Browsing by Author "Platte, Rodrigo B."
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Item Computing Eigenmodes of Elliptical Operators Using Radial Basis Functions(Department of Mathematical Sciences, 2003-03-21) Platte, Rodrigo B.; Driscoll, Tobin A.Radial basis function (RBF) approximations have been successfully used to solve boundary-value problems numerically. We show that RBFs can also be used to compute eigenmodes of elliptic operators. Special attention is given to the Laplacian operator in two dimensions. We include techniques to avoid degradation of the solution near the boundaries and corner singularities. Numerical results compare favorably to basic finite element methods.Item Polynomials and their Potential Theory for Gaussian Radial Basis Function Interpolation(Department of Mathematical Sciences, 2004) Driscoll, Tobin A.; Platte, Rodrigo B.We explore a connection between Gaussian radial basis functions and polynomials. Using standard tools of potential theory, we find that these radial functions are susceptible to the Runge phenomenon, not only in the limit of increasingly flat functions, but also in the finite shape parameter case. We show that there exist interpolation node distributions that prevent such phenomena and allow stable approximations. Using polynomials also provides an explicit interpolation formula that avoids the difficulties of inverting interpolation matrices, without imposing restrictions on the shape parameter or number of points.