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arXiv · 2605.08803

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

Wael Bahsoun·Gary Froyland·Maxence Phalempin
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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.

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Wael Bahsoun, Gary Froyland, Maxence Phalempin. 2026-05-09. Fixed-point approximation for self-consistent transfer operators with Newton's method. https://arxiv.org/abs/2605.08803

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