Q. T. Le Gia

Localized Linear Polynomial Operators and Quadrature Formulas on the Sphere

The purpose of this paper is to construct universal, auto-adaptive, localized, linear, polynomial (-valued) operators based on scattered data on the (hyper) sphere

Multiscale Analysis in Sobolev Spaces on the Sphere

\mathbb{S}^q

Continuous and discrete least-squares approximation by radial basis functions on spheres

(

Approximation of parabolic PDEs on spheres using spherical basis functions

q\ge2

Multiscale RBF collocation for solving PDEs on spheres

). The approximation and localization properties of our operators are studied theoretically in deterministic as well as probabilistic settings. Numerical experiments are presented to demonstrate their superiority over traditional least squares and discrete Fourier projection polynomial approximations. An essential ingredient in our construction is the construction of quadrature formulas based on scattered data, exact for integrating spherical polynomials of (moderately) high degree. Our formulas are based on scattered sites; i.e., in contrast to such well-known formulas as Driscoll-Healy formulas, we need not choose the location of the sites in any particular manner. While the previous attempts to construct such formulas have yielded formulas exact for spherical polynomials of degree at most 18, we are able to construct formulas exact for spherical polynomials of degree 178.

Polynomial operators and local approximation of solutions of pseudo-differential equations on the sphere

We consider a multiscale approximation scheme at scattered sites for functions in Sobolev spaces on the unit sphere

Preconditioners for pseudodifferential equations on the sphere with radial basis functions

\mathbb{S}^n

Galerkin approximation for elliptic PDEs on spheres

. The approximation is constructed using a sequence of scaled, compactly supported radial basis functions restricted to

Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere

\mathbb{S}^n

A pseudospectral quadrature method for Navier-Stokes equations on rotating spheres

. A convergence theorem for the scheme is proved, and the condition number of the linear system is shown to stay bounded by a constant from level to level, thereby establishing for the first time a mathematical theory for multiscale approximation with scaled versions of a single compactly supported radial basis function at scattered data points.