Graph-based prior and forward models for inverse problems on manifolds with boundaries. (1st March 2022)

• Record Type: Journal Article
• Authors: Harlim, John; Jiang, Shixiao W; Kim, Hwanwoo; Sanz-Alonso, Daniel
• Is Part Of: Inverse problems. Volume 38:Number 3(2022)
• Publication Date: 2022-03-01
• Subjects: Bayesian inverse problem -- manifold with boundary -- ghost point diffusion map -- graph-based prior; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/ac3994
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: This paper develops manifold learning techniques for the numerical solution of PDE-constrained Bayesian inverse problems on manifolds with boundaries. We introduce graphical Matérn-type Gaussian field priors that enable flexible modeling near the boundaries, representing boundary values by superposition of harmonic functions with appropriate Dirichlet boundary conditions. We also investigate the graph-based approximation of forward models from PDE parameters to observed quantities. In the construction of graph-based prior and forward models, we leverage the ghost point diffusion map algorithm to approximate second-order elliptic operators with classical boundary conditions. Numerical results validate our graph-based approach and demonstrate the need to design prior covariance models that account for boundary conditions.