Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/5883
Title: Minimizing the condition number of a Gram matrix
Authors: Chen, X 
Womersley, RS
Ye, JJ
Keywords: Condition number
Gram matrix
Least squares
Interpolation
Smoothing method
Generalized gradient
Semismooth
Spherical harmonics
Issue Date: 2011
Publisher: Society for Industrial and Applied Mathematics
Source: SIAM journal on optimization, 2011, v. 21, no. 1, p. 127–148 How to cite?
Journal: SIAM Journal on optimization 
Abstract: The condition number of a Gram matrix defined by a polynomial basis and a set of points is often used to measure the sensitivity of the least squares polynomial approximation. Given a polynomial basis, we consider the problem of finding a set of points and/or weights which minimizes the condition number of the Gram matrix. The objective function f in the minimization problem is nonconvex and nonsmooth. We present an expression of the Clarke generalized gradient of f and show that f is Clarke regular and strongly semismooth. Moreover, we develop a globally convergent smoothing method to solve the minimization problem by using the exponential smoothing function. To illustrate applications of minimizing the condition number, we report numerical results for the Gram matrix defined by the weighted Vandermonde-like matrix for least squares approximation on an interval and for the Gram matrix defined by an orthonormal set of real spherical harmonics for least squares approximation on the sphere.
URI: http://hdl.handle.net/10397/5883
ISSN: 1052-6234
EISSN: 1095-7189
DOI: 10.1137/100786022
Rights: © 2011 Society for Industrial and Applied Mathematics
Appears in Collections:Journal/Magazine Article

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