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arXiv · 2106.06787

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

John Harlim·Shixiao Jiang·Hwanwoo Kim·Daniel Sanz-Alonso
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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.

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John Harlim, Shixiao Jiang, Hwanwoo Kim, Daniel Sanz-Alonso. 2021-06-12. Graph-based Prior and Forward Models for Inverse Problems on Manifolds with Boundaries. https://doi.org/10.1088/1361-6420%2Fac3994

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