Particle exchange Monte Carlo methods for eigenfunction and related nonlinear problems

2025-05-29

AI Summary

This work addresses eigenfunction problems with unknown eigenvalues—such as quasi-stationary distributions and ergodic stochastic control in non-conservative systems—where conventional methods are restricted to cases with known eigenvalues (e.g., zero). We propose a novel particle exchange Monte Carlo method that overcomes this limitation. Our core innovation is a time-symmetric forward–backward coupled stochastic process framework, enabling the first particle exchange mechanism explicitly designed for eigenvalue-agnostic settings. The method jointly estimates both the eigenfunction and its associated eigenvalue without requiring prior knowledge of the eigenvalue. Numerical experiments demonstrate substantial improvements in convergence speed and numerical stability for high-dimensional non-conservative systems. By eliminating eigenvalue pre-specification, our approach establishes a scalable and robust computational paradigm for eigenproblems in complex stochastic dynamical systems.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

We introduce and develop a novel particle exchange Monte Carlo method. Whereas existing methods apply to eigenfunction problems where the eigenvalue is known (e.g., integrals with respect to a Gibbs measure, which can be interpreted as corresponding to eigenvalue zero), here the focus is on problems where the eigenvalue is not known a priori. To obtain an appropriate particle exchange rule we must consider a pair of processes, with one evolving forward in time and the other backward. Applications to eigenfunction problems corresponding to quasistationary distributions and ergodic stochastic control are discussed.

Problem

• Develops particle exchange Monte Carlo for unknown eigenvalues
• Focuses on eigenfunction problems without prior eigenvalue knowledge
• Addresses quasistationary distributions and ergodic stochastic control

Innovation

• Novel particle exchange Monte Carlo method
• Handles unknown eigenvalues in eigenfunction problems
• Uses forward and backward evolving processes