No abstract

The objective of this work is to provide a methodology for approximating globally optimal Fekete point configurations. This problem is of interest in numerical mathematics and scientific modeling. Following a brief discussion of the analytical background, Lipschitz global optimization (LOO) is applied to detem1ine-i.e., to numerically approximate-Fekete point configurations. Next to this optimization approach, an alternative strategy by formulating a set of differential-algebraic equations (DAEs) of index 2 will be considered. The steady states of the DAEs coincide with the optima of the function to be minimized. Illustrative numerical results with configurations of up to 150 Fekete points-are presented, to show the viability of both approaches.

For implicit Runge{Kutta methods intended for sti ODEs or DAEs, it is often dicult to embed a local error estimating method which gives realistic error estimates for sti/algebraic components. If the embedded method's stability function is unbounded at z = 1, sti error components are grossly overestimated. In practice some codes \improve" such inadequate error estimates by p remultiplying the estimate by a \lter" matrix which damps or removes the large, sti error components. Although improving computational performance, this technique is somewhat arbitrary and lacks a sound theoretical backing. In this scientic note we resolve this problem by introducing an implicit error estimator. It has the desired properties for sti/algebraic components without invoking articial improvements. The error estimator contains a free parameter which determines the magnitude of the error, and we show h o w this parameter is to be selected on the basis of method properties. The construction principles for the error estimator can be adapted to all implicit Runge{Kutta methods, and a better agreement between actual and estimated errors is achieved, resulting in better performance.

No abstract

No abstract