Emulation of Higher-Order Tensors in Manifold Monte Carlo Methods for Bayesian Inverse Problems

Abstract

The Bayesian approach to Inverse Problems relies predominantly on Markov Chain Monte Carlo methods for posterior inference. The typical nonlinear concentration of posterior measure observed in many such Inverse Problems presents severe challenges to existing simulation based inference methods. Motivated by these challenges the exploitation of local geometric information in the form of covariant gradients, metric tensors, Levi-Civita connections, and local geodesic flows, have been introduced to more effectively locally explore the configuration space of the posterior measure. However, obtaining such geometric quantities usually requires extensive computational effort and despite their effectiveness affect the applicability of these geometrically-based Monte Carlo methods. In this paper we explore one way to address this issue by the construction of an emulator of the model from which all geometric objects can be obtained in a much more computationally feasible manner. The main concept is to approximate the geometric quantities using a Gaussian Process emulator which is conditioned on a carefully chosen design set of configuration points, which also determines the quality of the emulator. To this end we propose the use of statistical experiment design methods to refine a potentially arbitrarily initialized design online without destroying the convergence of the resulting Markov chain to the desired invariant measure. The practical examples considered in this paper provide a demonstration of the significant improvement possible in terms of computational loading suggesting this is a promising avenue of further development.

46 pages, 11 figures

References in corpus (1)

• Solving Large-Scale PDE-constrained Bayesian Inverse Problems with Riemann Manifold Hamiltonian Monte Carlo

Cited by in corpus (14)

• Calibrate, Emulate, Sample
• Geometric MCMC for Infinite-Dimensional Inverse Problems
• Stein variational gradient descent with local approximations
• Adaptive Dimension Reduction to Accelerate Infinite-Dimensional Geometric Markov Chain Monte Carlo
• Multi-stage splitting integrators for sampling with modified Hamiltonian Monte Carlo methods
• Neural Network Gradient Hamiltonian Monte Carlo
• Variational Inference for Nonlinear Inverse Problems via Neural Net Kernels: Comparison to Bayesian Neural Networks, Application to Topology Optimization
• Probabilistic Constrained Bayesion Inversion for Transpiration Cooling
• Scaling Up Bayesian Uncertainty Quantification for Inverse Problems using Deep Neural Networks
• Approximate Bayesian inference from noisy likelihoods with Gaussian process emulated MCMC
• Optimal Experimental Design for Reliable Learning of History-Dependent Constitutive Laws
• Hamiltonian Monte Carlo Acceleration Using Surrogate Functions with Random Bases
• Multimodal, high-dimensional, model-based, Bayesian inverse problems with applications in biomechanics
• An acceleration strategy for randomize-then-optimize sampling via deep neural networks