Natasha Flyer publishes SIAM monograph on RBFs

By Brian Bevirt
12/07/2015 - 12:00am
Natasha Flyer and Bengt Fornberg
Natasha Flyer and Bengt Fornberg are a wife-and-husband team. Their SIAM monograph on applying Radial Basis Functions to the geosciences is a landmark in their work to bring this important computational method into practical use.

In its 47-year history, SIAM (Society of Industrial and Applied Mathematics) monographs have always carried great prestige because only a few are published each year by the leading researchers in a major field of interest. Natasha Flyer and Bengt Fornberg are world-renowned authorities working at the frontier of applying Radial Basis Functions (RBFs) to the geosciences. Their monograph, A Primer on Radial Basis Functions with Applications to the Geosciences, is based on a series of ten lectures and encompasses a decade of research and development by the authors. Some of the chapters are so up-to-date that the content is still awaiting publication in research journals. Designed to be used as a textbook for graduate courses, the book is also a primer for computational scientists and numerical analysts, as well as researchers in atmospheric modeling and geosciences who use large-scale PDE-based simulations on modern supercomputers.

Monograph cover
This is the cover of the SIAM monograph on Radial Basis Functions.

RBFs are a relatively new numerical approach for solving partial differential equations (PDEs) to high accuracy in any number of dimensions. This method applies to problems across a wide range of PDEs arising in fluid mechanics, wave motions, geosciences, and mathematical biology. Recently, RBFs have been shown to outperform state-of-the-art alternative approaches on some large benchmark problems. Being a meshless method, RBFs excel in solving problems that require geometric flexibility, local refinement for small features, and with little increase in programming complexity when extended to higher dimensional spaces. A key advantage of RBFs for geophysical modeling is that they do not depend on any grid, mesh, or coordinate system, but only the Euclidean distance between node locations in any dimensional space.

The book traces the algorithmic evolution of RBFs, starting with brief introductions to finite difference (FD) and pseudospectral (PS) methods. It then follows a logical progression to global RBFs and RBF-generated FD (RBF-FD) methods. The RBF-FD method, conceived in 2000, has proven to be a leading candidate for numerical simulations in an increasingly wide range of applications, including weather and climate modeling, seismic exploration for oil and gas, and electromagnetics, among others. This is the first survey of the RBF-FD methodology in book form.

Natasha received her Ph.D. from the University of Michigan, Ann Arbor. She is a staff scientist in CISL and IMAGe. Her research interests include developing computational methods for solar physics and geosciences and hybrid analytical-numerical methods for solving PDEs with singularities. Bengt received his Ph.D. from Uppsala University in Sweden. Following positions at the European Organization for Nuclear Research (CERN), California Institute of Technology, and Exxon Corporate Research, he has been on the faculty of Applied Mathematics at the University of Colorado, Boulder since 1995. His research focus is on numerical methods for solving PDEs and computational methods for analytic functions.