Nichtlineare Tschebyscheff Approximation mit Nebenbedingungen: Anwendungen

Summary In this paper we give some applications resulting from the theory of nonlinear approximation under side-conditions developed in a preceding paper [3]. Particularly a necessary condition for best approximation in terms of a generalized alternant is discussed, the approximating functions having restricted ranges. As special cases of this kind we deduce theorems for one-sided approximation and for approximations by positive functions. We conclude with a result in the theory of nonlinear programming.

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Zweiter Teil einer gekürzten Fassung der Dissertation des Verfassers [3].     

Variationsprobleme in Orliczräumen und Splines

Two convex variational problems in Orlicz spaces are considered. We give sufficient conditions for existence and uniqueness of solutions and present several characterizations of these solutions. We show that the best interpolation property of certain nonlinear classes of spline functions is a special case of our results. As an application we consider the problem of Hermite-Birkhoff-interpolation with linear inequality constraints and illustrate the results by a simple example.

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Diese Arbeit ist eine gekürzte Fassung des zweiten Teils der Dissertation des Verfassers (Fakultät für Mathematik der Ludwig-Maximilians-Universität).     

A priori Fehlerschranken für sukzessiv abgespaltene Polynomnullstellen

Summary The errors of the approximations to the zeros of a polynomial are analyzed, supposing these approximations have been found successively using factorization of the polynomial. We deduce an error bound depending only of the degree of the polynomial and the values of the reduced polynomials at the approximation being factored. The same method may be used to calculate error bounds in the case where round-off is involved.

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Für zahlreiche Diskussionen und Verbesserungen bin ich den Herren Prof. Dr. P. Henrici und dipl. Math. Rolf Jeltsch zu Dank verpflichtet.     

Approximating Networks and Extended Ritz Method for the Solution of Functional Optimization Problems

Functional optimization problems can be solved analytically only if special assumptions are verified; otherwise, approximations are needed. The approximate method that we propose is based on two steps. First, the decision functions are constrained to take on the structure of linear combinations of basis functions containing free parameters to be optimized (hence, this step can be considered as an extension to the Ritz method, for which fixed basis functions are used). Then, the functional optimization problem can be approximated by nonlinear programming problems. Linear combinations of basis functions are called approximating networks when they benefit from suitable density properties. We term such networks nonlinear (linear) approximating networks if their basis functions contain (do not contain) free parameters. For certain classes of d-variable functions to be approximated, nonlinear approximating networks may require a number of parameters increasing moderately with d, whereas linear approximating networks may be ruled out by the curse of dimensionality. Since the cost functions of the resulting nonlinear programming problems include complex averaging operations, we minimize such functions by stochastic approximation algorithms. As important special cases, we consider stochastic optimal control and estimation problems. Numerical examples show the effectiveness of the method in solving optimization problems stated in high-dimensional settings, involving for instance several tens of state variables.     

Analysis of Tensor Approximation Schemes for Continuous Functions

In this article, we analyze tensor approximation schemes for continuous functions. We assume that the function to be approximated lies in an isotropic Sobolev space and discuss the cost when approximating this function in the continuous analogue of the Tucker tensor format or of the tensor train format. We especially show that the cost of both approximations are dimension-robust when the Sobolev space under consideration provides appropriate dimension weights.

     

Characterization of strict approximations in subspaces of spline functions

The problem of approximating a given function by spline functions with fixed knots is discussed. Strict approximations which are particular unique best Chebyshev approximations are considered. The chief purpose is to develop a characterization theorem for these strict approximations.     

An algorithm for the computation of strict approximations in subspaces of spline functions

The problem of approximating a given function by spline functions with fixed knots is discussed. Strict approximations which are particular unique best Chebyshev approximations are considered. The chief purpose is to develop a characterization theorem for these strict approximations.     

A high order projection method for nonlinear two point boundary value problems

Summary Approximations to the solutions of a general class of 2m-th order nonlinear boundary value problems are developed in spaces of polynomial splines of degree 2m+1 by requiring the residual to be orthogonal to a class of polynomial splines of degree 2m–1 over the same mesh. Conditions are given for existence and uniqueness of approximations along with theoretical error rates. In some cases these rates are shown to be of the same order as the best approximation to the solution over the approximating spline spaces. Some computational notes and the results of numerical experiments are also given.     

Complexity of Neural Network Approximation with Limited Information: A Worst Case Approach

In neural network theory the complexity of constructing networks to approximate input-output functions is of interest. We study this in the more general context of approximating elements f of a normed space F using partial information about f. We assume information about f and the size of the network are limited, as is typical in radial basis function networks. We show complexity can be essentially split into two independent parts, information ε-complexity and neural ε-complexity. We use a worst case setting, and integrate elements of information-based complexity and nonlinear approximation. We consider deterministic and/or randomized approximations using information possibly corrupted by noise. The results are illustrated by examples including approximation by piecewise polynomial neural networks.     

On the best approximations and rate of convergence of decompositions in the root vectors of an operator

We establish upper bounds of the best approximations of elements of a Banach space B by the root vectors of an operator A that acts in B. The corresponding estimates of the best approximations are expressed in terms of a K-functional associated with the operator A. For the operator of differentiation with periodic boundary conditions, these estimates coincide with the classical Jackson inequalities for the best approximations of functions by trigonometric polynomials. In terms of K-functionals, we also prove the abstract Dini-Lipschitz criterion of convergence of partial sums of the decomposition of f from B in the root vectors of the operator A to f     

Nonlinear Chebyshev fitting from the solution of ordinary differential equations

The nonlinear Chebyshev approximation of real-valued data is considered where the approximating functions are generated from the solution of parameter dependent initial value problems in ordinary differential equations. A theory for this process applied to the approximation of continuous functions on a continuum is developed by the authors in [17]. This is briefly described and extended to approximation on a discrete set. A much simplified proof of the local Haar condition is given. Some algorithmic details are described along with numerical examples of best approximations computed by the Exchange algorithm and a Gauss-Newton type method.     

