Professor of Applied Mathematics, University of Colorado
Department of Applied Mathematics, 526 UCB
Faculty member, Center for Research on Training,
University of Colorado
Research interests:My main research interests are in developing, analyzing, and implementing numerical methods, in particular for solving PDEs to high orders of accuracy. Such methods include pseudospectral and high accuracy finite difference methods and, in particular, methods based on radial basis functions (RBFs). The main application areas include computational fluid dynamics, geophysical and astrophysical flows, different types of wave phenomena, and seismic exploration.
|This book is adapted from a series of lectures first given by the
authors at a CBMS/NSF conference held at University of
Massachusetts, Dartmouth. It 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. It is suitable as the text for a one-semester first-year graduate class. The book is primarily written for graduate students and researchers in application areas such as atmospheric modeling and geosciences. It is also suited for numerical analysts and computational scientists interested in large-scale PDE-based simulations on modern computer architectures.
Click on image of book cover for more information from SIAM bookstore.
The articles Solving PDEs with radial basis functions (BF and N. Flyer), Acta Numerica 24 (2015), 215-258, and Radial basis function-generated finite differences: A mesh-free method for computational geosciences (accepted manuscript) (N. Flyer, G.B. Wright and BF), Handbook of Geomathematics, 2014, Springer, both briefly cover some of the materials that are discussed further in the book.
Some recent presentations
|Michelle Ghrist|| Assoc. Prof. at USAF Academy,
Thesis: High-Order Finite Difference Methods for Wave Equations
|Grady Wright|| Assoc. Prof. at Boise State University
Thesis: Radial Basis Function Interpolation: Numerical and Analytical Developments
|| Adjunct Lecturer at Santa Clara University, Santa
Thesis: Recent Advances in Numerical PDEs
|| Assist. Prof. Michigan
Technological University, Houghton, MI.
Thesis: Analytical and Numerical Advances in Radial Basis Functions
|| Assist. Prof. Air Force
Institute of Technology, Wright-Patterson AFB, OH.
Thesis: A Computational Study of the Fourth Painlevé Equation and a Discussion of Adams Predictor-Corrector methods
|Greg Barnett||Thesis: A Robust RBF-FD Formulation based on Polyharmonic Splines and Polynomials|
|Natasha Flyer|| Scientist III at NCAR,
|Amik St-Cyr||Scientist at Royal Dutch Shell, Rijswijk, Holland|
|Jonah Reeger||Assist. Prof. Air Force Institute of Technology, Wright-Patterson AFB, OH.|
|Elisabeth Larsson||Senior Lecturer at Uppsala University|
|André Weideman||Prof. at Stellenbosch University|