High-performance implementation of Chebyshev filter diagonalization for interior eigenvalue computations

TL;DR

This paper presents a high-performance implementation of Chebyshev filter diagonalization, enabling efficient computation of many interior eigenvalues of large sparse matrices, demonstrated on a billion-dimensional quantum physics problem.

Contribution

It introduces an optimized parallel implementation of Chebyshev filter diagonalization for large-scale eigenvalue problems, surpassing traditional rational function methods.

Findings

Efficient computation of 100 interior eigenpairs for a matrix of size 10^9.

Analysis of damping kernel, search space, and polynomial degree effects.

Successful large-scale quantum physics application demonstration.

Abstract

We study Chebyshev filter diagonalization as a tool for the computation of many interior eigenvalues of very large sparse symmetric matrices. In this technique the subspace projection onto the target space of wanted eigenvectors is approximated with filter polynomials obtained from Chebyshev expansions of window functions. After the discussion of the conceptual foundations of Chebyshev filter diagonalization we analyze the impact of the choice of the damping kernel, search space size, and filter polynomial degree on the computational accuracy and effort, before we describe the necessary steps towards a parallel high-performance implementation. Because Chebyshev filter diagonalization avoids the need for matrix inversion it can deal with matrices and problem sizes that are presently not accessible with rational function methods based on direct or iterative linear solvers. To demonstrate the potential of Chebyshev filter diagonalization for large-scale problems of this kind we include as an example the computation of the $10^2$ innermost eigenpairs of a topological insulator matrix with dimension $10^9$ derived from quantum physics applications.