Gregory Fasshauer, Ph.D.
Research & Accomplishments
Meshfree methods for multivariate approximation (radial basis functions, moving least squares and related methods), Applications of meshfree methods (surface and solid model compression, image compression, options pricing, vibration analysis of composite beams and plates, gas-dynamics), Numerical analysis (solution of differential and integral equations, Nash iteration, multilevel algorithms, fast summation methods, meshfree pseudospectral methods, parallel computing), Approximation theory (approximation on spheres, approximate approximation), Computer-aided geometric design (minimal energy surfaces, geometric continuity), Spline theory (approximation with multivariate splines, dimensions of multivariate spline spaces).
2012 - Distinguished Teaching Fellow, College of Science and Letters
2008 - AMS Menger Prize Committee
2007 - Dean's Excellence Award for Teaching, College of Science and Letters, IIT.
2003 - Fellow of Wessex Institute of Technology, Great Britain.
1995 - Bjarni Jonsson Prize for Research, Vanderbilt University.
Approximation Theory XIV: San Antonio 2013 (edited with Larry Schumaker), Proceedings in Mathematics & Statistics, Vol. 83, Springer, New York, 2014
Progress on Meshless Methods (with A.J.M. Ferreira, E.J. Kansa and V.M.A. Leitao), Springer, New York, 2008
Meshfree Approximation Methods with MATLAB, World Scientific, 2007
An Introduction to the Hilbert-Schmidt SVD using Iterated Brownian Bridge Kernels, with R. Cavoretto and M. McCourt, Numerical Algorithms, DOI: 10.1007/s11075-014-9850-z.
Solving support vector machines in reproducing kernel Banach spaces with positive definite functions, with Q. Ye and F. Hickernell, Applied & Computational Harmonic Analysis, DOI: 10.1016/j.acha.2014.03.007.
Reproducing kernels of Sobolev spaces via a Green function approach with differential operators & boundary operators, with Q. Ye , Adv. Comp. Math. 38/4 (2013), 891-921.
Approximation of stochastic partial differential equations by a kernel-based collocation method, with I. Cialenco and Q. Ye, Int. J. Comput. Math. 89/18 (2012), 2543-2561.
On dimension-independent rates of convergence for function approximation with Gaussian kernels, with F.J. Hickernell and H. Wozniakowski, SIAM J. Numer. Anal. 50/1 (2012), 247-271.
Stable evaluation of Gaussian radial basis function interpolants, with M.J. McCourt, SIAM J. Sci. Comput. 34/2 (2012), A737-A762.
Positive definite kernels: past, present, future, Dolomites Research Notes on Approximation 4 (2011), 21-63.
Reproducing kernels of generalized Sobolev spaces via a Green function apporach with distributional operators, with Q. Ye, Numer. Math 119 (2011), 585-611.
On choosing "optimal" shape parameters for RBF approximation, with J.G. Zhang, Numerical Algorithms 45 (2007), 345-368.
Newton iteration with multiquadrics for the solution of nonlinear PDEs, Comput. Math. Applic. 43 (2002), 423—438.
Multistep approximation algorithms: improved convergence rates through postconditioning with smoothing kernels, with Joe Jerome, Adv. Comput. Math. 10 (1999), 1—27.
Scattered data fitting on the sphere, with LL Schumaker, M. Daehlen, and T. Lyche, in Mathematical Methods for Curves and Surfaces II, Vanderbilt University Press, pg. 117-166, 1998.
Solving partial differential equations by collocation with radial basis functions, with A Le Mehaute, C. Rabut, and LL Schumaker, in Surface Fitting and Multiresolution Methods, Vanderbilt University Press, pg. 131-138, 1997.