Zooming from global to local: a multiscale RBF approach

Abstract

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 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.

A Multiscale RBF Collocation Method for the Numerical Solution of Partial Differential Equations

“…Applications of this algorithm in solving PDEs on spheres are proposed in [13], on bounded domains were proposed in [14][15][16]. However, a series of global interpolation or approximation problems must be solved on different levels, although two kinds of local algorithms have been derived in [17,18] recently.…”

“…The approximation method can produce a sparse discrete algebraic system, because hierarchical radial basis functions are derived from CSRBFs with different support radii. Compared with compactly supported radial basis functions approximation [7,8] and stationary multilevel approximation [11][12][13][14][15][16][17][18], the new method can solve the present problem on a single level with higher accuracy and lower computational cost. The effectiveness of H-RBFs collocation method will be conformed by several numerical observations.…”

A Nonstationary Designer Space-Time Kernel

“…However, they also yield stationary kriging models, which is not always ideal. Some work has been done to try to build nonstationary models from stationary kernels (see, e.g., [15] or [13]).…”