Smolyak's Construction of Cubature Formulas of Arbitrary Trigonometric Degree

Abstract.We study cubature formulas for d-dimensional integrals with a high trigonometric degree. To obtain a trigonometric degree $\ell$ in dimension d, we need about $d^\ell/\ell !$ function values if d is large. Only a small number of arithmetical operations is needed to construct the cubature formulas using Smolyak's technique. We also compare different methods to obtain formulas with high trigonometric degree.

[1]  Franz-Jürgen Delvos,et al.  Multivariate Boolean Trapezoidal Rules , 1994 .

[2]  M. V. Noskov.  Cubature formulae for the approximate integration of functions of three variables , 1990 .

[3]  Alan Genz,et al.  Fully symmetric interpolatory rules for multiple integrals , 1986 .

[4]  S. B. Stechkin.  Approximation of periodic functions , 1974 .

[5]  N. Temirgaliev.  APPLICATION OF DIVISOR THEORY TO THE NUMERICAL INTEGRATION OF PERIODIC FUNCTIONS OF SEVERAL VARIABLES , 1991 .

[6]  K. Ritter,et al.  Simple Cubature Formulas with High Polynomial Exactness , 1999 .

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

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

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

[10]  K. Ritter,et al.  The Curse of Dimension and a Universal Method For Numerical Integration , 1997 .

[11]  Henryk Wozniakowski,et al.  Explicit Cost Bounds of Algorithms for Multivariate Tensor Product Problems , 1995, J. Complex..

[12]  Vladimir Temlyakov,et al.  ON A WAY OF OBTAINING LOWER ESTIMATES FOR THE ERRORS OF QUADRATURE FORMULAS , 1992 .

[13]  V. N. Temli︠a︡kov.  Approximation of periodic functions , 1993 .

[14]  I. Sloan.  Lattice Methods for Multiple Integration , 1994 .

[15]  Karin Frank,et al.  Computing Discrepancies Related to Spaces of Smooth Periodic Functions , 1998 .

[16]  T. Rella.  Lehrbuch der Kombinatorik , 1929 .

[17]  Ian H. Sloan,et al.  Cubature Rules of Prescribed Merit , 1997 .

[18]  K. Ritter,et al.  High dimensional integration of smooth functions over cubes , 1996 .

[19]  Ronald Cools,et al.  Minimal cubature formulae of trigonometric degree , 1996, Math. Comput..