350 research outputs found
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
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 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
Logarithmic and Riesz Equilibrium for Multiple Sources on the Sphere --- the Exceptional Case
We consider the minimal discrete and continuous energy problems on the unit
sphere in the Euclidean space in the presence
of an external field due to finitely many localized charge distributions on
, where the energy arises from the Riesz potential (
is the Euclidean distance) for the critical Riesz parameter if and the logarithmic potential if . Individually, a
localized charge distribution is either a point charge or assumed to be
rotationally symmetric. The extremal measure solving the continuous external
field problem for weak fields is shown to be the uniform measure on the sphere
but restricted to the exterior of spherical caps surrounding the localized
charge distributions. The radii are determined by the relative strengths of the
generating charges. Furthermore, we show that the minimal energy points solving
the related discrete external field problem are confined to this support. For
, we show that for point sources on the sphere, the equilibrium
measure has support in the complement of the union of specified spherical caps
about the sources. Numerical examples are provided to illustrate our results.Comment: 23 pages, 4 figure
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