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

Hybrid Inflation and Particle Physics

The prototype hybrid SUSY SU(5) inflation models, while well motivated from particle physics, and while allowing an acceptable inflationary phase with little or no fine tuning, are shown to have two fundamental phenomenological problems. (1) They inevitably result in the wrong vacuum after inflation is over; and (2) they do not solve the monopole problem. In order to get around the first problem the level of complexity of these models must be increased. One can also avoid the second problem in this way. We also demonstate another possibility by proposing a new general mechanism to avoid the monopole problem with, or without inflation

Multiscale approximation for functions in arbitrary Sobolev spaces by scaled radial basis functions on the unit sphere

AbstractIn this paper, we prove convergence results for multiscale approximation using compactly supported radial basis functions restricted to the unit sphere, for target functions outside the reproducing kernel Hilbert space of the employed kernel