Particle systems and kinetic equations modeling interacting agents in high dimension

TL;DR

This paper introduces a novel approach combining random projections and compressed sensing to efficiently simulate high-dimensional dynamical systems with many interacting agents, addressing computational challenges.

Contribution

It proposes a method that applies Johnson-Lindenstrauss embeddings and compressed sensing to simulate high-dimensional agent-based models and their kinetic equations.

Findings

Effective dimensionality reduction for agent simulations

Reduced computational cost in high-dimensional systems

Generalization to kinetic equations with delayed curse of dimension

Abstract

In this paper we explore how concepts of high-dimensional data compression via random projections onto lower-dimensional spaces can be applied for tractable simulation of certain dynamical systems modeling complex interactions. In such systems, one has to deal with a large number of agents (typically millions) in spaces of parameters describing each agent of high dimension (thousands or more). Even with today's powerful computers, numerical simulations of such systems are prohibitively expensive. We propose an approach for the simulation of dynamical systems governed by functions of adjacency matrices in high dimension, by random projections via Johnson-Lindenstrauss embeddings, and recovery by compressed sensing techniques. We show how these concepts can be generalized to work for associated kinetic equations, by addressing the phenomenon of the delayed curse of dimension, known in information-based complexity for optimal numerical integration problems in high dimensions.