Trends and Applications in Constructive Approximation

We provide closed form solutions, in an asymptotic sense, to the problem of best choice of the Poisson parameter in Poisson approximation of Poisson mixtures, with respect to the Kolmogorov and the Fortet-Mourier metrics. To do this, we apply a differential calculus based on different Taylor’s formulae for the Poisson process, either in terms of the forward differences of the function under consideration or in terms of the Charlier polynomials, which allows us to give simple unified proofs. This approach also shows that the zeros of a suitable linear combination of the first and second Charlier polynomials play a key role in determining the leading coefficient of the main term of the approximation.

Of concern is a class of second-order differential operators on the unit interval. The C₀-semigroup generated by them is approximated by iterates of positive linear operators that are introduced here as a modification of Bernstein operators. Finally, the corresponding stochastic differential equations are also investigated, leading, in particular to the evaluation of the asymptotic behaviour of the semigroup.

In 1945, W. Taylor discovered the barycentric formula for evaluating the interpolating polynomial. In 1984, W. Werner has given first consequences of the fact that the formula usually is a rational interpolant. We review some advances since the latter paper in the use of the formula for rational interpolation.

Inequalities of Jackson and Bernstein type are derived for polynomial approximation on simplices with respect to Sobolev norms. Although we do not find simple bases when looking at 120 years of research of orthogonal polynomials on triangles, sharp estimates are obtained from a decomposition into orthogonal subspaces. The formulas reflect the symmetries of simplices, but analogous estimates on rectangles show that we cannot expect rotational invariance of the terms with derivatives. An essential tool are selfadjoint differential operators that have already been used by other authors for the study of various approximation properties.

The methods of proof of regularity for interpolation problems often are dependent on the problem at hand. In case of given pairs of node generating polynomials the method of deriving an ordinary differential equation for the interpolating polynomial or that of exploiting the specific form of the node generator have mainly been used up to now.

Recently another method was used in the case of Pál-type interpolation where ‘only’ one of the node generators is fixed in advance: a ‘general’ method of deriving a companion generator that leads to a regular interpolation problem. Using (0, 2) Pál-type interpolation, it is shown that each of the methods has its merits and for sake of simplicity we will restrict ourselves to the case that the nodes are the zeros of pairs of polynomials of the following form: {p(z)q(z), p(z)} with p, q co-prime and both having simple zeros.

We give here a Kantorovich-type result, which provides suffcient conditions ensuring the convergence of the accelerated iterates.

We extend the Shepard operator by combining it with the Lidstone bivariate operator. We study this combined operator and give some error bounds.

In this paper we continue our earlier research [4] aimed at developing efficient methods of local approximation suitable for the first stage of a spline based two-stage scattered data fitting algorithm. As an improvement to the pure polynomial local approximation method used in [5], a hybrid polynomial/radial basis scheme was considered in [4], where the local knot locations for the RBF terms were selected using a greedy knot insertion algorithm. In this paper standard radial local approximations based on interpolation or least squares are considered and a faster procedure is used for knot selection, significantly reducing the computational cost of the method. Error analysis of the method and numerical results illustrating its performance are given.

We consider the synthesis of neural networks by evolutionary algorithms, which are randomized direct optimization methods inspired by neo-Darwinian evolution theory. Evolutionary algorithms in general as well as special variants for real-valued optimization and for search in the space of graphs are introduced. We put an emphasis on strategy adaptation, a feature of evolutionary methods that allows for the control of the search strategy during the optimization process.

Three recent applications of evolutionary optimization of neural systems are presented: topology optimization of multi-layer neural networks for face detection, weight optimization of recurrent networks for solving reinforcement learning tasks, and hyperparameter tuning of support vector machines.

Here we give an approach which makes use of the scattered data interpolation abilities of Radial Basis Function Networks to handle this problem.

Spline subdivision suggests the necessity of possibly using non nested grids to analyse the convergence of subdivision schemes, and also of changing grids to prove the smoothness of the limit curves. Inspired by this example, we introduce both non nested binary grids and equivalent grids. We give a sufficient condition for convergence, and we show how to use it through changes of grids to guarantee smoothness.

Let s ≥ 1 be an integer. A Gaussian network is a function on \(\mathbb{R}^s \) of the form \(g\left( x \right) = \Sigma _{k = 1}^N a_k \exp \left( { - \left\| {x - x_k } \right\|{}^2} \right)\) The minimal separation among the centers, defined by \(\min _{1 \leqslant j \ne k \leqslant N} \left\| {X_j - X_k } \right\|\) , is an important characteristic of the network that determines the stability of interpolation by Gaussian networks, the degree of approximation by such networks, etc. We prove that if \(g\left( x \right) = \Sigma _{k = 1}^N a_k \exp \left( { - \left\| {x - x_k } \right\|{}^2} \right)\) , the minimal separation of g exceeds 1/m, and log N = O(m²) then for any integer r ≥ 1, any partial derivative Dg of order r of g satisfies \(\left\| {Dg} \right\|_{p,\mathbb{R}^s } \leqslant cm^r \left\| g \right\|_{p,\mathbb{R}^s } \).

