Bayesian model inversion using stochastic spectral embedding

In this paper we propose a new sampling-free approach to solve Bayesian inverse problems that extends the recently introduced spectral likelihood expansions (SLE) method. The latter solves the inverse problem by expanding the likelihood function onto a global polynomial basis orthogonal w.r.t. the prior distribution. This gives rise to analytical expressions for key statistics of the Bayesian posterior distribution, such as evidence, posterior moments and posterior marginals by simple post-processing of the expansion coefficients. It is well known that in most practically relevant scenarios, likelihood functions have close-to-compact support, which causes the global SLE approach to fail due to the high polynomial degree required for an accurate spectral representation. To solve this problem, we herein replace the global polynomial expansion from SLE with a recently proposed method for local spectral expansion refinement called stochastic spectral embedding (SSE). This surrogate-modeling method was developed for functions with high local complexity. To increase the efficiency of SSE, we enhance it with an adaptive sample enrichment scheme. We show that SSE works well for likelihood approximations and retains the relevant spectral properties of SLE, thus preserving analytical expressions of posterior statistics. To assess the performance of our approach, we include three case studies ranging from low to high dimensional model inversion problems that showcase the superiority of the SSE approach compared to SLE and present the approach as a promising alternative to existing inversion frameworks.