On a map from the splines into a positive cone with applications

Résumé Dans cet article nous généralisons quelques caractérisations du meilleur approximant dans l'espace des fonctions splines, qui ont été récemment obtenues, indépendemment par John Rice et Larry Schumaker. Cette généralisation englobe quelques aspects théoriques et appliqués de l'approximation simultanée qui ont été étudiés dans des cadres différents par A. Bacopoulos. De plus, une estimation des vitesses de convergence des meilleures approximations splines est donnée en employant quelques résultats obtenus par M. Marsden et M. Marsden-I. J. Shoenberg.

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This research was partially supported by NRC grant No. A 8108.     

An error analysis for absolute and relative approximation

We consider the vectorial algorithm for finding best polynomial approximationsp P _n to a given functionf C[a, b], with respect to the norm · _s , defined byp – f _s =w ₁ (p – f)+w ₂ (p – f) A bound for the modulus of continuity of the best vectorial approximation operator is given, and using the floating point calculus of J. H. Wilkinson, a bound for the rounding error in the algorithm is derived. For givenf, these estimates provide an indication of the conditioning of the problem, an estimate of the obtainable accuracy, and a practical method for terminating the iteration.This paper was supported in part by the Canadian NCR A-8108, FCAC 74-09 and G.E.T.M.A.Part of this research was done during the first-named author's visit to theB! Chair of Applied Mathematics, University of Athens, Spring term, 1975.     

Computation of Best Simultaneous Approximations with a Singularity

Simultaneous approximation errors are generally discontinuous when the function to be approximated contains a zero in its domain of definition. In this article we indicate how the presence of such a zero (or, equivalently, the resulting singularity in the error expression) affects the computational schemata for finding all the best approximations. In particular, we develop an algorithm and show that its convergence rate is “best possible expected” in the sense that it is quadratic, as in the case for continuous errors. Numerical examples are provided.     

Mixed Frank–Wolfe Penalty Method with Applications to Nonconvex Optimal Control Problems

We consider a general optimization problem which is an abstract formulation of a broad class of state-constrained optimal control problems in relaxed form. We describe a generalized mixed Frank–Wolfe penalty method for solving the problem and prove that, under appropriate assumptions, accumulation points of sequences constructed by this method satisfy the necessary conditions for optimality. The method is then applied to relaxed optimal control problems involving lumped as well as distributed parameter systems. Numerical examples are given.     

Approximation of relaxed nonlinear parabolic optimal control problems

We consider a relaxed optimal control problem for systems defined by nonlinear parabolic partial differential equations with distributed control. The problem is completely discretized by using a finite-element approximation scheme with piecewise linear states and piecewise constant controls. Existence of optimal controls and necessary conditions for optimality are derived for both the continuous and the discrete problem. We then prove that accumulation points of sequences of discrete optimal [resp. extremal] controls are optimal [resp. extremal] for the continuous problem.     

Discrete approximation of relaxed optimal control problems

We consider a general nonlinear optimal control problem for systems governed by ordinary differential equations with terminal state constraints. No convexity assumptions are made. The problem, in its so-called relaxed form, is discretized and necessary conditions for discrete relaxed optimality are derived. We then prove that discrete optimality [resp., extremality] in the limit carries over to continuous optimality [resp., extremality]. Finally, we prove that limits of sequences of Gamkrelidze discrete relaxed controls can be approximated by classical controls.     

On convex vectorial optimization in linear spaces

We give a new method of scalarization for convex vectorial optimization problems, with applications to best vectorial approximation and to scalar problems of optimization and best approximation.This research was supported in part by NCR A-8108, FCAC 74-09, and GETMA. The results of this paper have been obtained during the second author's visit, from May to September 1974, to the Département d'Informatique, Université de Montréal. The authors thank Dr. G. Godini for valuable remarks which simplified the original proof of the main result of this paper.     

On approximation by sums of Chebyshev norms

A theory of sums of Chebyshev approximations is useful for the problem of simultaneous minimization of the absolute and relative errors of an approximation. In this paper some of the important properties of the Chebyshev alternation theory are studied from the point of view of extending them to sums of Chebyshev norms. Both positive and negative results are obtained. Specifically, it is shown that the sum of Chebyshev approximations with different weight functions is not a Chebyshev approximation.