Glyn-Davies, Alex, Duffin, Connor, Kazlauskaite, Ieva 
ORCID: 0000-0001-9690-0887, Girolami, Mark and Akyildiz, Ö. Deniz
(2025)
Statistical finite elements via interacting particle Langevin dynamics.
SIAM/ASA Journal on Uncertainty Quantification, 13 (3).
1200 - 1227.
ISSN 2166-2525
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.
| Item Type: | Article |
|---|---|
| Additional Information: | © 2025 Society for Industrial and Applied Mathematics and American Statistical Association |
| Divisions: | Statistics |
| Subjects: | Q Science > QA Mathematics |
| Date Deposited: | 01 Sep 2025 10:30 |
| Last Modified: | 26 Sep 2025 11:00 |
| URI: | http://eprints.lse.ac.uk/id/eprint/129337 |
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