Open AccessSharp multidimensional numerical integration for strongly convex functions on convex polytopes
Author(s) -
Osama Alabdali,
Allal Guessab
Publication year - 2020
Publication title -
filomat
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 34
eISSN - 2406-0933
pISSN - 0354-5180
DOI - 10.2298/fil2002601a
Subject(s) - mathematics , convexity , polytope , bounded function , quadratic equation , regular polygon , class (philosophy) , characterization (materials science) , numerical integration , function (biology) , convex function , mathematical analysis , combinatorics , geometry , materials science , artificial intelligence , evolutionary biology , computer science , financial economics , economics , biology , nanotechnology
This paper introduces and studies a new class of multidimensional numerical integration, which we call ?strongly positive definite cubature formulas?. We establish, among others, a characterization theorem providing necessary and sufficient conditions for the approximation error (based on such cubature formulas) to be bounded by the approximation error of the quadratic function. This result is derived as a consequence of two characterization results, which are of independent interest, for linear functionals obtained in a more general seeting. Thus, this paper extends some result previously reported in [2, 3] when convexity in the classical sense is only assumed. We also show that the centroidal Voronoi Tesselations provide an efficient way for constructing a class of optimal of cubature formulas. Numerical results for the two-dimensional test functions are given to illustrate the efficiency of our resulting cubature formulas.