GaussQR - A Framework for Stable Kernel Computation

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A joint project with Mike McCourt, with additional contributions by Salvatore Ganci and Haocheng Bian. The website serves as the home for the software on stable algorithms for kernel computation built upon the Hilbert-Schmidt SVD. This MATLAB library is freely available and allows others to experiment with our code. Thus far, the library contains routines for Gaussian

Fasshauer, GE, Ye Q.  2013.  Kernel-based Collocation Methods versus Galerkin Finite Element Methods for Approximating Elliptic Stochastic Partial Differential Equations. Meshfree Methods for Partial Differential Equations VI. :155-170. Abstract
In this paper we compare kernel-based collocation methods (meshfree approximation methods) with Galerkin finite element methods for solving elliptic stochastic partial differential equations driven by Gaussian noises. The kernel-based collocation solution is a linear combination of reproducing kernels not only done with related differential and boundary operators but also centered at chosen collocation points. Its random coefficients are solved by a random linear equation systems. The finite element solution is a tensor product of triangular finite element bases and Lagrange polynomials defined on a finite dimensional probability space. Its coefficients are solved by serval deterministic finite element problems. For the kernel-based collocation methods, we can simulate the Gaussian noise at the collocation points directly. But we need to truncate the Gaussian noises into finite dimensional noises for the finite element methods. According to our numerical experiments, the finite element methods have the same convergent rate as the kernel-based collocation methods if the Gaussian noises are truncated by the suitable terms.
Berkaliev, Z, Devi S, Fasshauer GE, Hickernell FJ, Kartal O, Li X, McCray P, Whitney S, Zawojewski JS.  Submitted.  Getting Students To Think Computationally: Initiating a Programmatic Self-Study.
Ala, G, Fasshauer GE, Francomano E, Ganci S, McCourt MJ.  Submitted.  The Method of Fundamental Solutions in Solving Coupled Boundary Value Problems for M/EEG.
McCourt, MJ, Fasshauer GE.  Submitted.  Approximating Derivatives Stably Using Gaussians.
Fasshauer, GE, Ye Q.  2014.  A Kernel-based Collocation Method for Elliptic Partial Differential Equations with Random Coefficients. Monte Carlo and Quasi-Monte Carlo Methods 2012. :331-347.
Fasshauer, GE, Ye Q.  2013.  Reproducing kernels of Sobolev spaces via a Green kernel approach with differential operators and boundary operators. Advances in Computational Mathematics. 38(4):891-921.Website
Song, G, Riddle J, Fasshauer GE, Hickernell FJ.  2012.  Multivariate interpolation with increasingly flat radial basis functions of finite smoothness. Advances in Computational Mathematics. 36(3):485-501.Website