  In this paper, we propose a mesh-free numerical method for solving elliptic
PDEs on unknown manifolds, identified with randomly sampled point cloud data.
The PDE solver is formulated as a spectral method where the test function space
is the span of the leading eigenfunctions of the Laplacian operator, which are
approximated from the point cloud data. While the framework is flexible for any
test functional space, we will consider the eigensolutions of a weighted
Laplacian obtained from a symmetric Radial Basis Function (RBF) method induced
by a weak approximation of a weighted Laplacian on an appropriate Hilbert
space. Especially, we consider a test function space that encodes the geometry
of the data yet does not require us to identify and use the sampling density of
the point cloud. To attain a more accurate approximation of the expansion
coefficients, we adopt a second-order tangent space estimation method to
improve the RBF interpolation accuracy in estimating the tangential
derivatives. This spectral framework allows us to efficiently solve the PDE
many times subjected to different parameters, which reduces the computational
cost in the related inverse problem applications. In a well-posed elliptic PDE
setting with randomly sampled point cloud data, we provide a theoretical
analysis to demonstrate the convergent of the proposed solver as the sample
size increases. We also report some numerical studies that show the convergence
of the spectral solver on simple manifolds and unknown, rough surfaces. Our
numerical results suggest that the proposed method is more accurate than a
graph Laplacian-based solver on smooth manifolds. On rough manifolds, these two
approaches are comparable. Due to the flexibility of the framework, we
empirically found improved accuracies in both smoothed and unsmoothed Stanford
bunny domains by blending the graph Laplacian eigensolutions and RBF
interpolator.
