Piecewise Deterministic Markov Processes for Bayesian Inference of PDE Coefficients

TL;DR

The paper addresses the challenge of performing Bayesian inference for PDE coefficient fields when likelihood evaluations are expensive. It introduces a surrogate-assisted thinning framework for PDMP samplers, enabling efficient event-time generation via surrogates with a dynamic offset that enforces the upper-bound condition, and applies affine preconditioning to improve geometry. Gaussian process surrogates, including gradient-informed and adaptive variants, are explored, and the approach is tested on a 1D elasticity problem where the coefficient field is inferred from noisy observations. Results show that GP-based surrogates substantially improve accuracy and effective sample size per forward-model evaluation compared with RWM and NUTS, with the Bouncy Particle Sampler offering the strongest overall efficiency and scaling. The work demonstrates the viability of PDMP sampling for PDE-governed Bayesian inverse problems and outlines future directions for higher dimensions, neural surrogates, and theoretical convergence analysis.

Abstract

We develop a general framework for piecewise deterministic Markov process (PDMP) samplers that enables efficient Bayesian inference in non-linear inverse problems with expensive likelihoods. The key ingredient is a surrogate-assisted thinning scheme in which a surrogate model provides a proposal event rate and a robust correction mechanism enforces an upper bound on the true rate by dynamically adjusting an additive offset whenever violations are detected. This construction is agnostic to the choice of surrogate and PDMP, and we demonstrate it for the Zig-Zag sampler and the Bouncy particle sampler with constant, Laplace, and Gaussian process (GP) surrogates, including gradient-informed and adaptively refined GP variants. As a representative application, we consider Bayesian inference of a spatially varying Young's modulus in a one-dimensional linear elasticity problem. Across dimensions, PDMP samplers equipped with GP-based surrogates achieve substantially higher accuracy and effective sample size per forward model evaluation than Random Walk Metropolis algorithm and the No-U-Turn sampler. The Bouncy particle sampler exhibits the most favorable overall efficiency and scaling, illustrating the potential of the proposed PDMP framework beyond this particular setting.

Figures (17)

• Figure 1: Illustration of the Zig-Zag sampler on a two-dimensional Gaussian. Figures (a) and (c) show the target density with the current position (dot) and velocity (arrow). Figures (b) and (d) show the corresponding event rates along each coordinate direction.
• Figure 2: Illustration of the Bouncy particle sampler on a two-dimensional Gaussian. Figures (a) and (c) show the target density with the current position (dot) and velocity (arrow). Figures (b) and (d) show the resulting event rate. Note that the event rate remains zero for longer, but has a steeper slope than in the Zig-Zag sampler.
• Figure 3: Illustration of the thinning procedure for simulating event times. The vertical lines represent candidate event times drawn from a Poisson process with upper bound $\bar{\lambda}$. Rejected events are shown in blue, while accepted events are shown in orange. Fewer rejections is better.
• Figure 4: Linear elastic bar problem with 2 and 5 dimensional discretization of Young's modulus field in (a) and (b), respectively.
• Figure 5: Posterior distribution of the two-dimensional problem in the original (a) and the affine space (b).