View PDF HTML (experimental)

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 with Gaussian mixture approximation, together with an easy-to-use reduced version. 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 reduced version 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).

Submission history

From: Junxiong Jia [view email]
[v1] Thu, 20 Mar 2025 10:50:35 UTC (22,425 KB)
[v2] Sun, 23 Mar 2025 08:06:23 UTC (22,425 KB)
[v3] Wed, 1 Oct 2025 05:02:15 UTC (6,032 KB)
[v4] Sat, 7 Feb 2026 02:32:54 UTC (6,038 KB)
[v5] Mon, 10 Aug 2026 03:01:49 UTC (8,969 KB)