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This paper develops manifold learning techniques for the numerical solutionof PDE-constrained Bayesian inverse problems on manifolds with boundaries. Weintroduce graphical Mat\'ern-type Gaussian field priors that enable flexiblemodeling near the boundaries, representing boundary values by superposition ofharmonic functions with appropriate Dirichlet boundary conditions. We alsoinvestigate the graph-based approximation of forward models from PDE parametersto observed quantities. In the construction of graph-based prior and forwardmodels, we leverage the ghost point diffusion map algorithm to approximatesecond-order elliptic operators with classical boundary conditions. Numericalresults validate our graph-based approach and demonstrate the need to designprior covariance models that account for boundary conditions.

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