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Dive into the research topics where Robert S. Womersley is active.

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Featured researches published by Robert S. Womersley.


Mathematical Programming | 1993

Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices

Michael L. Overton; Robert S. Womersley

The sum of the largest eigenvalues of a symmetric matrix is a nonsmooth convex function of the matrix elements. Max characterizations for this sum are established, giving a concise characterization of the subdifferential in terms of a dual matrix. This leads to a very useful characterization of the generalized gradient of the following convex composite function: the sum of the largest eigenvalues of a smooth symmetric matrix-valued function of a set of real parameters. The dual matrix provides the information required to either verify first-order optimality conditions at a point or to generate a descent direction for the eigenvalue sum from that point, splitting a multiple eigenvalue if necessary. Connections with the classical literature on sums of eigenvalues and eigenvalue perturbation theory are discussed. Sums of the largest eigenvalues in the absolute value sense are also addressed.


Advances in Computational Mathematics | 2004

Extremal systems of points and numerical integration on the sphere

Ian H. Sloan; Robert S. Womersley

This paper considers extremal systems of points on the unit sphere Sr⊂Rr+1, related problems of numerical integration and geometrical properties of extremal systems. Extremal systems are systems of dn=dim Pn points, where Pn is the space of spherical polynomials of degree at most n, which maximize the determinant of an interpolation matrix. Extremal systems for S2 of degrees up to 191 (36,864 points) provide well distributed points, and are found to yield interpolatory cubature rules with positive weights. We consider the worst case cubature error in a certain Hilbert space and its relation to a generalized discrepancy. We also consider geometrical properties such as the minimal geodesic distance between points and the mesh norm. The known theoretical properties fall well short of those suggested by the numerical experiments.


Advances in Computational Mathematics | 2001

How good can polynomial interpolation on the sphere be

Robert S. Womersley; Ian H. Sloan

This paper explores the quality of polynomial interpolation approximations over the sphere Sr−1⊂Rr in the uniform norm, principally for r=3. Reimer [17] has shown there exist fundamental systems for which the norm ‖Λn‖ of the interpolation operator Λn, considered as a map from C(Sr−1) to C(Sr−1), is bounded by dn, where dn is the dimension of the space of all spherical polynomials of degree at most n. Another bound is dn1/2(λavg/λmin )1/2, where λavg and λmin  are the average and minimum eigenvalues of a matrix G determined by the fundamental system of interpolation points. For r=3 these bounds are (n+1)2 and (n+1)(λavg/λmin )1/2, respectively. In a different direction, recent work by Sloan and Womersley [24] has shown that for r=3 and under a mild regularity assumption, the norm of the hyperinterpolation operator (which needs more point values than interpolation) is bounded by O(n1/2), which is optimal among all linear projections. How much can the gap between interpolation and hyperinterpolation be closed?For interpolation the quality of the polynomial interpolant is critically dependent on the choice of interpolation points. Empirical evidence in this paper suggests that for points obtained by maximizing λmin , the growth in ‖Λn‖ is approximately n+1 for n<30. This choice of points also has the effect of reducing the condition number of the linear system to be solved for the interpolation weights. Choosing the points to minimize the norm directly produces fundamental systems for which the norm grows approximately as 0.7n+1.8 for n<30. On the other hand, ‘minimum energy points’, obtained by minimizing the potential energy of a set of (n+1)2 points on S2, turn out empirically to be very bad as interpolation points.This paper also presents numerical results on uniform errors for approximating a set of test functions, by both interpolation and hyperinterpolation, as well as by non-polynomial interpolation with certain global basis functions.


SIAM Journal on Matrix Analysis and Applications | 1992

On the sum of the largest eigenvalues of a symmetric matrix

Michael L. Overton; Robert S. Womersley

The sum of the largest k eigenvalues of a symmetric matrix has a well-known extremal property that was given by Fan in 1949 [Proc. Nat. Acad. Sci., 35 (1949), pp. 652–655]. A simple proof of this property, which seems to have been overlooked in the vast literature on the subject and its many generalizations, is discussed. The key step is the observation, which is neither new nor well known, that the convex hull of the set of projection matrices of rank k is the set of symmetric matrices with eigenvalues between 0 and 1 and summing to k. The connection with the well-known Birkhoff theorem on doubly stochastic matrices is also discussed. This approach provides a very convenient characterization for the subdifferential of the eigenvalue sum, described in a separate paper.


Archive | 1995

Recent Advances in Nonsmooth Optimization

Ding-Zhu Du; Liqun Qi; Robert S. Womersley

Nonsmooth optimization covers the minimization or maximization of functions which do not have the differentiability properties required by classical methods. This book includes papers on theory, algorithms and applications for problems with fast-order nondifferentiability (the usual sense of nonsmooth optimization), second-order nondifferentiability, nonsmooth equations, nonsmooth variational inequalities and other problems related to nonsmooth optimization.


SIAM Journal on Matrix Analysis and Applications | 1995

Second Derivatives for Optimizing Eigenvalues of Symmetric Matrices

Michael L. Overton; Robert S. Womersley

Let


Journal of Optimization Theory and Applications | 1986

An algorithm for composite nonsmooth optimization problems

Robert S. Womersley; R. Fletcher

A


SIAM Journal on Matrix Analysis and Applications | 1988

On minimizing the spectral radius of a nonsymmetric matrix function: optimality conditions and duality theory

Michael L. Overton; Robert S. Womersley

denote an


Mathematical Programming | 1985

Local properties of algorithms for minimizing nonsmooth composite functions

Robert S. Womersley

n \times n


SIAM Journal on Numerical Analysis | 2006

Existence of Solutions to Systems of Underdetermined Equations and Spherical Designs

Xiaojun Chen; Robert S. Womersley

real symmetric matrix-valued function depending on a vector of real parameters,

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Ian H. Sloan

University of New South Wales

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Xiaojun Chen

Hong Kong Polytechnic University

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Yuguang Wang

University of New South Wales

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Liqun Qi

Hong Kong Polytechnic University

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Johann S. Brauchart

University of New South Wales

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Michael L. Overton

Courant Institute of Mathematical Sciences

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Quoc Thong Le Gia

University of New South Wales

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Danny Wong

University of New South Wales

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Kerstin Hesse

University of New South Wales

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