Statistical finite elements via interacting particle Langevin dynamics
File(s)
Accepted version
Author(s)
Glyn-Davies, Alex
Duffin, Connor
Kazlauskaite, Ieva
Girolami, Mark
Akyildiz, O Deniz
Type
Journal 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 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 the forcing for nonlinear Poisson PDE, and (iii) estimating
diffusivity for linear 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.
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 the forcing for nonlinear Poisson PDE, and (iii) estimating
diffusivity for linear 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.
Date Issued
2025-09-01
Date Acceptance
2025-05-21
Citation
SIAM/ASA Journal on Uncertainty Quantification, 2025, 13 (3), pp.1200-1227
ISSN
2166-2525
Publisher
Society for Industrial and Applied Mathematics
Start Page
1200
End Page
1227
Journal / Book Title
SIAM/ASA Journal on Uncertainty Quantification
Volume
13
Issue
3
Copyright Statement
Copyright © 2025 Society for Industrial and Applied Mathematics and American Statistical Association. This is the author’s accepted manuscript made available under a CC-BY licence in accordance with Imperial’s Research Publications Open Access policy (www.imperial.ac.uk/oa-policy)
License URL
Publication Status
Published
Date Publish Online
2025-08-21