Abstract

In this work, we develop a class of interacting particle Langevin algorithms to solve inverse problems for partial differential equations (PDEs). We leverage the statistical finite elements formulation to obtain a finite-dimensional statistical model, where the parameter is that of the forward map and the latent variable is the discretised solution of the PDE , assumed to be partially observed. We then adapt a recently proposed expectation-maximisation like scheme, the interacting particle Langevin algorithm (IPLA), for this problem and obtain a joint estimation procedure for the parameters and the latent variables. The estimation of an unknown source term is demonstrated for linear and nonlinear Poisson PDEs, as well as the diffusivity parameter for the linear Poisson PDE . We provide computational complexity estimates for forcing estimation in the linear case, including comprehensive numerical experiments and preconditioning strategies that significantly improve the performance.