Adapted from a series of lectures given by the authors, this monograph focuses on radial basis functions (RBFs), a powerful numerical methodology for solving 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, astro- and geosciences, mathematical biology, and other areas and has lately been shown to compete successfully against the very best previous approaches on some large benchmark problems. Using examples and heuristic explanations to create a practical and intuitive perspective, the authors address how, when, and why RBF-based methods work. The authors trace the algorithmic evolution of RBFs, starting with brief introductions to finite difference (FD) and pseudospectral (PS) methods and following a logical progression to global RBFs and then to 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 seismic exploration for oil and gas, weather and climate modeling, and electromagnetics, among others. This is the first survey in book format of the RBF-FD methodology and is suitable as the text for a one-semester first-year graduate class.
Bengt Fornberg received his PhD from Uppsala University in Sweden. Following positions at the European Organization for Nuclear Research (CERN), the 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 methodologies for analytic functions. Natasha Flyer received her PhD from the University of Michigan, Ann Arbor. She is a staff scientist at the National Center for Atmospheric Research in Boulder, Colorado. Her research interests include development of computational methods for solar physics and geosciences and hybrid analytical-numerical methods for the solution of PDEs with singularities.
Chapter 1: Brief Summary of Finite Difference Methods Chapter 2: Brief Summary of Pseudospectral Methods Chapter 3: Introduction to Radial Basis Functions Chapter 4: Global RBFs for Solving PDEs Chapter 5: RBF-Generated FD (RBF-FD) Methods Chapter 6: Global RBF Applications to Geo-Modeling: Spherical Domains Chapter 7: RBF-FD Applications to Geo-Modeling: Spherical Domains Chapter 8: RBF-FD Applications to Geo-Modeling: Limited-Area Domains Appendix A: Introduction to RBFs via Cubic Splines Appendix B: Spherical Harmonics Appendix C: Some Node Distribution Strategies Appendix D: Cartesian Vector Operators on a Sphere
The first book to survey Radial Basis Function methodology, valuable for graduates and researchers in many application areas.
