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

We introduce a methodology for nonlinear inverse problems using a variational Bayesian approach where the unknown quantity is a spatial field. A structured Bayesian Gaussian process latent variable model is used both to construct a low-dimensional generative model of the sample-based stochastic prior as well as a surrogate for the forward evaluation. Its Bayesian formulation captures epistemic uncertainty introduced by the limited number of input and output examples, automatically selects an appropriate dimensionality for the learned latent representation of the data, and rigorously propagates the uncertainty of the data-driven dimensionality reduction of the stochastic space through the forward model surrogate. The structured Gaussian process model explicitly leverages spatial information for an informative generative prior to improve sample efficiency while achieving computational tractability through Kronecker product decompositions of the relevant kernel matrices. Importantly, the Bayesian inversion is carried out by solving a variational optimization problem, replacing traditional computationally-expensive Monte Carlo sampling. The methodology is demonstrated on an elliptic PDE and is shown to return well-calibrated posteriors and is tractable with latent spaces with over 100 dimensions.