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We introduce a methodology for nonlinear inverse problems using a variationalBayesian approach where the unknown quantity is a spatial field. A structuredBayesian Gaussian process latent variable model is used both to construct alow-dimensional generative model of the sample-based stochastic prior as wellas a surrogate for the forward evaluation. Its Bayesian formulation capturesepistemic uncertainty introduced by the limited number of input and outputexamples, automatically selects an appropriate dimensionality for the learnedlatent representation of the data, and rigorously propagates the uncertainty ofthe data-driven dimensionality reduction of the stochastic space through theforward model surrogate. The structured Gaussian process model explicitlyleverages spatial information for an informative generative prior to improvesample efficiency while achieving computational tractability through Kroneckerproduct decompositions of the relevant kernel matrices. Importantly, theBayesian inversion is carried out by solving a variational optimizationproblem, replacing traditional computationally-expensive Monte Carlo sampling.The methodology is demonstrated on an elliptic PDE and is shown to returnwell-calibrated posteriors and is tractable with latent spaces with over 100dimensions.

Structured Bayesian Gaussian process latent variable model: applications to data-driven dimensionality reduction and high-dimensional inversion