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The aim of this paper is to show how rapidly decaying RBF Lagrange functions on the spheres can be used to create effective, stable finite difference methods based on radial basis functions (RBF-FD). For certain classes of PDEs this approach leads to precise convergence estimates for stencils which grow moderately with increasing discretization fineness.

We introduce intrinsic interpolatory bases for data structured on graphs and derive properties of those bases. Polyharmonic Lagrange functions are shown to satisfy exponential decay away from their centers. The decay depends on the density of the zeros of the Lagrange function, showing that they scale with the density of the data. These results indicate that Lagrange-type bases are ideal building blocks for analyzing data on graphs, and we illustrate their use in kernel-based machine learning applications., Comment: 17 pages

We introduce a meshless method for solving both continuous and discrete variational formulations of a volume constrained, nonlocal diffusion problem. We use the discrete solution to approximate the continuous solution. Our method is nonconforming and uses a localized Lagrange basis that is constructed out of radial basis functions. By verifying that certain inf-sup conditions hold, we demonstrate that both the continuous and discrete problems are well-posed, and also present numerical and theoretical results for the convergence behavior of the method. The stiffness matrix is assembled by a special quadrature routine unique to the localized basis. Combining the quadrature method with the localized basis produces a well-conditioned, symmetric matrix. This then is used to find the discretized solution., Comment: 23 pages, 2, figures, submitted to Math Comp

While inverse estimates in the context of radial basis function approximation on boundary-free domains have been known for at least ten years, such theorems for the more important and difficult setting of bounded domains have been notably absent. This article develops inverse estimates for finite dimensional spaces arising in radial basis function approximation and meshless methods. The inverse estimates we consider control Sobolev norms of linear combinations of a localized basis by the $L_p$ norm over a bounded domain. The localized basis is generated by forming local Lagrange functions for certain types of RBFs (namely Mat\'ern and surface spline RBFs). In this way it extends the boundary-free construction of Fuselier, Hangelbroek, Narcowich, Ward and Wright.

This article develops direct and inverse estimates for certain finite dimensional spaces arising in kernel approximation. Both the direct and inverse estimates are based on approximation spaces spanned by local Lagrange functions which are spatially highly localized. The construction of such functions is computationally efficient and generalizes the construction given by the authors for restricted surface splines on $\mathbb{R}^d$. The kernels for which the theory applies includes the Sobolev-Mat\'ern kernels for closed, compact, connected, $C^\infty$ Riemannian manifolds., Comment: 29 pages. To appear in Festschrift for the 80th Birthday of Ian Sloan

We present a novel Galerkin method for solving partial differential equations on the sphere. The problem is discretized by a highly localized basis which is easily constructed. The stiffness matrix entries are computed by a recently developed quadrature formula unique to the localized bases we consider. We present error estimates and investigate the stability of the discrete stiffness matrix. Implementation and numerical experiments are discussed., Comment: 22 pages, 1 figure. Changes to previous version include new numerical experiments, updated figures, new styling, and re-written sections, and additional theory to address the use of local Lagrange functions for the discretization basis