Kernel methods for Bayesian elliptic inverse problems on manifolds

John Harlim, Daniel Sanz-Alonso, Ruiyi Yang

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish (US)
Pages (from-to)1414-1445
Number of pages32
JournalSIAM-ASA Journal on Uncertainty Quantification
Volume8
Issue number4
DOIs
StatePublished - 2021

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Modeling and Simulation
  • Statistics, Probability and Uncertainty
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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