Data-driven combined state and parameter reduction for inverse problems

Himpe C, Ohlberger M

Abstract

In this contribution we present an accelerated optimization-based approach for combined state and parameter reduction of a parametrized forward model, which is used to construct a surrogate model in a Bayesian inverse problem setting. Following the ideas presented in Lieberman et al. (SIAM J. Sci. Comput.32(5), 2523–2542, 2010), our approach is based on a generalized data-driven optimization functional in the construction process of the reduced order model and the usage of a Monte-Carlo basis enrichment strategy that results in an additional speed-up of the overall method. In principal, the model reduction procedure is based on the offline construction of appropriate low-dimensional state and parameter spaces and an online inversion step using the resulting surrogate model that is obtained through projection of the underlying forward model onto the reduced spaces. The generalizations and enhancements presented in this work are shown to decrease overall computational time and thus allow an application to large-scale problems. Numerical experiments for a generic model and a fMRI connectivity model are presented in order to compare the computational efficiency of our improved method with the original approach.