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Constructing cubature formulae: the science behind the art

  • Ronald Cools (a1)
Abstract

In this paper we present a general, theoretical foundation for the construction of cubature formulae to approximate multivariate integrals. The focus is on cubature formulae that are exact for certain vector spaces of polynomials. Our main quality criteria are the algebraic and trigonometric degrees. The constructions using ideal theory and invariant theory are outlined. The known lower bounds for the number of points are surveyed and characterizations of minimal cubature formulae are given. We include references to all known minimal cubature formulae. Finally, some methods to construct cubature formulae illustrate the previously introduced concepts and theorems.

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Appell, P. (1890), ‘Sur une classe de polynômes à deux variables et le calcul approché des intégrales double’, Ann. Fac. Sci. Univ. Toulouse 4, H1–H20.
Appell, P. and Kampé de Fériet, J. (1926), Fonctions Hypergéométriques et Hypersphériques – Polynomes d'Hermite, Gauthier-Villars, Paris.
Becker, T. (1987), Konstruktion von interpolatorischen Kubaturformeln mit Anwendungen in der Finit-Element-Methode, PhD thesis, Technische Hochschule Darmstadt.
Beckers, M. and Cools, R. (1993), A relation between cubature formulae of trigonometric degree and lattice rules, in Numerical Integration IV (Brass, H. and Hämmerlin, G., eds), Birkhäuser, Basel, pp. 1324.
Beckers, M. and Haegemans, A. (1991), ‘The construction of three-dimensional invariant cubature formulae’, J. Comput. Appl. Math. 35, 109118.
Berens, H. and Schmid, H. J. (1992), On the number of nodes of odd degree cubature formulae for integrals with Jacobi weights on a simplex, in Numerical Integration – Recent Developments, Software and Applications (Espelid, T. and Genz, A., eds), Vol. 357 of NATO ASI Series C: Math. and Phys. Sciences, Kluwer, Dordrecht, pp. 3744.
Berens, H., Schmid, H. J. and Xu, Y. (1995), ‘Multivariate Gaussian cubature formulae’, Arch. Math. 64, 2632.
Buchberger, B. (1985), Gröbner bases: An algorithmic method in polynomial ideal theory, in Progress, Directions and Open Problems in Multidimensional Systems Theory (Bose, N., ed.), Reidel, Dordrecht, pp. 184232.
Cools, R. (1989), The construction of cubature formulae using invariant theory and ideal theory, PhD thesis, Katholieke Universiteit Leuven.
Cools, R. (1992), A survey of methods for constructing cubature formulae, in Numerical Integration – Recent Developments, Software and Applications (Espelid, T. and Genz, A., eds), Vol. 357 of NATO ASI Series C: Math. and Phys. Sciences, Kluwer, Dordrecht, pp. 124.
Cools, R. and Haegemans, A. (1987 a), ‘Automatic computation of knots and weights of cubature formulae for circular symmetric planar regions’, J. Comput. Appl. Math. 20, 153158.
Cools, R. and Haegemans, A. (1987 b), ‘Construction of fully symmetric cubature formulae of degree 4K – 3 for fully symmetric planar regions’, J. Comput. Appl. Math. 17, 173180.
Cools, R. and Haegemans, A. (1987 c), Construction of minimal cubature formulae for the square and the triangle using invariant theory, Report TW 96, Dept. of Computer Science, Katholieke Universiteit Leuven.
Cools, R. and Haegemans, A. (1988 a), ‘Another step forward in searching for cubature formulae with a minimal number of knots for the square’, Computing 40, 139146.
Cools, R. and Haegemans, A. (1988 b), Construction of symmetric cubature formulae with the number of knots (almost) equal to Möller's lower bound, in Numerical Integration III (Brass, H. and Hämmerlin, G., eds), Birkhäuser, Basel, pp. 2536.
Cools, R. and Haegemans, A. (1988 c), ‘Why do so many cubature formulae have so many positive weights?’, BIT 28, 792802.
Cools, R. and Rabinowitz, P. (1993), ‘Monomial cubature rules since ’Stroud‘: A compilation’, J. Comput. Appl. Math. 48, 309326.
Cools, R. and Reztsov, A. (1997), ‘Different quality indexes for lattice rules’, J. Complexity. To appear.
Cools, R. and Schmid, H. J. (1989), ‘Minimal cubature formulae of degree 2k – 1 for two classical functional’, Computing 43, 141157.
Cools, R. and Schmid, H. J. (1993), A new lower bound for the number of nodes in cubature formulae of degree 4n + 1 for some circularly symmetric integrals, in Numerical Integration IV (Brass, H. and Hämmerlin, G., eds), Birkhäuser, Basel, pp. 5766.
Cools, R. and Sloan, I. H. (1996), ‘Minimal cubature formulae of trigonometric degree’, Math. Comp. 65, 15831600.
Davis, P. J. (1967), ‘A construction of nonnegative approximate quadratures’, Math. Comp. 21, 578582.
Davis, P. J. and Rabinowitz, P. (1984), Methods of Numerical Integration, Academic, London.
