Optimal Transport, Timesteppers, Newton-Krylov Methods and Steady States of Collective Particle Dynamics

TL;DR

This paper extends matrix-free timesteppers to stochastic particle systems using optimal transport, enabling efficient steady-state distribution computation with high noise through smooth CDF-based methods and Newton-Krylov solvers.

Contribution

It introduces a novel framework combining optimal transport and smooth CDF timesteppers with Newton-Krylov methods for steady-state analysis of stochastic particle systems.

Findings

Efficient steady-state distribution computation under high stochastic noise.

Error analysis shows convergence is achievable despite noise.

Validated methods on a two-dimensional distribution example.

Abstract

Timesteppers constitute a powerful tool in modern computational science and engineering. Although they are typically used to advance the system forward in time, they can also be viewed as nonlinear mappings that implicitly encode steady states and stability information. In this work, we present an extension of the matrix-free framework for calculating, via timesteppers, steady states of deterministic systems to stochastic particle simulations, where intrinsic randomness prevents direct steady state extraction. By formulating stochastic timesteppers in the language of optimal transport, we reinterpret them as operators acting on probability measures rather than on individual particle trajectories. This perspective enables the construction of smooth cumulative- and inverse-cumulative-distribution-function ((I)CDF) timesteppers that evolve distributions rather than particles. Combined with matrix-free Newton-Krylov solvers, these smooth timesteppers allow efficient computation of steady-state distributions even under high stochastic noise. We perform an error analysis quantifying how noise affects finite-difference Jacobian action approximations, and demonstrate that convergence can be obtained even in high noise regimes. Finally, we introduce higher-dimensional generalizations based on smooth CDF-related representations of particles and validate their performance on a non-trivial two-dimensional distribution. Together, these developments establish a unified variational framework for computing meaningful steady states of both deterministic and stochastic timesteppers.