Non-Linear Least-Squares Optimization of Rational Filters for the Solution of Interior Eigenvalue Problems

TL;DR

This paper introduces a non-linear least-squares optimization framework for designing rational filters that significantly enhance the efficiency of contour-based eigensolvers in solving interior eigenvalue problems.

Contribution

It presents a novel optimization scheme for rational filters, exploiting symmetries and constraints, to improve convergence in eigenvalue computations, outperforming existing filters.

Findings

Optimized filters outperform existing ones on benchmark problems.

The framework exploits symmetries for Hermitian eigenproblems.

Effective starting points improve optimization outcomes.

Abstract

Rational filter functions can be used to improve convergence of contour-based eigensolvers, a popular family of algorithms for the solution of the interior eigenvalue problem. We present a framework for the optimization of rational filters based on a non-convex weighted Least-Squares scheme. When used in combination with the FEAST library, our filters out-perform existing ones on a large and representative set of benchmark problems. This work provides a detailed description of: (1) a set up of the optimization process that exploits symmetries of the filter function for Hermitian eigenproblems, (2) a formulation of the gradient descent and Levenberg-Marquardt algorithms that exploits the symmetries, (3) a method to select the starting position for the optimization algorithms that reliably produces effective filters, (4) a constrained optimization scheme that produces filter functions with specific properties that may be beneficial to the performance of the eigensolver that employs them.