  Performing stochastic inversion on a computationally expensive forward
simulation model with a high-dimensional uncertain parameter space (e.g. a
spatial random field) is computationally prohibitive even with gradient
information provided. Moreover, the `nonlinear' mapping from parameters to
observables generally gives rise to non-Gaussian posteriors even with Gaussian
priors, thus hampering the use of efficient inversion algorithms designed for
models with Gaussian assumptions. In this paper, we propose a novel Bayesian
stochastic inversion methodology, characterized by a tight coupling between a
gradient-based Langevin Markov Chain Monte Carlo (LMCMC) method and a kernel
principal component analysis (KPCA). This approach addresses the
`curse-of-dimensionality' via KPCA to identify a low-dimensional feature space
within the high-dimensional and nonlinearly correlated spatial random field.
Moreover, non-Gaussian full posterior probability distribution functions are
estimated via an efficient LMCMC method on both the projected low-dimensional
feature space and the recovered high-dimensional parameter space. We
demonstrate this computational framework by integrating and adapting recent
developments such as data-driven statistics-on-manifolds constructions and
reduction-through-projection techniques to solve inverse problems in linear
elasticity.