Davis, P. J., Rabinowitz, P. and Cools, R. (199x), Methods of Numerical Integration. Work in progress.
de Doncker, E. (1979), ‘New Euler-Maclaurin expansions and their application to quadrature over the s-dimensional simplex’, Math. Comp. 33, 10031018.
Duffy, M. G. (1982), ‘Quadrature over a pyramid or cube of integrands with a singularity at a vertex’, SIAM J. Numer. Anal. 19, 12601262.
Edwards, H. M. (1980), ‘The genesis of ideal theory’, Arch. Hist. Exact Sci. 23, 321378.
Engels, H. (1980), Numerical Quadrature and Cubature, Academic, London.
Fisher, C. S. (1967), ‘The death of a mathematical theory: a study in the sociology of knowledge’, Arch. Hist. Exact Sci. 3, 136159.
Flatto, L. (1978), ‘Invariants of finite reflection groups’, Enseign. Math. 24, 237292.
Fritsch, F. N. (1970), ‘On the existence of regions with minimal third degree integration formulas’, Math. Comp. 24, 855861.
Frolov, K. K. (1977), ‘On the connection between quadrature formulas and sublattices of the lattice of integral vectors’, Soviet Math. Dokl. 18, 3741.
Gatermann, K. (1988), ‘The construction of symmetric cubature formulas for the square and the triangle’, Computing 40, 229240.
Gatermann, K. (1992), Linear representations of finite groups and the ideal theoretical construction of G-invariant cubature formulas, in Numerical Integration – Recent Developments, Software and Applications (Espelid, T. and Genz, A., ), Vol. 357 of NATO ASI Series C: Math. and Phys. Sciences, Kluwer, Dordrecht, pp. 2535.
Gebauer, R. and Möller, H. M. (1988), ‘On an installation of Buchberger's algorithm’, J. Symb. Computation 6, 275286.
Gout, J. L. and Guessab, A. (1986), ‘Sur les formules de quadrature numérique à nombre minimal de noeuds d'intégration’, Numer. Math. 49, 439455.
Gröbner, W. (1949), Moderne Algebraische Geometrie, Springer, Wien.
Grundmann, A. and Möller, H. M. (1978), ‘Invariant integration formulas for the n-simplex by combinatorial methods’, SIAM J. Numer. Anal. 15, 282290.
Guessab, A. (1986), ‘Cubature formulae which are exact on spaces P, intermediate between Pk and QkNumer. Math. 49, 561576.
Haegemans, A. (1982), Construction of known and new cubature formulas of degree five for three-dimensional symmetric regions, using orthogonal polynomials, in Numerical Integration, Birkhäuser, Basel, pp. 119127.
Haegemans, A. and Piessens, R. (1976), ‘Construction of cubature formulas of degree eleven for symmetric planar regions, using orthogonal polynomials’, Numer. Math. 25, 139148.
Haegemans, A. and Piessens, R. (1977), ‘Construction of cubature formulas of degree seven and nine symmetric planar regions, using orthogonal polynomials’, SIAM J. Numer. Anal. 14, 492508.
Hilbert, D. (1890), ‘Über die Theorie der algebraischen Formen’, Math. Ann. 36, 473534.
Hillion, P. (1977), ‘Numerical integration on a triangle’, Internat. J. Numer. Methods Engrg. 11, 797815.
Jackson, D. (1936), ‘Formal properties of orthogonal polynomials in two variables’, Duke Math. J. 2, 423434.
Keast, P. and Lyness, J. N. (1979), ‘On the structure of fully symmetric multidimensional quadrature rules’, SIAM. J. Numer. Anal. 16, 1129.
Kepler, J. (1615), Nova stereometria doliorum vinariorum, in primis Austriaci, figuræ omnium aptissimæ, Authore Ioanne Kepplero, imp. Cæs. Matthiæ I. ejusq; fidd. Ordd. Austriæ supra Anasum Mathematico, Lincii, Anno MDCXV.
Konjaev, S. I. (1977), ‘Ninth-order quadrature formulas invariant with respect to the icosahedral group’, Soviet Math. Dokl. 18, 497501.
Korobov, N. M. (1959), ‘On approximate calculation of multiple integrals’, Dokl. Akad. Nauk SSSR 124, 12071210. Russian.
Lebedev, V. I. (1976), ‘Quadrature on a sphere’, USSR Comput. Math. and Math. Phys. 16, 1024.
Lebedev, V. I. (1995), ‘A quadrature formula for the sphere of 59th algebraic order of accuracy’, Dokl. Math. 50, 283286.
Lebedev, V. I. and Skorokhodov, A. L. (1992), ‘Quadrature formulas of orders 41, 47, and 53 for the sphere’, Dokl. Math. 45, 587592.
Li, T. Y. (1997), Numerical solution of multivariate polynomial systems by homotopy continuation methods, in Acta Numerica, Vol. 6, Cambridge University Press, pp. 399436.
Lyness, J. N. (1976), ‘An error functional expansion for N-dimensional quadrature with an integrand function singular at a point’, Math. Comp. 30, 123.
