Adaptive Polynomial Filtering for Hermitian Interior Eigenproblems: Convergence Analysis

TL;DR

This paper introduces an adaptive polynomial filtering method using Chebyshev expansion for efficient and robust interior eigenvalue computations of large Hermitian matrices, with convergence analysis and acceleration techniques.

Contribution

It presents a novel adaptive polynomial filtering strategy with convergence bounds, spurious eigenvalue detection, and implementation acceleration for interior eigenproblems.

Findings

The method achieves improved efficiency over classical approaches.

Convergence bounds are established for both undamped and damped cases.

Numerical results confirm robustness and speed of the proposed approach.

Abstract

Interior eigenvalue problems for large-scale sparse Hermitian matrices are fundamental in computational science. We propose an adaptive polynomial filtering strategy based on Chebyshev expansion of a step function, integrated into a filtered subspace iteration framework. We establish pointwise convergence bounds in both undamped and damped settings and incorporate an enhanced spurious eigenvalue detection technique to improve efficiency and robustness. At the implementation level, we employ MaSpMM to accelerate the polynomial filtering step. Numerical results demonstrate the efficiency and robustness of the proposed method compared with classical approaches.