Fuzzy controller can be approximated or generalized respectively by replacing the fuzzy sets in the rule conclusions by real numbers or functions. Such a controller is called a TS controller and can be seen as a classical gain scheduling controller. Therefore, TS control can be interpreted as fuzzy and classical control as well. Besides this, for this type of control during the last years there were methods developed, that make it interesting for practical applications.

The objective of this paper is the introduction of TS control and the discussion of its position at the border between fuzzy and classical control, but also the presentation of suitable methods, approaches and fields of application for this controller type.

Let S ⊂ \(\mathbb{R}\) denote a compact set with infinite cardinality and C(S) the set of real continuous functions on S. We investigate the problem of polynomial and orthogonal polynomial bases of C(S).

In case of S ={s0, sl, s2,…} ∪ {σ}, where (sk){skk=0/∞} is a monotone sequence with σ = limk→∞sk, we give a sufficient and necessary condition for the existence of a so-called Lagrange basis. Furthermore, we show that little q-Jacobi polynomials which fulfill a certain boundedness property constitute a basis in case of Sq, = {1,q, q²,…} ∪ {0}, 0<q<1.

Over the past decades, the field of chemical engineering has witnessed an increased interest in unsteady-state processes. Multifunctional, as well as intensified chemical processes, may exhibit instationary behaviour especially when based on periodical operating conditions. Ideally, instationary processes lead to a higher yield and increased selectivities compared to conventional steady-state fixed-bed processes. Typical candidates among these are the reverse-flow-reactor, the chromatographic reactor and the adsorptive reactor. Since the underlying regeneration strategy is nearly always based on cycles — e.g., a reaction cycle is followed by a regeneration cycle and so on — the overall temporal behaviour of such processes eventually develops into cyclic steady-states (after a transient phase). Experiments reveal a slow transient behaviour into the cyclic steady-state. This can also be observed in simulation based on conventional numerical treatment such as the method of lines. In addition to this problem many instationary processes exhibit sharp fronts or even shocks which require stabilisation of the convective terms. In this work we present a method of combining the idea of global discretisation with modern stabilisation techniques of type FEM-FCT and FEM-TVD in order to obtain an efficient, well approximating and robust tool for the general simulation of instationary and in particularly cyclic steady-state processes.

This short note contains some supplementary results concerning the operators introduced by D.H. Mache and the semigroup associated with them. Special attention is paid to the action of the operators and the semigroup on monomials.

Polynomial and spline quasi-interpolants (QIs) are practical and effective approximation operators. Among their remarkable properties, let us cite for example: good shape properties, easy computation and evaluation (no linear system to solve), uniform boundedness independently of the degree (polynomials) or of the partition (splines), good approximation order. We shall emphasize new results on various types of univariate and multivariate polynomial or spline QIs, depending on the nature of coefficient functionals, which can be differential, discrete or integral We shall also present some applications of QIs to numerical methods.

We prove that the weighted error of approximation by generalized Bernstein polynomials introduced in [1] is equivalent to the modulus of smoothness of the function. This result is analogous to a well-known theorem of Ditzian and Ivanov [2] for the classical Bernstein polynomials.

Gradient-based optimization algorithms are the standard methods for adapting the weights of neural networks. The natural gradient gives the steepest descent direction based on a non-uclidean, from a theoretical point of view more appropriate metric in the weight space. While the natural gradient has already proven to be advantageous for online learning, we explore its benefits for batch learning: We empirically compare Rprop (resilient back-propagation), one of the best performing first-order learning algorithms, using the Euclidean and the non-Euclidean metric, respectively. As batch steepest descent on the natural gradient is closely related to Levenberg-Marquardt optimization, we add this method to our comparison.

It turns out that the Rprop algorithm can indeed profit from the natural gradient: the optimization speed measured in terms of weight updates can increase significantly compared to the original version. Rprop based on the non-Euclidean metric shows at least similar performance as Levenberg-Marquardt optimization on the two benchmark problems considered and appears to be slightly more robust. However, in Levenberg-Marquardt optimization and Rprop using the natural gradient computing a weight update requires cubic time and quadratic space. Further, both methods have additional hyperparameters that are difficult to adjust. In contrast, conventional Rprop has linear space and time complexity, and its hyperparameters need no difficult tuning.