TITLE: Residual Updating Algorithms for Kernel Interpolation

SPEAKER: Greg Fasshauer

ABSTRACT:

I will first present two scattered data approximation methods from a numerical analysis point of view: radial basis function or kernel interpolation and moving least squares approximation. Then I will introduce the idea of approximate moving least squares approximation and connect all three methods via a residual updating algorithm. In parallel I will attempt to point out connections to an analogous set of methods (kriging, local polynomial regression and higher-order kernels for density estimation) in statistics. The idea of residual updating will be illuminated both from a more analytical perspective and at the numerical linear algebra level where we have rediscovered an old algorithm due to Riley [1].
[1] J.D. Riley. Solving systems of linear equations with a positive definite, symmetric, but possibly ill-conditioned matrix. Mathematical Tables and Other Aids to Computation 9/51 (1955), 96–101.

Brief bio:
Greg Fasshauer
Professor, Associate Chair and Director of Undergraduate Studies
Illinois Institute of Technology
Department of Applied Mathematics
Chicago, IL 60616

Education and Positions
* Since 1997: Assistant, associate and full professor, Department of Applied Mathematics, IIT
* Ralph P. Boas Visiting Assistant Professor: Department of Mathematics, Northwestern University (1995-1997)
* Ph.D. (Mathematics): Vanderbilt University with Larry L. Schumaker (1995)
* M.A. (Mathematics): Vanderbilt University with Larry L. Schumaker (1993)
* Diplom (Mathematics) & Staatsexamen (Mathematics and English): University of Stuttgart with Klaus Höllig (1991)

Research Interests (currently supported by NSF)
* Meshfree approximation methods for multivariate approximation and their application
* Radial basis functions and positive definite kernels
* Approximation theory
* Computer-aided geometric design
* Spline theory
* Numerical solution of PDEs

Books and 40+ papers
* Meshfree Approximation Methods with MATLAB
Interdisciplinary Mathematical Sciences - Vol. 6 World Scientific Publishers, Singapore, 2007
* Progress on Meshless Methods (edited with A.J.M. Ferreira, E.J. Kansa, and V.M.A. Leitão)
Computational Methods in Applied Sciences, Vol. 11 Springer, Berlin, 2009