Graph-based prior and forward models for inverse problems on manifolds with boundaries

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 boundaries, representing boundary values by superposition harmonic functions appropriate Dirichlet conditions. also investigate graph-based approximation forward models from PDE parameters to observed quantities. In construction prior and models, we leverage ghost point diffusion map algorithm approximate second-order elliptic operators classical Numerical results validate our approach demonstrate need design covariance account