Multilevel Sequential${}^2$ Monte Carlo for Bayesian Inverse Problems

Abstract

The identification of parameters in mathematical models using noisy observations is a common task in uncertainty quantification. We employ the framework of Bayesian inversion: we combine monitoring and observational data with prior information to estimate the posterior distribution of a parameter. Specifically, we are interested in the distribution of a diffusion coefficient of an elliptic PDE. In this setting, the sample space is high-dimensional, and each sample of the PDE solution is expensive. To address these issues we propose and analyse a novel Sequential Monte Carlo (SMC) sampler for the approximation of the posterior distribution. Classical, single-level SMC constructs a sequence of measures, starting with the prior distribution, and finishing with the posterior distribution. The intermediate measures arise from a tempering of the likelihood, or, equivalently, a rescaling of the noise. The resolution of the PDE discretisation is fixed. In contrast, our estimator employs a hierarchy of PDE discretisations to decrease the computational cost. We construct a sequence of intermediate measures by decreasing the temperature or by increasing the discretisation level at the same time. This idea builds on and generalises the multi-resolution sampler proposed in [P.S. Koutsourelakis, J. Comput. Phys., 228 (2009), pp. 6184-6211] where a bridging scheme is used to transfer samples from coarse to fine discretisation levels. Importantly, our choice between tempering and bridging is fully adaptive. We present numerical experiments in 2D space, comparing our estimator to single-level SMC and the multi-resolution sampler.

References in corpus (8)

• The Ensemble Kalman Filter for Inverse Problems
• Multilevel Sequential Monte Carlo Samplers
• Multilevel ensemble Kalman filtering
• Iterative Updating of Model Error for Bayesian Inversion
• Solving Large-Scale PDE-constrained Bayesian Inverse Problems with Riemann Manifold Hamiltonian Monte Carlo
• Multilevel Sequential Monte Carlo Samplers for Normalizing Constants
• Multilevel ensemble Kalman filtering for spatially extended models
• Multilevel Sequential Monte Carlo with Dimension-Independent Likelihood-Informed Proposals

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• Bayesian parameter identification in Cahn-Hilliard models for biological growth
• Multilevel Sequential Importance Sampling for Rare Event Estimation
• Multifidelity multilevel Monte Carlo to accelerate approximate Bayesian parameter inference for partially observed stochastic processes
• Generalized Parallel Tempering on Bayesian Inverse Problems
• Central limit theorems for multilevel Monte Carlo methods
• Multilevel adaptive sparse Leja approximations for Bayesian inverse problems
• Transform-based particle filtering for elliptic Bayesian inverse problems
• Multi-index ensemble Kalman filtering
• Adaptive regularisation for ensemble Kalman inversion
• Multilevel Markov Chain Monte Carlo with likelihood scaling for Bayesian inversion with high-resolution observations
• Multilevel Stein variational gradient descent with applications to Bayesian inverse problems