The main goal of this paper is to introduce a novel meshless kernel Galerkin method for numerically solving partial differential equations on the sphere. Specifically, we will use this method to treat the partial differential equation for stationary heat conduction on S 2 , in an inhomogeneous, anisotropic medium. The Galerkin method used to do this employs spatially well-localized, "small footprint", robust bases for the associated kernel space. The stiffness matrices arising in the problem have entries decaying exponentially fast away from the diagonal. Discretization is achieved by first zeroing out small entries, resulting in a sparse matrix, and then replacing the remaining entries by ones computed via a very efficient kernel quadrature formula for the sphere. Error estimates for the approximate Galerkin solution are also obtained.2010 Mathematics Subject Classification. 65M60, 65M12, 41A30, 41A55.
In Flury (1990) the k principal points of a random vector X are defined as the points p(1),... ,p(k) minimizing E{minHX-p(i)ll2; i= 1,..., k}. We extend this concept to that of k principal points with respect to a loss function L, and present an algorithm for their computation in the univariate case.
We introduce a meshfree Galerkin method for solving nonlocal diffusion problems. Radial basis functions are used to construct an approximation scheme that requires only scattered nodes with no triangulation. A quadrature scheme specific to radial basis functions is implemented to produce a Galerkin radial basis function method that yields fast assembly of a sparse stiffness matrix. We provide numerical evidence for convergence rates using one and two dimensional nonlocal problems.
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