Year: 2025
Type: article
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 element method (statFEM) formulation to obtain a finite-dimensional latent variable statistical model where the parameter is that of the (discretized) 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-maximization–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 a linear Poisson PDE, (ii) estimating diffusivity for a linear Poisson PDE, and (iii) estimating the forcing for a 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.
DOI: 10.1137/24m1693593
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: PREdictive Modelling with QuantIfication of UncERtainty for MultiphasE Systems (PREMIERE), Semantic Information Pursuit for Multimodal Data Analysis, Revolutionising Operational Safety and Economy for High-value Infrastructure using Population-based SHM (ROSEHIPS), G2574216409, G2972334111, Digital Roads, Delivering Enhanced Through-Life Nuclear Asset Management, G3525226743, G3690003352, G4062695751, Advancing Probabilistic Machine Learning to Deliver Safer, More Efficient, and Predictable Air Traffic Control, G7470475232, G926698369, CoSInES (COmputational Statistical INference for Engineering and Security)
OpenAlex ID: W4413370496