Accurate Solution of Bayesian Inverse Uncertainty Quantification Problems Combining Reduced Basis Methods and Reduction Error Models

Abstract

ReuseUnless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version -refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher's website. TakedownIf you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. Abstract Computational inverse problems related to partial differential equations (PDEs) often contain nuisance parameters that cannot be effectively identified but still need to be considered as part of the problem. The objective of this work is to show how to take advantage of a reduced order framework to speed up Bayesian inversion on the identifiable parameters of the system, while marginalizing away the (potentially large number of) nuisance parameters. The key ingredients are twofold. On the one hand, we rely on a reduced basis (RB) method, equipped with computable a posteriori error bounds, to speed up the solution of the forward problem. On the other hand, we develop suitable reduction error models (REMs) to quantify in an inexpensive way the error between the full-order and the reduced-order approximation of the forward problem, in order to gauge the effect of this error on the posterior distribution of the identifiable parameters. Numerical results dealing with inverse problems governed by elliptic PDEs in the case of both scalar parameters and parametric fields highlight the combined role played by RB accuracy and REM effectivity.

Machine-learning error models for approximate solutions to parameterized systems of nonlinear equations

“…This work demonstrated significant improvements in accuracy with respect to the a priori multifidelity correction error model in high-dimensional parameter spaces. Follow-on work also demonstrated the promise of this kind of a posteriori error model in an uncertainty-quantification context [40].…”

A posteriori stochastic correction of reduced models in delayed‐acceptance MCMC, with application to multiphase subsurface inverse problems

“…The backbone of the approximation developed here is a given reduced model, denoted F * (·), built using existing techniques, including grid coarsening, 13,16,35,36 linearization of the forward model, 11 and projection-based methods. [37][38][39][40][41] Our key contribution here is to present a new way to improve the approximation to the likelihood function by considering the posterior statistics of the numerical error of the reduced model. This leads to the ADA algorithm with substantial improvement in the computational efficiency compared to the classical DA or surrogate transition algorithms.…”

A transport-based multifidelity preconditioner for Markov chain Monte Carlo

“…Under certain assumptions, it has been shown that the posterior distribution induced by the low-fidelity model converges to the original/highfidelity posterior distribution as the low-fidelity approximation is refined [39,18]. Other approaches adapt low-fidelity models [19,32] over a finite interval of posterior exploration, or quantify the error introduced by sampling the low-fidelity posterior distribution [21,35]. Yet another family of approaches incrementally and infinitely refines approximations of the forward model on-the-fly during MCMC sampling [16,15]; under appropriate conditions, these schemes guarantee that the MCMC chain asymptotically samples the highfidelity posterior distribution.…”