  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.
