Geometric MCMC for Infinite-Dimensional Inverse Problems

Abstract

Bayesian inverse problems often involve sampling posterior distributions on infinite-dimensional function spaces. Traditional Markov chain Monte Carlo (MCMC) algorithms are characterized by deteriorating mixing times upon mesh-refinement, when the finite-dimensional approximations become more accurate. Such methods are typically forced to reduce step-sizes as the discretization gets finer, and thus are expensive as a function of dimension. Recently, a new class of MCMC methods with mesh-independent convergence times has emerged. However, few of them take into account the geometry of the posterior informed by the data. At the same time, recently developed geometric MCMC algorithms have been found to be powerful in exploring complicated distributions that deviate significantly from elliptic Gaussian laws, but are in general computationally intractable for models defined in infinite dimensions. In this work, we combine geometric methods on a finite-dimensional subspace with mesh-independent infinite-dimensional approaches. Our objective is to speed up MCMC mixing times, without significantly increasing the computational cost per step (for instance, in comparison with the vanilla preconditioned Crank-Nicolson (pCN) method). This is achieved by using ideas from geometric MCMC to probe the complex structure of an intrinsic finite-dimensional subspace where most data information concentrates, while retaining robust mixing times as the dimension grows by using pCN-like methods in the complementary subspace. The resulting algorithms are demonstrated in the context of three challenging inverse problems arising in subsurface flow, heat conduction and incompressible flow control. The algorithms exhibit up to two orders of magnitude improvement in sampling efficiency when compared with the pCN method.

37 pages, 15 figures

References in corpus (1)

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

Cited by in corpus (38)

• Earth System Modeling 2.0: A Blueprint for Models That Learn From Observations and Targeted High-Resolution Simulations
• A practical and efficient approach for Bayesian quantum state estimation
• 2022 Review of Data-Driven Plasma Science
• The statistical finite element method (statFEM) for coherent synthesis of observation data and model predictions
• Data Assimilation: The Schrödinger Perspective
• Demonstration of the Relationship between Sensitivity and Identifiability for Inverse Uncertainty Quantification
• Solving Bayesian Inverse Problems via Variational Autoencoders
• Bayesian inference of heterogeneous epidemic models: Application to COVID-19 spread accounting for long-term care facilities
• Variational Bayesian Approximation of Inverse Problems using Sparse Precision Matrices
• Bayesian Poroelastic Aquifer Characterization from InSAR Surface Deformation Data Part II: Quantifying the Uncertainty
• Learning thermoacoustic interactions in combustors using a physics-informed neural network
• Dimension-Robust MCMC in Bayesian Inverse Problems
• Adaptive Dimension Reduction to Accelerate Infinite-Dimensional Geometric Markov Chain Monte Carlo
• Data-Free Likelihood-Informed Dimension Reduction of Bayesian Inverse Problems
• Consistency of Bayesian inference with Gaussian process priors for a parabolic inverse problem
• Multi-core parallel tempering Bayeslands for basin and landscape evolution
• Generalized Parallel Tempering on Bayesian Inverse Problems
• Where to Drill Next? A Dual-Weighted Approach to Adaptive Optimal Design of Groundwater Surveys
• Variational Inference for Nonlinear Inverse Problems via Neural Net Kernels: Comparison to Bayesian Neural Networks, Application to Topology Optimization
• Ensemble sampler for infinite-dimensional inverse problems
• Deep Markov Chain Monte Carlo
• Novel Deep neural networks for solving Bayesian statistical inverse
• Multimodal Bayesian Registration of Noisy Functions using Hamiltonian Monte Carlo
• Continuum limit and preconditioned Langevin sampling of the path integral molecular dynamics
• Dimension-independent Markov chain Monte Carlo on the sphere
• Physics-Informed Machine Learning with Conditional Karhunen-Loève Expansions
• Adaptive inference over Besov spaces in the white noise model using $p$-exponential priors
• Non-centered parametric variational Bayes' approach for hierarchical inverse problems of partial differential equations
• Scalable optimization-based sampling on function space
• Scaling Up Bayesian Uncertainty Quantification for Inverse Problems using Deep Neural Networks
• Spatiotemporal Besov Priors for Bayesian Inverse Problems
• Stability of Gibbs Posteriors from the Wasserstein Loss for Bayesian Full Waveform Inversion
• MCMC for a hyperbolic Bayesian inverse problem in traffic flow modelling
• Stein variational reduced basis Bayesian inversion
• Optimal Experimental Design for Mathematical Models of Hematopoiesis
• An acceleration strategy for randomize-then-optimize sampling via deep neural networks
• hIPPYlib-MUQ: A Bayesian Inference Software Framework for Integration of Data with Complex Predictive Models under Uncertainty
• A persistent-homology-based Bayesian prior for potential coefficient reconstruction in an elliptic PDE