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A New Quasi-Monte Carlo Technique Based on Nonnegative Least Squares and Approximate Fekete Points

Part of: Numerical approximation and computational geometry Approximations and expansions Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  17 November 2016

Claudia Bittante ,
Stefano De Marchi  and
Giacomo Elefante
Claudia Bittante*
Affiliation:
Department of Mathematics, University of Padova, Italy
Stefano De Marchi*
Affiliation:
Department of Mathematics, University of Padova, Italy
Giacomo Elefante*
Affiliation:
Department of Mathematics, University of Fribourg, Switzerland
*
*Corresponding author. Email addresses:cbittant@math.unipd.it (C. Bittante), demarchi@math.unipd.it (S. De Marchi), giacomo.elefante@unifr.ch (G. Elefante)
*Corresponding author. Email addresses:cbittant@math.unipd.it (C. Bittante), demarchi@math.unipd.it (S. De Marchi), giacomo.elefante@unifr.ch (G. Elefante)
*Corresponding author. Email addresses:cbittant@math.unipd.it (C. Bittante), demarchi@math.unipd.it (S. De Marchi), giacomo.elefante@unifr.ch (G. Elefante)
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Abstract

The computation of integrals in higher dimensions and on general domains, when no explicit cubature rules are known, can be ”easily” addressed by means of the quasi-Monte Carlo method. The method, simple in its formulation, becomes computationally inefficient when the space dimension is growing and the integration domain is particularly complex. In this paper we present two new approaches to the quasi-Monte Carlo method for cubature based on nonnegative least squares and approximate Fekete points. The main idea is to use less points and especially good points for solving the system of the moments. Good points are here intended as points with good interpolation properties, due to the strict connection between interpolation and cubature. Numerical experiments show that, in average, just a tenth of the points should be used mantaining the same approximation order of the quasi-Monte Carlo method. The method has been satisfactory applied to 2 and 3-dimensional problems on quite complex domains.

Keywords

Cubaturequasi-Monte Carlo methodnonnegative least squaresapproximate Fekete points

MSC classification

Secondary: 11K45: Pseudo-random numbers; Monte Carlo methods 41A45: Approximation by arbitrary linear expressions 65D30: Numerical integration 65D32: Quadrature and cubature formulas
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 9 , Issue 4 , November 2016 , pp. 640 - 663
Copyright
Copyright © Global-Science Press 2016 

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