19 research outputs found
Experiments on a Parallel Nonlinear Jacobi–Davidson Algorithm
AbstractThe Jacobi–Davidson (JD) algorithm is very well suited for the computation of a few eigen-pairs of large sparse complex symmetric nonlinear eigenvalue problems. The performance of JD crucially depends on the treatment of the so-called correction equation, in particular the preconditioner, and the initial vector. Depending on the choice of the spectral shift and the accuracy of the solution, the convergence of JD can vary from linear to cubic. We investigate parallel preconditioners for the Krylov space method used to solve the correction equation.We apply our nonlinear Jacobi–Davidson (NLJD) method to quadratic eigenvalue problems that originate from the time-harmonic Maxwell equation for the modeling and simulation of resonating electromagnetic structures
Efficient computation of matrix power-vector products: application for space-fractional diffusion problems
A novel algorithm is proposed for computing matrix-vector products A^\alpha v, where A is a symmetric positive semidefinite sparse matrix and \alpha > 0. The method can be applied for the efficient implementation of the matrix transformation method to solve space-fractional diffusion problems. The performance of the new algorithm is studied in a comparison with the conventional MATLAB subroutines to compute matrix powers
A multigrid accelerated eigensolver for the Hermitian Wilson-Dirac operator in lattice QCD
Eigenvalues of the Hermitian Wilson-Dirac operator are of special interest in
several lattice QCD simulations, e.g., for noise reduction when evaluating
all-to-all propagators. In this paper we present a Davidson-type eigensolver
that utilizes the structural properties of the Hermitian Wilson-Dirac operator
to compute eigenpairs of this operator corresponding to small eigenvalues.
The main idea is to exploit a synergy between the (outer) eigensolver and its
(inner) iterative scheme which solves shifted linear systems. This is achieved
by adapting the multigrid DD-AMG algorithm to a solver for shifted
systems involving the Hermitian Wilson-Dirac operator. We demonstrate that
updating the coarse grid operator using eigenvector information obtained in the
course of the generalized Davidson method is crucial to achieve good
performance when calculating many eigenpairs, as our study of the local
coherence shows. We compare our method with the commonly used software-packages
PARPACK and PRIMME in numerical tests, where we are able to achieve significant
improvements, with speed-ups of up to one order of magnitude and a near-linear
scaling with respect to the number of eigenvalues. For illustration we compare
the distribution of the small eigenvalues of on a lattice
with what is predicted by the Banks-Casher relation in the infinite volume
limit
A new stopping criterion for Krylov solvers applied in Interior Point Methods
A surprising result is presented in this paper with possible far reaching
consequences for any optimization technique which relies on Krylov subspace
methods employed to solve the underlying linear equation systems. In this paper
the advantages of the new technique are illustrated in the context of Interior
Point Methods (IPMs). When an iterative method is applied to solve the linear
equation system in IPMs, the attention is usually placed on accelerating their
convergence by designing appropriate preconditioners, but the linear solver is
applied as a black box solver with a standard termination criterion which asks
for a sufficient reduction of the residual in the linear system. Such an
approach often leads to an unnecessary 'oversolving' of linear equations. In
this paper a new specialized termination criterion for Krylov methods used in
IPMs is designed. It is derived from a deep understanding of IPM needs and is
demonstrated to preserve the polynomial worst-case complexity of these methods.
The new criterion has been adapted to the Conjugate Gradient (CG) and to the
Minimum Residual method (MINRES) applied in the IPM context. The new criterion
has been tested on a set of linear and quadratic optimization problems
including compressed sensing, image processing and instances with partial
differential equation constraints. Evidence gathered from these computational
experiments shows that the new technique delivers significant improvements in
terms of inner (linear) iterations and those translate into significant savings
of the IPM solution time
Convergence analysis of a block preconditioned steepest descent eigensolver with implicit deflation
Gradient-type iterative methods for solving Hermitian eigenvalue problems can
be accelerated by using preconditioning and deflation techniques. A
preconditioned steepest descent iteration with implicit deflation (PSD-id) is
one of such methods. The convergence behavior of the PSD-id is recently
investigated based on the pioneering work of Samokish on the preconditioned
steepest descent method (PSD). The resulting non-asymptotic estimates indicate
a superlinear convergence of the PSD-id under strong assumptions on the initial
guess. The present paper utilizes an alternative convergence analysis of the
PSD by Neymeyr under much weaker assumptions. We embed Neymeyr's approach into
the analysis of the PSD-id using a restricted formulation of the PSD-id. More
importantly, we extend the new convergence analysis of the PSD-id to a
practically preferred block version of the PSD-id, or BPSD-id, and show the
cluster robustness of the BPSD-id. Numerical examples are provided to validate
the theoretical estimates.Comment: 26 pages, 10 figure
On Inner Iterations in the Shift-Invert Residual Arnoldi Method and the Jacobi--Davidson Method
Using a new analysis approach, we establish a general convergence theory of
the Shift-Invert Residual Arnoldi (SIRA) method for computing a simple
eigenvalue nearest to a given target and the associated eigenvector.
In SIRA, a subspace expansion vector at each step is obtained by solving a
certain inner linear system. We prove that the inexact SIRA method mimics the
exact SIRA well, that is, the former uses almost the same outer iterations to
achieve the convergence as the latter does if all the inner linear systems are
iteratively solved with {\em low} or {\em modest} accuracy during outer
iterations. Based on the theory, we design practical stopping criteria for
inner solves. Our analysis is on one step expansion of subspace and the
approach applies to the Jacobi--Davidson (JD) method with the fixed target
as well, and a similar general convergence theory is obtained for it.
Numerical experiments confirm our theory and demonstrate that the inexact SIRA
and JD are similarly effective and are considerably superior to the inexact
SIA.Comment: 20 pages, 8 figure