Bengt Fornberg

Professor of Applied MathematicsUniversity of Colorado

Regular mail:  
University of Colorado
Department of Applied Mathematics, 526 UCB
Boulder , CO 80309

Tel:        303-4925915

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.

Earlier book:

B. Fornberg: A Practical Guide to Pseudospectral Methods

(Cambridge University Press, 1996)

New book:

B. Fornberg and N. Flyer: A Primer on Radial Basis Functions with Applications to the Geosciences

(SIAM, 2015)

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

Some former Ph.D. students 

Michelle Ghrist   Assoc. Prof. at USAF Academy, Colorado Springs, CO.
    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
Julia Zuev

   Adjunct Lecturer at Santa Clara University, Santa Clara, CA. 
    Thesis: Recent Advances in Numerical PDEs
Cecile Piret

  Assist. Prof. Michigan Technological University, Houghton, MI.
    Thesis: Analytical and Numerical Advances in Radial Basis Functions
Jonah Reeger

  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 
Brad Martin     Thesis: Application of RBF-FD to Wave and Heat Transport Problems in Domains with Interfaces

Some present collaborators 

Natasha Flyer   Scientist III at NCAR, Boulder
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
Victor Bayona   Scientist at ECMWF, Reading, UK.

Recent honors

2014       Fellow of SIAM (Society for Industrial and Applied Mathematics)
2014       University of Colorado Boulder Faculty Assembly Award for Excellence in Research, Scholarly and Creative Work.


Recent papers:

For the actually published articles, please refer directly to the respective journals. In many cases, the links below lead to earlier manuscript versions than the actually published ones. When the links below lead to the published version, please use the file only in ways that are consistent with relevant copyright regulations. 

A few less recent papers: