rpsolve : callable
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Right preconditioner R
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CU : list of (ndarray, ndarray)
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Columns of matrices C and U in GCROT
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outer_v : list of ndarrays
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Augmentation vectors in LGMRES
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prepend_outer_v : bool, optional
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Whether augmentation vectors come before or after
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Krylov iterates
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Raises
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------
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LinAlgError
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If nans encountered
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Returns
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-------
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Q, R : ndarray
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QR decomposition of the upper Hessenberg H=QR
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B : ndarray
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Projections corresponding to matrix C
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vs : list of ndarray
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Columns of matrix V
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zs : list of ndarray
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Columns of matrix Z
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y : ndarray
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Solution to ||H y - e_1||_2 = min!
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res : float
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The final (preconditioned) residual norm
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"""
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if lpsolve is None:
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lpsolve = lambda x: x
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if rpsolve is None:
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rpsolve = lambda x: x
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axpy, dot, scal, nrm2 = get_blas_funcs(['axpy', 'dot', 'scal', 'nrm2'], (v0,))
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vs = [v0]
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zs = []
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y = None
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res = np.nan
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m = m + len(outer_v)
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# Orthogonal projection coefficients
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B = np.zeros((len(cs), m), dtype=v0.dtype)
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# H is stored in QR factorized form
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Q = np.ones((1, 1), dtype=v0.dtype)
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R = np.zeros((1, 0), dtype=v0.dtype)
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eps = np.finfo(v0.dtype).eps
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breakdown = False
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# FGMRES Arnoldi process
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for j in range(m):
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# L A Z = C B + V H
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if prepend_outer_v and j < len(outer_v):
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z, w = outer_v[j]
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elif prepend_outer_v and j == len(outer_v):
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z = rpsolve(v0)
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w = None
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elif not prepend_outer_v and j >= m - len(outer_v):
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z, w = outer_v[j - (m - len(outer_v))]
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else:
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z = rpsolve(vs[-1])
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w = None
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if w is None:
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w = lpsolve(matvec(z))
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else:
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# w is clobbered below
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w = w.copy()
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w_norm = nrm2(w)
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# GCROT projection: L A -> (1 - C C^H) L A
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# i.e. orthogonalize against C
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for i, c in enumerate(cs):
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alpha = dot(c, w)
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B[i,j] = alpha
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w = axpy(c, w, c.shape[0], -alpha) # w -= alpha*c
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# Orthogonalize against V
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hcur = np.zeros(j+2, dtype=Q.dtype)
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for i, v in enumerate(vs):
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alpha = dot(v, w)
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hcur[i] = alpha
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w = axpy(v, w, v.shape[0], -alpha) # w -= alpha*v
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hcur[i+1] = nrm2(w)
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with np.errstate(over='ignore', divide='ignore'):
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# Careful with denormals
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alpha = 1/hcur[-1]
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if np.isfinite(alpha):
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w = scal(alpha, w)
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if not (hcur[-1] > eps * w_norm):
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# w essentially in the span of previous vectors,
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# or we have nans. Bail out after updating the QR
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# solution.
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breakdown = True
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vs.append(w)
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zs.append(z)
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# Arnoldi LSQ problem
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# Add new column to H=Q*R, padding other columns with zeros
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Q2 = np.zeros((j+2, j+2), dtype=Q.dtype, order='F')
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Q2[:j+1,:j+1] = Q
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Q2[j+1,j+1] = 1
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R2 = np.zeros((j+2, j), dtype=R.dtype, order='F')
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R2[:j+1,:] = R
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Q, R = qr_insert(Q2, R2, hcur, j, which='col',
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overwrite_qru=True, check_finite=False)
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# Transformed least squares problem
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# || Q R y - inner_res_0 * e_1 ||_2 = min!
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# Since R = [R'; 0], solution is y = inner_res_0 (R')^{-1} (Q^H)[:j,0]
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# Residual is immediately known
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res = abs(Q[0,-1])
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# Check for termination
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if res < atol or breakdown:
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break
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if not np.isfinite(R[j,j]):
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# nans encountered, bail out
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raise LinAlgError()
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# -- Get the LSQ problem solution
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# The problem is triangular, but the condition number may be
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# bad (or in case of breakdown the last diagonal entry may be
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# zero), so use lstsq instead of trtrs.
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y, _, _, _, = lstsq(R[:j+1,:j+1], Q[0,:j+1].conj())
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B = B[:,:j+1]
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return Q, R, B, vs, zs, y, res
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def gcrotmk(A, b, x0=None, tol=1e-5, maxiter=1000, M=None, callback=None,
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m=20, k=None, CU=None, discard_C=False, truncate='oldest',
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atol=None):
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"""
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Solve a matrix equation using flexible GCROT(m,k) algorithm.
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Parameters
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----------
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A : {sparse matrix, dense matrix, LinearOperator}
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The real or complex N-by-N matrix of the linear system.
