Lanczos-based Low-Rank Correction Method for Solving the Dyson Equation in Inhomogenous Dynamical Mean-Field Theory

TL;DR

This paper introduces a Lanczos-based low-rank correction algorithm that significantly accelerates the computation of local Green's functions in inhomogeneous dynamical mean-field theory, enabling efficient solutions for large sparse matrices.

Contribution

The paper presents a novel low-rank algorithm based on Lanczos methods that improves the efficiency of solving Dyson's equation in inhomogeneous DMFT, outperforming traditional matrix inversion techniques.

Findings

At least 25-fold performance improvement over explicit matrix inversion.

Comparable scaling of the low-rank method and domain decomposition methods.

Effective for large sparse matrices in strongly interacting systems.

Abstract

Inhomogeneous dynamical mean-field theory has been employed to solve many interesting strongly interacting problems from transport in multilayered devices to the properties of ultracold atoms in a trap. The main computational step, especially for large systems, is the problem of calculating the inverse of a large sparse matrix to solve Dyson's equation and determine the local Green's function at each lattice site from the corresponding local self-energy. We present a new efficient algorithm, the Lanczos-based low-rank algorithm, for the calculation of the inverse of a large sparse matrix which yields this local (imaginary time) Green's function. The Lanczos-based low-rank algorithm is based on a domain decomposition viewpoint, but avoids explicit calculation of Schur complements and relies instead on low-rank matrix approximations derived from the Lanczos algorithm, for solving the Dyson equation. We report at least a 25-fold improvement of performance compared to explicit decomposition (such as sparse LU) of the matrix inverse. We also report that scaling relative to matrix sizes, of the low-rank correction method on the one hand and domain decomposition methods on the other, are comparable.