Sequential Monte Carlo with Gaussian Mixture Approximation for Infinite-Dimensional Statistical Inverse Problems

TL;DR

This work tackles infinite-dimensional Bayesian inverse problems for PDEs, where the posterior is costlier and often multimodal. It proposes SMC-GM, an algorithm that combines a likelihood-informed Gaussian-mixture mutation (pCN-GM) with an infinite-dimensional SMC framework, and proves convergence under weakened assumptions. A key theoretical contribution is showing the denseness of Gaussian-mixture measures in infinite-dimensional spaces, providing a solid foundation for posterior approximation via Gaussian mixtures. Numerically, SMC-GM demonstrates strong multi-modality probing, significant computational savings, and discretization-invariant behavior across linear and nonlinear PDE inverse problems, including Darcy flow and sparse-measurement scenarios.

Abstract

By formulating the inverse problem of partial differential equations (PDEs) as a statistical inference problem, the Bayesian approach provides a general framework for quantifying uncertainties. In the inverse problem of PDEs, parameters are defined on an infinite-dimensional function space, and the PDEs induce a computationally intensive likelihood function. Additionally, sparse data tends to lead to a multi-modal posterior. These features make it difficult to apply existing sequential Monte Carlo (SMC) algorithms. To overcome these difficulties, we propose new conditions for the likelihood functions, construct a Gaussian mixture based preconditioned Crank-Nicolson transition kernel, and demonstrate the universal approximation property of the infinite-dimensional Gaussian mixture probability measure. By combining these three novel tools, we propose a new SMC algorithm, named SMC-GM. For this new algorithm, we obtain a convergence theorem that allows Gaussian priors, illustrating that the sequential particle filter actually reproduces the true posterior distribution. Furthermore, the proposed new algorithm is rigorously defined on the infinite-dimensional function space, naturally exhibiting the discretization-invariant property. Numerical experiments demonstrate that the new approach has a strong ability to probe the multi-modality of the posterior, significantly reduces the computational burden, and numerically exhibits the discretization-invariant property (important for large-scale problems).

Figures (6)

• Figure 1: (a): Posterior samples obtained from the SMC-GM method. (b): Posterior samples obtained from the SMC-pCN method.
• Figure 2: (a): The background truth of $u$. (b): The posterior mean estimated using SMC-GM. (c): The posterior mean estimated using SMC-pCN.
• Figure 3: Posterior densities of the 1st, 4th, 7th, 10th, 13th, 16th Fourier coefficients.
• Figure 4: In the SMC method, we always need to find a sequence $\{h_j\}_{j=1}^J$ that ranges from 0 to 1 to determine $\{\mu_j\}_{j=1}^J$, the intermediate measures. (a): $\{h_j\}_{j=1}^J$ found by the SMC-GM algorithm under different dimensional grid discretizations. (b): $\{h_j\}_{j=1}^J$ found by SMC-RW under different dimensional grid discretizations.
• Figure 5: (a)-(d) display four out of the eight cluster means from the SMC-GM sampling results, while (e) and (f) show two out of the five cluster means from the SMC-pCN sampling results.

Theorems & Definitions (13)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Lemma 2.1
• Remark 2.4
• Theorem 2.2
• Remark 2.5
• Theorem 2.3
• Theorem 2.4
• Remark 2.6