2019/10/23 by John Harlim, Harlim, John, Daniel Sanz-Alonso +3 · 4 citations
Computer Science · Decision Sciences · Mathematics · #FOS: Mathematics #Gaussian Processes and Bayesian Inference #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1910.10669
openalex publication_date 2019/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper investigates the formulation and implementation of Bayesian inverse problems to learn input parameters of partial differential equations (PDEs) defined on manifolds. Specifically, we study the inverse problem of determining the diffusion coefficient of a second-order elliptic PDE on a closed manifold from noisy measurements of the solution. Inspired by manifold learning techniques, we approximate the elliptic differential operator with a kernel-based integral operator that can be discretized via Monte-Carlo without reference to the Riemannian metric. The resulting computational method is mesh-free and easy to implement, and can be applied without full knowledge of the underlying manifold, provided that a point cloud of manifold samples is available. We adopt a Bayesian perspective to the inverse problem, and establish an upper-bound on the total variation distance between the true posterior and an approximate posterior defined with the kernel forward map. Supporting numerical results show the effectiveness of the proposed methodology.