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On the Use of a Stochastic Galerkin Surrogate to Accelerate the Solution of PDE-Based Bayesian Inverse Problems

Rynn, James Andrew John

[Thesis]. Manchester, UK: The University of Manchester; 2020.

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Abstract

In this thesis, we are interested in the efficient numerical solution of PDE-based Bayesian inverse problems, where the aim is to approximate the posterior distribution for the unknown inputs to a PDE model given indirect, noisy observations of the output. Typical sampling approaches for investigating the posterior distribution rely on producing repeated approximations, using deterministic finite element methods or similar, to the corresponding forward problem. The computational expense of constructing these repeated approximations often makes such approaches infeasible, even when a sophisticated sampling method is employed. We examine the use of a surrogate approach to expedite the sampling of the posterior using a Markov chain Monte Carlo (MCMC) sampling algorithm. Our particular choice of surrogate is a stochastic Galerkin finite element method (SGFEM) approximation to a parametric form of the forward problem, which approximates the forward solution for a range of possible unknown inputs as a function of parameters representing the uncertain inputs. By solving a single large linear system in the offline stage, we avoid the need to compute repeated approximations during the online (MCMC) stage and thus save a significant amount of computational effort. We investigate the accuracy and efficiency of this surrogate approach through solution of both a model elliptic problem and an industry inspired parabolic inverse problem. There are two novel contributions in this thesis. Firstly, through a series of numerical experiments, we perform a numerical demonstration of theoretical results on the convergence of our approximations to the posterior obtained when solving a model elliptic inverse problem using an SGFEM surrogate in an MCMC routine. That is, we have shown the numerical convergence of the approximate posterior distribution as the discretisation parameters of the SGFEM approximation are refined. Secondly, we have demonstrated the rapid solution of a parabolic inverse problem of interest to the industrial sponsors of this project, the National Physical Laboratory. This problem concerns the determination of the unknown thermal conductivity of a material given thermogram data from a laser flash experiment, where a sample of the material is heated and measurements of its temperature are taken. Our approach combines a stochastic Galerkin method with a time stepping routine to allow rapid evaluation of the approximate posterior density. We perform numerical investigations into both the accuracy of our approach and the features of the physical problem. Much of the work involved in solving the industrial problem would be beyond any realistic computational budget without the use of our efficient sampling approach.