Graph-based Prior and Forward Models for Inverse Problems on Manifolds with Boundaries

TL;DR

This paper introduces graph-based manifold learning techniques with boundary-aware Gaussian priors and PDE forward models, improving inverse problem solutions on manifolds with boundaries.

Contribution

It develops boundary-adapted Gaussian priors and graph-based PDE approximations using ghost point diffusion maps for inverse problems on manifolds with boundaries.

Findings

Graph-based priors effectively model boundary conditions.

Numerical results validate the approach.

Boundary-aware models outperform boundary-agnostic ones.

Abstract

This paper develops manifold learning techniques for the numerical solution of PDE-constrained Bayesian inverse problems on manifolds with boundaries. We introduce graphical Mat\'ern-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.