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Alternatively, ``A`` can be a linear operator which can
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produce ``Ax`` using, e.g.,
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``scipy.sparse.linalg.LinearOperator``.
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b : {array, matrix}
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Right hand side of the linear system. Has shape (N,) or (N,1).
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x0 : {array, matrix}
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Starting guess for the solution.
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tol, atol : float, optional
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Tolerances for convergence, ``norm(residual) <= max(tol*norm(b), atol)``.
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The default for ``atol`` is `tol`.
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.. warning::
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The default value for `atol` will be changed in a future release.
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For future compatibility, specify `atol` explicitly.
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maxiter : int, optional
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Maximum number of iterations. Iteration will stop after maxiter
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steps even if the specified tolerance has not been achieved.
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M : {sparse matrix, dense matrix, LinearOperator}, optional
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Preconditioner for A. The preconditioner should approximate the
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inverse of A. gcrotmk is a 'flexible' algorithm and the preconditioner
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can vary from iteration to iteration. Effective preconditioning
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dramatically improves the rate of convergence, which implies that
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fewer iterations are needed to reach a given error tolerance.
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callback : function, optional
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User-supplied function to call after each iteration. It is called
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as callback(xk), where xk is the current solution vector.
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m : int, optional
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Number of inner FGMRES iterations per each outer iteration.
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Default: 20
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k : int, optional
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Number of vectors to carry between inner FGMRES iterations.
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According to [2]_, good values are around m.
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Default: m
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CU : list of tuples, optional
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List of tuples ``(c, u)`` which contain the columns of the matrices
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C and U in the GCROT(m,k) algorithm. For details, see [2]_.
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The list given and vectors contained in it are modified in-place.
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If not given, start from empty matrices. The ``c`` elements in the
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tuples can be ``None``, in which case the vectors are recomputed
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via ``c = A u`` on start and orthogonalized as described in [3]_.
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discard_C : bool, optional
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Discard the C-vectors at the end. Useful if recycling Krylov subspaces
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for different linear systems.
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truncate : {'oldest', 'smallest'}, optional
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Truncation scheme to use. Drop: oldest vectors, or vectors with
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smallest singular values using the scheme discussed in [1,2].
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See [2]_ for detailed comparison.
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Default: 'oldest'
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Returns
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-------
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x : array or matrix
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The solution found.
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info : int
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Provides convergence information:
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* 0 : successful exit
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* >0 : convergence to tolerance not achieved, number of iterations
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References
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----------
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.. [1] E. de Sturler, ''Truncation strategies for optimal Krylov subspace
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methods'', SIAM J. Numer. Anal. 36, 864 (1999).
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.. [2] J.E. Hicken and D.W. Zingg, ''A simplified and flexible variant
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of GCROT for solving nonsymmetric linear systems'',
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SIAM J. Sci. Comput. 32, 172 (2010).
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.. [3] M.L. Parks, E. de Sturler, G. Mackey, D.D. Johnson, S. Maiti,
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''Recycling Krylov subspaces for sequences of linear systems'',
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SIAM J. Sci. Comput. 28, 1651 (2006).
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"""
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A,M,x,b,postprocess = make_system(A,M,x0,b)
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if not np.isfinite(b).all():
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raise ValueError("RHS must contain only finite numbers")
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if truncate not in ('oldest', 'smallest'):
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raise ValueError("Invalid value for 'truncate': %r" % (truncate,))
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if atol is None:
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warnings.warn("scipy.sparse.linalg.gcrotmk called without specifying `atol`. "
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"The default value will change in the future. To preserve "
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"current behavior, set ``atol=tol``.",
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category=DeprecationWarning, stacklevel=2)
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atol = tol
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matvec = A.matvec
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psolve = M.matvec
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if CU is None:
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CU = []
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if k is None:
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k = m
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axpy, dot, scal = None, None, None
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r = b - matvec(x)
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axpy, dot, scal, nrm2 = get_blas_funcs(['axpy', 'dot', 'scal', 'nrm2'], (x, r))
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b_norm = nrm2(b)
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if discard_C:
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CU[:] = [(None, u) for c, u in CU]
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# Reorthogonalize old vectors
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if CU:
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# Sort already existing vectors to the front
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CU.sort(key=lambda cu: cu[0] is not None)
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# Fill-in missing ones
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C = np.empty((A.shape[0], len(CU)), dtype=r.dtype, order='F')
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us = []
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j = 0
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while CU:
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# More memory-efficient: throw away old vectors as we go
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c, u = CU.pop(0)
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if c is None:
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c = matvec(u)
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C[:,j] = c
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j += 1
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us.append(u)
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# Orthogonalize
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Q, R, P = qr(C, overwrite_a=True, mode='economic', pivoting=True)
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del C
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# C := Q
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cs = list(Q.T)
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# U := U P R^-1, back-substitution
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new_us = []
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for j in range(len(cs)):
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u = us[P[j]]
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for i in range(j):
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u = axpy(us[P[i]], u, u.shape[0], -R[i,j])
