Accurate solution of Bayesian inverse uncertainty quantification problems combining reduced basis methods and reduction error models

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.