Preconditioning of discrete Helmholtz operators perturbed by a diagonal complex matrix

Abstract

Incomplete factorizations are popular preconditioning techniques for solving large and sparse linear systems. In the case of highly indeÿnite complex-symmetric linear systems, the convergence of Krylov subspace methods sometimes degrades with increasing level of ÿll-in. The reasons for this disappointing behaviour are twofold. On the one hand, the eigenvalues of the preconditioned system tend to 1, but the 'convergence' is not monotonous. On the other hand, the eigenvalues with negative real part, on their move towards 1 have to cross the origin, whence the risk of clustering eigenvalues around 0 while 'improving' the preconditioner. This makes it risky to predict any gain when passing from a level to a higher one. We examine a remedy which consists in slightly moving the spectrum of the original system matrix along the imaginary axis. Theoretical analysis that motivates our approach and experimental results are presented, which displays the e ciency of the new preconditioning techniques.

A fast method for the solution of the Helmholtz equation

“…In the absence of discretization and round-off errors, this preconditioner is exact! This is the fundamental difference between our approach and other approaches known to us, particularly the family of shifted-Laplace preconditioners [7,12,18,21]. Of course, in the presence of discretization errors, we expect this preconditioner to be only approximate.…”

A fast method for the solution of the Helmholtz equation

“…In the absence of discretization and round-off errors, this preconditioner is exact! This is the fundamental difference between our approach and other approaches known to us, particularly the family of shifted-Laplace preconditioners [7,12,18,21]. Of course, in the presence of discretization errors, we expect this preconditioner to be only approximate.…”

Three‐dimensional parallel frequency‐domain visco‐acoustic wave modelling based on a hybrid direct/iterative solver

“…We computed the preconditioner from the damped Helmholtz equation, following the original idea of Magolu monga Made (2000) further investigated by Erlangga et al (2006) and Riyanti et al (2006) in iterative‐solver modelling. The damped wave equation reads where ι=.…”

Modification and Compensation Strategies for Threshold-based Incomplete Factorizations

“…Previous work, such as in [18,23,30,40], has shown that purely imaginary shifts of symmetric and indefinite problems have the effect of clustering eigenvalues on a circle in the right half-plane, with an accumulation point near one. This results in a more stable factorization, which leads to a more effective ILU-based preconditioner.…”

A fast method for the solution of the Helmholtz equation

“…In the absence of discretization and round-off errors, this preconditioner is exact! This is the fundamental difference between our approach and other approaches known to us, particularly the family of shifted-Laplace preconditioners [7,12,18,21]. Of course, in the presence of discretization errors, we expect this preconditioner to be only approximate.…”

Three‐dimensional parallel frequency‐domain visco‐acoustic wave modelling based on a hybrid direct/iterative solver

“…We computed the preconditioner from the damped Helmholtz equation, following the original idea of Magolu monga Made (2000) further investigated by Erlangga et al (2006) and Riyanti et al (2006) in iterative‐solver modelling. The damped wave equation reads where ι=.…”

Modification and Compensation Strategies for Threshold-based Incomplete Factorizations

“…Previous work, such as in [18,23,30,40], has shown that purely imaginary shifts of symmetric and indefinite problems have the effect of clustering eigenvalues on a circle in the right half-plane, with an accumulation point near one. This results in a more stable factorization, which leads to a more effective ILU-based preconditioner.…”

A fast method for the solution of the Helmholtz equation

“…In the absence of discretization and round-off errors, this preconditioner is exact! This is the fundamental difference between our approach and other approaches known to us, particularly the family of shifted-Laplace preconditioners [7,12,18,21]. Of course, in the presence of discretization errors, we expect this preconditioner to be only approximate.…”

Three‐dimensional parallel frequency‐domain visco‐acoustic wave modelling based on a hybrid direct/iterative solver

“…We computed the preconditioner from the damped Helmholtz equation, following the original idea of Magolu monga Made (2000) further investigated by Erlangga et al (2006) and Riyanti et al (2006) in iterative‐solver modelling. The damped wave equation reads where ι=.…”

Modification and Compensation Strategies for Threshold-based Incomplete Factorizations

“…Previous work, such as in [18,23,30,40], has shown that purely imaginary shifts of symmetric and indefinite problems have the effect of clustering eigenvalues on a circle in the right half-plane, with an accumulation point near one. This results in a more stable factorization, which leads to a more effective ILU-based preconditioner.…”