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if abs(R[j,j]) < 1e-12 * abs(R[0,0]):
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# discard rest of the vectors
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break
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u = scal(1.0/R[j,j], u)
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new_us.append(u)
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# Form the new CU lists
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CU[:] = list(zip(cs, new_us))[::-1]
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if CU:
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axpy, dot = get_blas_funcs(['axpy', 'dot'], (r,))
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# Solve first the projection operation with respect to the CU
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# vectors. This corresponds to modifying the initial guess to
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# be
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#
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# x' = x + U y
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# y = argmin_y || b - A (x + U y) ||^2
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#
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# The solution is y = C^H (b - A x)
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for c, u in CU:
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yc = dot(c, r)
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x = axpy(u, x, x.shape[0], yc)
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r = axpy(c, r, r.shape[0], -yc)
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# GCROT main iteration
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for j_outer in range(maxiter):
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# -- callback
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if callback is not None:
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callback(x)
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beta = nrm2(r)
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# -- check stopping condition
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beta_tol = max(atol, tol * b_norm)
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if beta <= beta_tol and (j_outer > 0 or CU):
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# recompute residual to avoid rounding error
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r = b - matvec(x)
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beta = nrm2(r)
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if beta <= beta_tol:
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j_outer = -1
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break
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ml = m + max(k - len(CU), 0)
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cs = [c for c, u in CU]
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try:
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Q, R, B, vs, zs, y, pres = _fgmres(matvec,
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r/beta,
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ml,
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rpsolve=psolve,
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atol=max(atol, tol*b_norm)/beta,
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cs=cs)
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y *= beta
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except LinAlgError:
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# Floating point over/underflow, non-finite result from
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# matmul etc. -- report failure.
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break
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#
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# At this point,
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#
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# [A U, A Z] = [C, V] G; G = [ I B ]
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# [ 0 H ]
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#
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# where [C, V] has orthonormal columns, and r = beta v_0. Moreover,
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#
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# || b - A (x + Z y + U q) ||_2 = || r - C B y - V H y - C q ||_2 = min!
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#
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# from which y = argmin_y || beta e_1 - H y ||_2, and q = -B y
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#
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#
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# GCROT(m,k) update
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#
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# Define new outer vectors
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# ux := (Z - U B) y
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ux = zs[0]*y[0]
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for z, yc in zip(zs[1:], y[1:]):
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ux = axpy(z, ux, ux.shape[0], yc) # ux += z*yc
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by = B.dot(y)
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for cu, byc in zip(CU, by):
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c, u = cu
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ux = axpy(u, ux, ux.shape[0], -byc) # ux -= u*byc
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# cx := V H y
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hy = Q.dot(R.dot(y))
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cx = vs[0] * hy[0]
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for v, hyc in zip(vs[1:], hy[1:]):
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cx = axpy(v, cx, cx.shape[0], hyc) # cx += v*hyc
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# Normalize cx, maintaining cx = A ux
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# This new cx is orthogonal to the previous C, by construction
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try:
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alpha = 1/nrm2(cx)
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if not np.isfinite(alpha):
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raise FloatingPointError()
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except (FloatingPointError, ZeroDivisionError):
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# Cannot update, so skip it
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continue
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cx = scal(alpha, cx)
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ux = scal(alpha, ux)
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# Update residual and solution
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gamma = dot(cx, r)
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r = axpy(cx, r, r.shape[0], -gamma) # r -= gamma*cx
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x = axpy(ux, x, x.shape[0], gamma) # x += gamma*ux
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# Truncate CU
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if truncate == 'oldest':
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while len(CU) >= k and CU:
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del CU[0]
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elif truncate == 'smallest':
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if len(CU) >= k and CU:
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# cf. [1,2]
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D = solve(R[:-1,:].T, B.T).T
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W, sigma, V = svd(D)
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# C := C W[:,:k-1], U := U W[:,:k-1]
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new_CU = []
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for j, w in enumerate(W[:,:k-1].T):
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c, u = CU[0]
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c = c * w[0]
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u = u * w[0]
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for cup, wp in zip(CU[1:], w[1:]):
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cp, up = cup
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c = axpy(cp, c, c.shape[0], wp)
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u = axpy(up, u, u.shape[0], wp)
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# Reorthogonalize at the same time; not necessary
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# in exact arithmetic, but floating point error
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# tends to accumulate here
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for cp, up in new_CU:
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alpha = dot(cp, c)
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c = axpy(cp, c, c.shape[0], -alpha)
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u = axpy(up, u, u.shape[0], -alpha)
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alpha = nrm2(c)
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c = scal(1.0/alpha, c)
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u = scal(1.0/alpha, u)
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new_CU.append((c, u))
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CU[:] = new_CU
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# Add new vector to CU
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CU.append((cx, ux))
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# Include the solution vector to the span
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CU.append((None, x.copy()))
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if discard_C:
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CU[:] = [(None, uz) for cz, uz in CU]
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return postprocess(x), j_outer + 1
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