Über die Extremaleigenschaft nichtlinearer interpolierender Splines

Summary In this paper we consider interpolation problems for nonlinear classes of spline functions. It is the aim to establish an extremal property for the solution of such a problem. Using this property, sufficient conditions for the existence of a solution of the interpolation problem are given. Some numerical examples illustrate the results.

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Die vorliegende Arbeit ist eine gekürzte Fassung des ersten Teils der Dissertation des Verfassers (Fakultät für Mathematik der Ludwig-Maximilians-Universität).     

Approximation of classes of periodic functions with small smoothness

We prove that the approximations of classes of periodic functions with small smoothness in the metrics of the spaces C and L by different linear summation methods for Fourier series are asymptotically equal to the least upper bounds of the best approximations of these classes by trigonometric polynomials of degree not higher than (n - 1). We establish that the Fejér method is asymptotically the best among all positive linear approximation methods for these classes.     

Approximations in spaces of locally integrable functions

We study approximations of functions from the sets $\hat L_\beta ^\psi \mathfrak{N}$ , which are determined by convolutions of the following form: $$f\left( x \right) = A_0 + \int\limits_{ - \infty }^\infty {\varphi \left( {x + t} \right)\hat \psi _\beta \left( t \right)dt, \varphi \in \mathfrak{N}, \hat \psi _\beta \in L\left( { - \infty ,\infty } \right),} $$ where η is a fixed subset of functions with locally integrablepth powers (p≥1). As approximating aggregates, we use the so-called Fourier operators, which are entire functions of exponential type ≤ σ. These functions turn into trigonometric polynomials if the function ?(·) is periodic (in particular, they may be the Fourier sums of the function approximated). The approximations are studied in the spacesL _p determined by local integral norms ∥·∥_-p . Analogs of the Lebesgue and Favard inequalities, wellknown in the periodic case, are obtained and used for finding estimates of the corresponding best approximations which are exact in order. On the basis of these inequalities, we also establish estimates of approximations by Fourier operators, which are exact in order and, in some important cases, exact with respect to the constants of the principal terms of these estimates.     

Characterization of semicontinuity of solutions to abstract optimization problems

Abstract

We develop a theory of best simultaneous approximations for closed downward sets in a conditionally complete lattice Banach space X with a strong unit. We study best simultaneous approximation in X by elements of downward and normal sets, and give necessary and sufficient conditions for any element of best simultaneous approximation by a closed subset of X. We prove that a downward subset of X is strictly downward if and only if each its boundary point is simultaneous Chebyshev.     

Optimal convergence rates for semidiscrete approximations of parabolic problems with nonsmooth boundary data

We consider semidiscrete approximations of parabolic boundary value problems based on an elliptic approximation by J. Nitsche, in which the approximating subspaces are not subject to any boundary conditions. Optimal L_p (L ₂) error estimates are derived for both smooth and nonsmooth boundary data. The approach is

based on semigroup theory combined with the theory of singular integrals.     

Minimum relative error approximations for 1/t

Summary We give explicit solutions to the problem of minimizing the relative error for polynomial approximations to 1/t on arbitrary finite subintervals of (0, ). We give a simple algorithm, using synthetic division, for computing practical representations of the best approximating polynomials. The resulting polynomials also minimize the absolute error in a related functional equation. We show that, for any continuous function with no zeros on the interval of interest, the geometric convergence rates for best absolute error and best relative error approximants must be equal. The approximation polynomials for 1/t are useful for finding suitably precise initial approximations in iterative methods for computing reciprocals on computers.     

Zur rationalen Tschebyscheff Approximation differenzierbarer und analytischer Funktionen

Summary In this paper we investigate Tschebyscheff-Approximations for differentiable and analytical functions by generalized rational functions. We derive necessary and sufficient conditions for the dimension of the set of all best approximations for a given function ƒ to be bounded by a constant independent of ƒ.     

A class of algorithms for large scale nonlinear minimax optimization

Sequential linear programming and sequential quadratic programming based algorithms are often used to solve nonlinear minimax problems. In case of large scale problems, however, these algorithms can be quite tedious, since linear approximations of every nonlinear function are utilized in the mathematical program approximating the original problem (at any iteration). This paper is concerned with algorithms that require, at each iteration, approximations of only a small fraction of the functions. Such methods are thus well suited for large scale problems. Global convergence of this class of algorithms is proven.     

Programming with a special class of nonlinear functionals

Summary In this paper we consider programming problems in which the constraints are linear and the objective function is the product or the quotient of two functions, each function being a homogeneous form of first degree with a constant added to it.With the proper assumptions of concavity or convexity of the homogeneous forms, this nonlinear programming problem is reduced to that of maximization of a concave function over a convex constraint set.

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Zusammenfassung In der vorliegenden Arbeit werden Programme untersucht, bei denen die Nebenbedingungen linear sind und die Zielfunktion als Produkt bzw. Quotient zweier Funktionen darstellbar ist, die bis auf additive Konstanten homogen von 1. Grad sind. Bei geeigneten Konvexitäts- oder Konkavitätsannahmen für diese Funktionen lassen sich solche Programme auf die Maximierung einer konkaven Funktion in einem konvexen Gebiet zurückführen.

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Prepared with the partial support of the C.S.I.R., India.

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Vorgel. v.:J. Nitsche.