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Numerical integration using integrals over hyperplane sections of simplices in a triangulation of a polytope

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Abstract

In this paper, we consider the problem of approximating a definite integral of a given function f when, rather than its values at some points, a number of integrals of f over some hyperplane sections of simplices in a triangulation of a polytope P in \(\mathbb {R}^d\) are only available. We present several new families of “extended” integration formulas, all of which are a weighted sum of integrals over some hyperplane sections of simplices, and which contain in a special case of our result multivariate analogues of the midpoint rule, the trapezoidal rule and the Simpson’s rule. Along with an efficient algorithm for their implementations, several illustrative numerical examples are provided comparing these cubature formulas among themselves. The paper also presents the best possible explicit constants for their approximation errors. We perform numerical tests which allow the comparison of the new cubature formulas. Finally, we will discuss a conjecture suggested by the numerical results.

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Authors and Affiliations

  1. Laboratoire de Mathématiques et de leurs Applications, UMR CNRS 4152, Université de Pau et des Pays de l’Adour, 64000, Pau, France

    Allal Guessab

  2. Novosibirsk State University, 630090, Novosibirsk, Russia

    Boris Semisalov

  3. Institute of Computational Technologies SB RAS, 630090, Novosibirsk, Russia

    Boris Semisalov

Authors
  1. Allal Guessab
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  2. Boris Semisalov
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Correspondence to Allal Guessab.

Additional information

Communicated by Jan Hesthaven.

One of the authors, Boris Semisalov would like to acknowledge Russian Science Foundation (Project No. 17-71-10135) for financial support.

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Guessab, A., Semisalov, B. Numerical integration using integrals over hyperplane sections of simplices in a triangulation of a polytope. Bit Numer Math 58, 613–660 (2018). https://doi.org/10.1007/s10543-018-0703-3

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  • DOI: https://doi.org/10.1007/s10543-018-0703-3

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