3,700 research outputs found
The convergence of Jacobi-Davidson for Hermitian eigenproblems
Rayleigh Quotient iteration is an iterative method with some attractive convergence properties for nding (interior) eigenvalues of large sparse Hermitian matrices. However, the method requires the accurate (and, hence, often expensive) solution of a linear system in every iteration step. Unfortunately, replacing the exact solution with a cheaper approximation may destroy the convergence. The (Jacobi-)Davidson correction equation can be seen as a solution for this problem. In this paper we deduce quantitative results to support this viewpoint and we relate it to other methods. This should make some of the experimental observations in practice more quantitative in the Hermitian case. Asymptotic convergence bounds are given for xed preconditioners and for the special case if the correction equation is solved with some xed relative residual precision. A new dynamic tolerance is proposed and some numerical illustration is presented
Improved Accuracy and Parallelism for MRRR-based Eigensolvers -- A Mixed Precision Approach
The real symmetric tridiagonal eigenproblem is of outstanding importance in
numerical computations; it arises frequently as part of eigensolvers for
standard and generalized dense Hermitian eigenproblems that are based on a
reduction to tridiagonal form. For its solution, the algorithm of Multiple
Relatively Robust Representations (MRRR) is among the fastest methods. Although
fast, the solvers based on MRRR do not deliver the same accuracy as competing
methods like Divide & Conquer or the QR algorithm. In this paper, we
demonstrate that the use of mixed precisions leads to improved accuracy of
MRRR-based eigensolvers with limited or no performance penalty. As a result, we
obtain eigensolvers that are not only equally or more accurate than the best
available methods, but also -in most circumstances- faster and more scalable
than the competition
Stability Estimates and Structural Spectral Properties of Saddle Point Problems
For a general class of saddle point problems sharp estimates for
Babu\v{s}ka's inf-sup stability constants are derived in terms of the constants
in Brezzi's theory. In the finite-dimensional Hermitian case more detailed
spectral properties of preconditioned saddle point matrices are presented,
which are helpful for the convergence analysis of common Krylov subspace
methods. The theoretical results are applied to two model problems from optimal
control with time-periodic state equations. Numerical experiments with the
preconditioned minimal residual method are reported
Sharp error bounds for Ritz vectors and approximate singular vectors
We derive sharp bounds for the accuracy of approximate eigenvectors (Ritz
vectors) obtained by the Rayleigh-Ritz process for symmetric eigenvalue
problems. Using information that is available or easy to estimate, our bounds
improve the classical Davis-Kahan theorem by a factor that can be
arbitrarily large, and can give nontrivial information even when the
theorem suggests that a Ritz vector might have no accuracy at all.
We also present extensions in three directions, deriving error bounds for
invariant subspaces, singular vectors and subspaces computed by a
(Petrov-Galerkin) projection SVD method, and eigenvectors of self-adjoint
operators on a Hilbert space
Dissecting the FEAST algorithm for generalized eigenproblems
We analyze the FEAST method for computing selected eigenvalues and
eigenvectors of large sparse matrix pencils. After establishing the close
connection between FEAST and the well-known Rayleigh-Ritz method, we identify
several critical issues that influence convergence and accuracy of the solver:
the choice of the starting vector space, the stopping criterion, how the inner
linear systems impact the quality of the solution, and the use of FEAST for
computing eigenpairs from multiple intervals. We complement the study with
numerical examples, and hint at possible improvements to overcome the existing
problems.Comment: 11 Pages, 5 Figures. Submitted to Journal of Computational and
Applied Mathematic
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