Fixed-point approximation for self-consistent transfer operators with Newton's method

TL;DR

This paper develops a novel numerical approach using nonlinear Fourier-Fejér discretisation and Newton's method to efficiently approximate fixed points of self-consistent transfer operators in dynamical systems.

Contribution

It introduces a new discretisation technique and a Newton framework with quadratic convergence for solving self-consistent transfer operators.

Findings

Proves convergence of the discretisation to the true fixed point.

Establishes exponential convergence of the iterative scheme.

Demonstrates efficiency and flexibility through numerical examples.

Abstract

Self consistent transfer operators arise naturally in the study of mean-field coupled dynamical systems and are closely related to kinetic PDEs such as the Vlasov equation. Despite substantial progress on existence and uniqueness of fixed points for self-consistent transfer operators, the development of fast, reliable, and provably accurate numerical methods remains largely unresolved. In this work, we construct a nonlinear Fourier-Fej\'er discretisation and establish convergence of the resulting finite-dimensional fixed point to that of the original self-consistent transfer operator. Further, using the nonlinear Fourier-Fej\'er discretisation, we prove exponential convergence of a sequential iteration scheme and develop a Newton framework with quadratic convergence. We present numerical examples demonstrating the efficiency and flexibility of the above methods.