Constructing Positive Interpolatory Cubature Formulas

Positive interpolatory cubature formulas (CFs) are constructed for quite general integration domains and weight functions. These CFs are exact for general vector spaces of continuous real-valued functions that contain constants. At the same time, the number of data points -- all of which lie inside the domain of integration -- and cubature weights -- all positive -- is less or equal to the dimension of that vector space. The existence of such CFs has been ensured by Tchakaloff in 1957. Yet, to the best of the author's knowledge, this work is the first to provide a procedure to successfully construct them.

[1]  A. Stroud.  Approximate calculation of multiple integrals , 1973 .

[2]  James Clerk Maxwell,et al.  On Approximate Multiple Integration between Limits by Summation , 2011 .

[3]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[4]  I. P. Mysovskikh.  Cubature formulas that are exact for trigonometric polynomials , 1998 .

[5]  R. Cools,et al.  Monomial cubature rules since “Stroud”: a compilation , 1993 .

[6]  M. Mahoney,et al.  History of Mathematics , 1924, Nature.

[7]  I. P. Mysovskikh.  THE APPROXIMATION OF MULTIPLE INTEGRALS BY USING INTERPOLATORY CUBATURE FORMULAE , 1980 .

[8]  E. Hlawka.  Funktionen von beschränkter Variatiou in der Theorie der Gleichverteilung , 1961 .

[9]  Ronald Cools,et al.  Constructing cubature formulae: the science behind the art , 1997, Acta Numerica.

[10]  Daan Huybrechs,et al.  Stable high-order quadrature rules with equidistant points , 2009, J. Comput. Appl. Math..

[11]  Jean-François Richard,et al.  Methods of Numerical Integration , 2000 .

[12]  Jan Glaubitz.  Stable High-Order Cubature Formulas for Experimental Data , 2020, ArXiv.

[13]  Lauwerens Kuipers,et al.  Uniform distribution of sequences , 1974 .

[14]  Nonnegative interpolation formulas for uniformly elliptic equations , 1968 .

[15]  A. W. Wymore,et al.  Numerical Evaluation of Multiple Integrals I , 2010 .

[16]  H. Weyl.  Über die Gleichverteilung von Zahlen mod. Eins , 1916 .

[17]  M. Wayne Wilson.  Necessary and Sufficient Conditions for Equidistant Quadrature Formula , 1970 .

[18]  Jan Glaubitz,et al.  Stable discretisations of high-order discontinuous Galerkin methods on equidistant and scattered points , 2020, ArXiv.

[19]  P. Gruber,et al.  Funktionen von beschränkter Variation in der Theorie der Gleichverteilung , 1990 .

[20]  M. W. Wilson,et al.  Discrete least squares and quadrature formulas , 1970 .

[21]  J. Halton.  On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals , 1960 .

[22]  R. E. Cline,et al.  $l_2 $-Solutions to Underdetermined Linear Systems , 1976 .

[23]  Jan Glaubitz,et al.  Stable High Order Quadrature Rules for Scattered Data and General Weight Functions , 2020, SIAM J. Numer. Anal..

[24]  E. Steinitz.  Bedingt konvergente Reihen und konvexe Systeme. , 1913 .

[25]  Adi Ben-Israel,et al.  Generalized inverses: theory and applications , 1974 .

[26]  Philip J. Davis.  A Construction of Nonnegative Approximate Quadratures , 1967 .

[27]  Philipp Birken,et al.  Numerical Linear Algebra , 2011, Encyclopedia of Parallel Computing.

[28]  W. Gautschi.  Orthogonal Polynomials: Computation and Approximation , 2004 .

[29]  Hermann Engles,et al.  Numerical quadrature and cubature , 1980 .

[30]  Lloyd N. Trefethen,et al.  Cubature, Approximation, and Isotropy in the Hypercube , 2017, SIAM Rev..

[31]  R. Cools.  Monomial cubature rules since “Stroud”: a compilation—part 2 , 1999 .

[32]  Ronald Cools,et al.  An encyclopaedia of cubature formulas , 2003, J. Complex..

[33]  J. Miller.  Numerical Analysis , 1966, Nature.