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Colloquium April 25

Date: Tuesday April 25

Time: 16:00 to 17:00

Place: 219 McBryde

Speaker: Heidi Thornquist of Sandia National Laboratories

Title: Fixed-Polynomial Approximate Spectral Transformations for Preconditioning the Eigenvalue Problem

Abstract

Arnoldi's method is often used to compute a few eigenvalues and eigenvectors of large, sparse matrices. When the eigenvalues of interest are not dominant or well-separated, this method may suffer from slow convergence. Spectral transformations are a common acceleration technique that address this issue by introducing a modified eigenvalue problem that is easier to solve than the original. This modified problem accentuates the eigenvalues of interest, but requires solving a linear system, which is computationally expensive for large-scale eigenvalue problems.

In this talk, we will discuss how this expense can be reduced through a preconditioning scheme that uses a fixed-polynomial operator to approximate the spectral transformation. Three different constructions for a fixed-polynomial operator are derived from some common iterative methods for non-Hermitian linear systems. The implementation details and numerical behavior of these three operators are compared. Accuracy heuristics for employing a fixed-polynomial operator with Arnoldi's method are presented. Numerical experiments demonstrate that this preconditioning scheme is a competitive approach for solving large-scale eigenvalue problems. The results will illustrate the effectiveness of this technique using several practical eigenvalue problems from science and engineering ranging from hundreds to more than a million unknowns.

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