Volume 9, Issue 4
A New Quasi-Monte Carlo Technique Based on Nonnegative Least Squares and Approximate Fekete Points

Claudia Bittante, Stefano De Marchi & Giacomo Elefante

Numer. Math. Theor. Meth. Appl., 9 (2016), pp. 640-663.

Published online: 2016-09

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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 to maintain the same approximation order of the quasi-Monte Carlo method. The method has been satisfactorily applied to 2- and 3-dimensional problems on quite complex domains.

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@Article{NMTMA-9-640, author = {}, title = {A New Quasi-Monte Carlo Technique Based on Nonnegative Least Squares and Approximate Fekete Points}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2016}, volume = {9}, number = {4}, pages = {640--663}, 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 to maintain the same approximation order of the quasi-Monte Carlo method. The method has been satisfactorily applied to 2- and 3-dimensional problems on quite complex domains.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2016.m1516}, url = {http://global-sci.org/intro/article_detail/nmtma/12393.html} }
TY - JOUR T1 - A New Quasi-Monte Carlo Technique Based on Nonnegative Least Squares and Approximate Fekete Points JO - Numerical Mathematics: Theory, Methods and Applications VL - 4 SP - 640 EP - 663 PY - 2016 DA - 2016/09 SN - 9 DO - http://doi.org/10.4208/nmtma.2016.m1516 UR - https://global-sci.org/intro/article_detail/nmtma/12393.html KW - AB -

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 to maintain the same approximation order of the quasi-Monte Carlo method. The method has been satisfactorily applied to 2- and 3-dimensional problems on quite complex domains.

Claudia Bittante, Stefano De Marchi & Giacomo Elefante. (2020). A New Quasi-Monte Carlo Technique Based on Nonnegative Least Squares and Approximate Fekete Points. Numerical Mathematics: Theory, Methods and Applications. 9 (4). 640-663. doi:10.4208/nmtma.2016.m1516
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