Springer, New York (1999) Google Scholar Digital Library Conway, J., Sloane, N.: Sphere Packings, Lattices and Groups, 3rd edn.Metrika 33, 337-347 (1986) Google Scholar Cross Ref Böhning, D.: A vertex-exchange-method in $$D$$D-optimal design theory.Inference 11, 57-69 (1985) Google Scholar Cross Ref Böhning, D.: Numerical estimation of a probability measure.Audze, P., Eglais, V.: New approach for planning out of experiments.Armijo, L.: Minimization of functions having Lipschitz continuous first partial derivatives.An algorithm is proposed for the construction of n-point designs. The results suggest that designs minimizing the regularized dispersion for suitable values of the regularization parameter should have good space-filling properties. They are often close to the uniform measure but do not coincide with it. Using recent results and algorithms from experimental design theory, we show how to construct optimal measures numerically. The example of design in the unit ball is considered in details and some analytic results are presented. Using results from potential theory, we investigate properties of optimal measures. We show that the criterion is convex for a certain range of the regularization parameter (depending on space dimension) and give a necessary and sufficient condition characterizing the optimal distribution of design points. We consider a continuous extension of a regularized version of the minimax, or dispersion, criterion widely used in space-filling design for computer experiments and quasi-Monte Carlo methods.
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