Numerical solutions of a boundary value problem on the sphere using radial basis functions

Boundary value problems on the unit sphere arise naturally in geophysics and oceanography when scientists model a physical quantity on large scales. Robust numerical methods play an important role in solving these problems. In this article, we construct numerical solutions to a boundary value problem defined on a spherical sub-domain (with a sufficiently smooth boundary) using radial basis functions (RBF). The error analysis between the exact solution and the approximation is provided. Numerical experiments are presented to confirm theoretical estimates

Fast iterative solvers for boundary value problems on a local spherical region

Boundary value problems on local spherical regions arise naturally in geophysics and oceanography when scientists model a physical quantity on large scales. Meshless methods using radial basis functions provide a simple way to construct numerical solutions with high accuracy. However, the linear systems arising from these methods are usually ill-conditioned, which poses a challenge for iterative solvers. We construct preconditioners based on an additive Schwarz method to accelerate the solution process for solving boundary value problems on local spherical regions. References D. Crowdy. Point vortex motion on the surface of a sphere with impenetrable boundaries. Physics of Fluids, 18:036602 (2006). doi:10.1063/1.2183627. A. E. Gill. Atmosphere-Ocean Dynamics, International Geophysics Series Volume 30. Academic, New York (1982). R. Kidambi and P. K. Newton. Point vortex motion on a sphere with solid boundaries. Physics of Fluids, 12:581 (2000). doi:10.1063/1.870263. Q. T. Le Gia, I. H. Sloan, and T. Tran. Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere. Math. Comp., 78:79--101 (2009). doi:10.1090/S0025-5718-08-02150-9. C. Muller. Spherical Harmonics, Vol. 17 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (1966). M. V. Nezlin. Some remarks on coherent structures out of chaos in planetary atmospheres and oceans. Chaos, 4:109--111 (1994). doi:10.1063/1.165997. T. Tran, Q. T. Le Gia, I. H. Sloan, and E. P. Stephan. Preconditioners for pseudodifferential equations on the sphere with radial basis functions. Numer. Math., 115:141--163 (2009). doi:10.1007/s00211-009-0269-8

Zooming from Global to Local: A Multiscale RBF Approach

Because physical phenomena on Earth's surface occur on many different length scales, it makes sense when seeking an efficient approximation to start with a crude global approximation, and then make a sequence of corrections on finer and finer scales. It also makes sense eventually to seek fine scale features locally, rather than globally. In the present work, we start with a global multiscale radial basis function (RBF) approximation, based on a sequence of point sets with decreasing mesh norm, and a sequence of (spherical) radial basis functions with proportionally decreasing scale centered at the points. We then prove that we can "zoom in" on a region of particular interest, by carrying out further stages of multiscale refinement on a local region. The proof combines multiscale techniques for the sphere from Le Gia, Sloan and Wendland, SIAM J. Numer. Anal. 48 (2010) and Applied Comp. Harm. Anal. 32 (2012), with those for a bounded region in $\mathbb{R}^d$ from Wendland, Numer. Math. 116 (2012). The zooming in process can be continued indefinitely, since the condition numbers of matrices at the different scales remain bounded. A numerical example illustrates the process

Approximation of linear partial differential equations on spheres

The theory of interpolation and approximation of solutions to differential and integral equations on spheres has attracted considerable interest in recent years; it has also been applied fruitfully in fields such as physical geodesy, potential theory, oceanography, and meteorology. In this dissertation we study the approximation of linear partial differential equations on spheres, namely a class of elliptic partial differential equations and the heat equation on the unit sphere. The shifts of a spherical basis function are used to construct the approximate solution. In the elliptic case, both the finite element method and the collocation method are discussed. In the heat equation, only the collocation method is considered. Error estimates in the supremum norms and the Sobolev norms are obtained when certain regularity conditions are imposed on the spherical basis functions

Multi-level higher order QMC Galerkin discretization for affine parametric operator equations

We develop a convergence analysis of a multi-level algorithm combining higher order quasi-Monte Carlo (QMC) quadratures with general Petrov-Galerkin discretizations of countably affine parametric operator equations of elliptic and parabolic type, extending both the multi-level first order analysis in [\emph{F.Y.~Kuo, Ch.~Schwab, and I.H.~Sloan, Multi-level quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficient} (in review)] and the single level higher order analysis in [\emph{J.~Dick, F.Y.~Kuo, Q.T.~Le~Gia, D.~Nuyens, and Ch.~Schwab, Higher order QMC Galerkin discretization for parametric operator equations} (in review)]. We cover, in particular, both definite as well as indefinite, strongly elliptic systems of partial differential equations (PDEs) in non-smooth domains, and discuss in detail the impact of higher order derivatives of {\KL} eigenfunctions in the parametrization of random PDE inputs on the convergence results. Based on our \emph{a-priori} error bounds, concrete choices of algorithm parameters are proposed in order to achieve a prescribed accuracy under minimal computational work. Problem classes and sufficient conditions on data are identified where multi-level higher order QMC Petrov-Galerkin algorithms outperform the corresponding single level versions of these algorithms. Numerical experiments confirm the theoretical results