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

  • Claudia Bittante (a1), Stefano De Marchi (a1) and Giacomo Elefante (a2)
Abstract
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.

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*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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[2] M. Briani , A. Sommariva , M. Vianello , Computing Fekete and Lebesgue points: simplex, square, disk, J. Comput. Appl. Math. 236 (2012), no. 9, 24772486.

[3] L. Bos , J.-P. Calvi , N. Levenberg , A. Sommariva and M. Vianello , Geometric weakly admissible meshes, discrete least squares approximation and approximate Fekete points, Math. Comp. 80(275) (2011), 16231638.

[6] B. Bojanov and G. Petrova , Numerical integration over a disc. A new Gaussian quadrature formula, Numer. Math. 80 (1998), 3959.

[10] J. P. Calvi and N. Levenberg , Uniform approximation by discrete least squares polynomials, J. Approx. Theory 152 (2008), 82100.

[11] A. Civril and M. Magdon-Ismail , On selecting a maximum volume sub-matrix of a matrix and related problems, Theoretical Computer Science 410 (2009), 48014811.

[15] J. Dick and F. Pillichshammer , Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration, Cambridge University Press, Cambridge, 2010.

[16] M. Drmota and R. F. Tichy , Sequences, discrepancies and applications, Lecture Notes in Math., 1651, Springer, Berlin, 1997.

[18] A. Kroó , On optimal polynomial meshes, J. Approx. Theory 163 (2011), 11071124.

[20] C. Lemieux , Monte Carlo and Quasi-Monte Carlo Sampling, Springer 2009.

[22] W. J. Morokoff and R. E. Caflisch , Quasi-random sequences and their discrepancies, SIAM J. Sci. Comput. 15 (1994), no. 6, 12511279.

[24] H. G. Niederreiter , Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc. 84 (1978), no. 6, 9571041.

[27] A. Sommariva , M. Vianello , Product Gauss cubature over polygons based on Green's integration formula, BIT Num. Mathematics 47 (2007), 441453.

[32] A. Sommariva and M. Vianello , Gauss-Green cubature and moment computation over arbitrary geometries, J. Comput. Appl. Math. 231 (2009), 886896.

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Numerical Mathematics: Theory, Methods and Applications
  • ISSN: 1004-8979
  • EISSN: 2079-7338
  • URL: /core/journals/numerical-mathematics-theory-methods-and-applications
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