G. A. Watson

Globally convergent methods for semi-infinite programming

Recently developed methods for nonlinear semi-infinite programming problems have only local convergence properties. In this paper, we show how the convergence can be globalized by the use of an exact penalty function. Both convergence and rate of convergence results are established.

Approximation in normal linear spaces

Abstract A historical account is given of the development of methods for solving approximation problems set in normed linear spaces. Approximation of both real functions and real data is considered, with particular reference to Lp (or lp) and Chebyshev norms. As well as coverage of methods for the usual linear problems, an account is given of the development of methods for approximation by functions which are nonlinear in the free parameters, and special attention is paid to some particular nonlinear approximating families.

Data Fitting Problems with Bounded Uncertainties in the Data

An analysis of a class of data fitting problems, where the data uncertainties are subject to known bounds, is given in a very general setting. It is shown how such problems can be posed in a computationally convenient form, and the connection with other more conventional data fitting problems is examined. The problems have attracted interest so far in the special case when the underlying norm is the least squares norm. Here the special structure can be exploited to computational advantage, and we include some observations which contribute to algorithmic development for this particular case. We also consider some variants of the main problems and show how these too can be posed in a form which facilitates their numerical solution.

On two methods for discreteL p approximation

A recent paper by Merle and Späth [3] gives some computational experience with two algorithms for linear discreteLp approximation. In this note, we establish a theoretical basis for some convergence properties observed by these authors.ZusammenfassungMerle und Späth [3] beschrieben 1974 numerische Erfahrungen mit zwei Algorithmen zur linearen diskretenLp Approximation. In der vorliegenden Arbeit wird eine theoretische Basis für einige Konvergenzeigenschaften gegeben, die von den Autoren beobachtet wurden.

On matrix approximation problems with Ky Fank norms

LetA be a realm xn matrix whose elements depend onl free parameters forming the vectorx. Then a class of approximation problems can be defined by the requirement thatx be chosen to minimize ∥A(x)∥, for a given matrix normon m ×n matrices. For example, it may be required to approximate a given matrix by a particular type of matrix, or by a linear combination of matrices. In the derivation of effective algorithms for such problems, a prerequisite is the provision of appropriate conditions satisfied by a solution, and the subdifferential of the matrix norm plays a crucial role in this. Therefore a characterization of the subdifferential is important, and this is considered for a class of orthogonally invariant norms known as Ky Fank norms, which include as special cases the spectral norm and the trace norm. The results lead to a consideration of efficient algorithms.

Least Squares Fitting of Circles and Ellipses to Measured Data

Fitting conic sections to data is an important problem with many applications. Often the data are obtained from a physical object using a coordinate measuring machine with a touch probe. The probe directions, relative to a particular frame of reference, are an important part of the problem, although conventional methods make no use of these. Here an approach is used for fitting circles and ellipses which takes account of the measurement design. Algorithms are developed, and some numerical results are given.

Numerical analysis : A.R. Mitchell 75th birthday volume

Some biographical and mathematical notes, D. F. Griffiths et al the finite element method and computer aided geometric design, R. E. Barnhill fixed points and spurious modes of a nonlinear infinite-step map, M. Aves et al Runge-Kutta methods as mathematical objects, J. C. Butcher Runge-Kutta methods on manifolds, M. Calvo et al numerical solutions of one and two dimensional hyperbolic systems modelling a fluidized bed, I. Christie et al numerical experiments for a nonoverlapping domain decomposition method for partial differential equations, J. Douglas and D-Q Yang a piecewise uniform adaptive grid algorithm for nonlinear dispersive wave equations, E. S. Fraga and J. L. Morris diagonal dominance and positive definiteness of upwind approximations for advection diffusion problems, G. H. Golum et al chaos in numerics, B. M. Herbst et al exact difference formulas for linear differential operators, D. P. Laurie and A. Craig (part contents).

Best simultaneous approximation of an infinite set of functions

Abstract Some recent results concerning characterization and uniqueness for a class of simultaneous approximation problems are extended to the case when the number of functions being approximated simultaneously is infinite.

A class of best simultaneous approximation problems

For a general class of best simultaneous approximation problems, characterization and uniqueness results are established.

On the solution of the errors in variables problem using the l 1 norm

A fundamental problem in data analysis is that of fitting a given model to observed data. It is commonly assumed that only the dependent variable values are in error, and the least squares criterion is often used to fit the model. When significant errors occur in all the variables, then an alternative approach which is frequently suggested for this errors in variables problem is to minimize the sum of squared orthogonal distances between each data point and the curve described by the model equation. It has long been recognized that the use of least squares is not always satisfactory, and thel1 criterion is often superior when estimating the true form of data which contain some very inaccurate observations. In this paper the measure of goodness of fit is taken to be thel1 norm of the errors. A Levenberg-Marquardt method is proposed, and the main objective is to take full advantage of the structure of the subproblems so that they can be solved efficiently.