Ghost Point Diffusion Maps for solving elliptic PDE's on Manifolds with Classical Boundary Conditions
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In this paper, we extend the class of kernel methods, the so-called diffusion maps (DM), and its local kernel variants, to approximate second-order differential operators defined on smooth manifolds with boundary that naturally arise in elliptic PDE models. To achieve this goal, we introduce the Ghost Point Diffusion Maps (GPDM) estimator on an extended manifold, identified by the set of point clouds on the unknown original manifold together with a set of ghost points specified along the tangential direction at the boundary of the original manifold. The resulting GPDM estimator restricts the standard DM matrix to a set of extrapolation equations that estimates the function values on the ghost points. This adjustment is analogous to the usual correction on the matrix components near the boundary in classical finite-difference methods. As opposed to the classical DM which diverges near the boundary, the proposed GPDM estimator converges pointwise even near the boundary, assuming that the extended manifold is smooth. Applying the consistent GPDM estimator to solve the well-posed elliptic PDEs with classical boundary conditions (Dirichlet, Neumann, and Robin), we establish the convergence of the approximate solution under appropriate smoothness assumptions. We validate the proposed mesh-free PDE solver with supporting numerical examples on various problems, defined on simple manifolds embedded in 2D to 5D Euclidean spaces as well as on an unknown manifold. Numerically, we also find that the GPDM is more accurate compared to DM in solving eigenvalue problems associated with the second-order differential operators on bounded smooth manifolds.
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