A spectrum adaptive kernel polynomial method

TL;DR

This paper introduces a spectrum adaptive kernel polynomial method that leverages the Lanczos algorithm to dynamically select parameters, reducing initial estimates and computational costs in spectral density approximation.

Contribution

It proposes a novel spectrum adaptive KPM using Lanczos without reorthogonalization, enabling parameter selection after computation and improving efficiency.

Findings

Decouples computation from approximation for practical benefits

Theoretically justified Lanczos-based approach in finite precision

Numerical examples demonstrate efficiency and pedagogical advantages

Abstract

The kernel polynomial method (KPM) is a powerful numerical method for approximating spectral densities. Typical implementations of the KPM require an a prior estimate for an interval containing the support of the target spectral density, and while such estimates can be obtained by classical techniques, this incurs addition computational costs. We propose an spectrum adaptive KPM based on the Lanczos algorithm without reorthogonalization which allows the selection of KPM parameters to be deferred to after the expensive computation is finished. Theoretical results from numerical analysis are given to justify the suitability of the Lanczos algorithm for our approach, even in finite precision arithmetic. While conceptually simple, the paradigm of decoupling computation from approximation has a number of practical and pedagogical benefits which we highlight with numerical examples.