Faber approximation to the Mori-Zwanzig equation

TL;DR

This paper introduces a novel Faber series-based approximation method for the Mori-Zwanzig equation, providing asymptotically optimal convergence for linear systems and demonstrating its effectiveness through numerical experiments.

Contribution

The paper develops a new Faber series approximation for the Mori-Zwanzig equation with proven superlinear convergence, advancing the theoretical understanding and practical application of operator series expansions.

Findings

Superlinear convergence of the Faber series approximation

Effective numerical applications to wave propagation and oscillator chains

Theoretical analysis confirming optimal approximation properties

Abstract

We develop a new effective approximation of the Mori-Zwanzig equation based on operator series expansions of the orthogonal dynamics propagator. In particular, we study the Faber series, which yields asymptotically optimal approximations converging at least $R$-superlinearly with the polynomial order for linear dynamical systems. We provide a through theoretical analysis of the new method and present numerical applications to random wave propagation and harmonic chains of oscillators interacting on the Bethe lattice and on graphs with arbitrary topology.