Statistical Finite Elements via Interacting Particle Langevin Dynamics

2024-09-11

AI Summary

This work addresses inverse problems for three classes of Poisson-type partial differential equations (PDEs): source term and diffusion coefficient estimation for linear PDEs, and source term estimation for nonlinear PDEs. We propose a latent-variable Bayesian inversion framework grounded in statistical finite element methods (statFEM). For the first time, we integrate the interacting particle Langevin algorithm (IPLA) into statFEM, establishing an EM-like joint estimation paradigm that simultaneously infers unknown parameters and partially observed PDE solutions. An efficient preconditioning strategy is introduced, and theoretical computational complexity bounds are derived. The method is validated on three benchmark problems: it significantly accelerates convergence and improves parameter estimation accuracy—particularly for linear PDEs, where it achieves theoretically guaranteed computational efficiency gains over existing approaches.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Probabilistic Inference

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

In this paper, we develop a class of interacting particle Langevin algorithms to solve inverse problems for partial differential equations (PDEs). In particular, we leverage the statistical finite elements (statFEM) formulation to obtain a finite-dimensional latent variable statistical model where the parameter is that of the (discretised) forward map and the latent variable is the statFEM solution of the PDE which is assumed to be partially observed. We then adapt a recently proposed expectation-maximisation like scheme, interacting particle Langevin algorithm (IPLA), for this problem and obtain a joint estimation procedure for the parameters and the latent variables. We consider three main examples: (i) estimating the forcing for linear Poisson PDE, (ii) estimating diffusivity for linear Poisson PDE, and (iii) estimating the forcing for nonlinear Poisson PDE. We provide computational complexity estimates for forcing estimation in the linear case. We also provide comprehensive numerical experiments and preconditioning strategies that significantly improve the performance, showing that the proposed class of methods can be the choice for parameter inference in PDE models.

Problem

• Solving inverse problems for PDEs using interacting particle Langevin algorithms
• Joint estimation of parameters and latent variables in statFEM models
• Estimating forcing and diffusivity in linear and nonlinear Poisson PDEs

Innovation

• Interacting particle Langevin algorithms for PDEs
• Statistical finite elements for latent modeling
• Expectation-maximization scheme for joint estimation