Lyness, J. N. (1992), On handling singularities in finite elements, in Numerical Integration – Recent Developments, Software and Applications (Espelid, T. and Genz, A., eds), Vol. 357 of NATO ASI Series C: Math. and Phys. Sciences, Kluwer, Dordrecht, pp. 219233.
Lyness, J. N. and Cools, R. (1994), A survey of numerical cubature over triangles, in Mathematics of Computation 1943–1993: A Half-Century of Computational Mathematics (Gautschi, W., ed.), Vol. 48 of Proceedings of Symposia in Applied Mathematics, AMS, Providence, RI, pp. 127150.
Lyness, J. N. and de Doncker, E. (1993), ‘Quadrature error expansions, II. The full corner singularity’, Numer. Math. 64, 355370.
Lyness, J. N. and de Doncker-Kapenga, E. (1987), ‘On quadrature error expansions, part I’, J. Comput. Appl. Math. 17, 131149.
Lyness, J. N. and Jespersen, D. (1975), ‘Moderate degree symmetric quadrature rules for the triangle’, J. Inst. Math. Appl. 15, 1932.
Lyness, J. N. and McHugh, B. J. J. (1970), ‘On the remainder term in the N-dimensional Euler-Maclaurin expansion’, Numer. Math. 15, 333344.
Lyness, J. N. and Monegato, G. (1980), ‘Quadrature error functional expansion for the simplex when the integrand has singularities at vertices’, Math. Comp. 34, 213225.
Lyness, J. N. and Puri, K. K. (1973), ‘The Euler-Maclaurin expansion for the simplex’, Math. Comp. 27, 273293.
Maeztu, J. I. and Sainz de la Maza, E. (1995), ‘Consistent structures of invariant quadrature rules for the n-simplex’, Math. Comp. 64, 11711192.
Mantel, F. and Rabinowitz, P. (1977), ‘The application of integer programming to the computation of fully symmetric integration formulas in two and three dimensions’, SIAM J. Numer. Anal. 14, 391425.
Maxwell, J. C. (1877), ‘On approximate multiple integration between limits of summation’, Proc. Cambridge Philos. Soc. 3, 3947.
Möller, H. M. (1973), Polynomideale und Kubaturformeln, PhD thesis, Universität Dortmund.
Möller, H. M. (1976), ‘Kubaturformeln mit minimaler Knotenzahl’, Numer. Math. 25, 185200.
M, H. M.öller (1979), Lower bounds for the number of nodes in cubature formulae, in Numerische Integration, Vol. 45 of ISNM, Birkhäuser, Basel, pp. 221230.
Möller, H. M. (1987), On the construction of cubature formulae with few nodes using Gröbner bases, in Numerical Integration (Keast, P. and Fairweather, G., eds), Reidel, Dordrecht, pp. 177192.
Möller, H. M. and Mora, F. (1986), ‘New constructive methods in classical ideal theory’, J. Algebra 100, 138178.
Morrow, C. R. and Patterson, T. N. L. (1978), ‘Construction of algebraic cubature rules using polynomial ideal theory’, SIAM J. Numer. Anal. 15, 953976.
Mysovskikh, I. P. (1966), ‘A proof of minimality of the number of nodes of a cubature formula for a hypersphere’, Zh. Vychisl. Mat. Mat. Fiz. 6, 621630. Russian. Published as I. P. Mysovskih 1966.
Mysovskikh, I. P. (1968), ‘On the construction of cubature formulas with fewest nodes’, Soviet Math. Dokl. 9, 277280.
Mysovskikh, I. P. (1975), ‘On Chakalov's theorem’, USSR Comput. Math. and Math. Phys. 15, 221227.
Mysovskikh, I. P. (1977), ‘On the evaluation of integrals over the surface of a sphere’, Soviet Math. Dokl. 18, 925929. Published as I. P. Mysovskih 1977.
Mysovskikh, I. P. (1980), The approximation of multiple integrals by using interpolatory cubature formulae, in Quantitative Approximation (Vore, R. D. and Scherer, K., eds), Academic, New York, pp. 217243.
Mysovskikh, I. P. (1981), Interpolatory Cubature Formulas, Izdat. ‘Nauka’, Moscow-Leningrad. Russian. See I. P. Mysovskikh 1992.
Mysovskikh, I. P. (1988), ‘Cubature formulas that are exact for trigonometric polynomials’, Metody Vychisl. 15, 719. Russian.
Mysovskikh, I. P. (1990), On the construction of cubature formulas that are exact for trigonometric polynomials, in Numerical Analysis and Mathematical Modelling (Wakulicz, A., ed.), Vol. 24 of Banach Center Publications, PWN – Polish Scientific Publishers, Warsaw, pp. 2938. Russian.
Mysovskikh, I. P. (1992), Interpolatorische Kubaturformeln, Bericht Nr. 74, Institut für Geometrie und Praktische Mathematik der RWTH Aachen. Translated from the Russian by Dietrich, I. and Engels, H.. Published as J. P. Mysovskih 1992.
Mysovskikh, I. P. and Ja Černicina, V. (1971), ‘The answer to a question of Radon’, Soviet Math. Dokl. 12, 852854. Published as I. P. Mysovskih and V. Ja Černicina 1971.
Niederreiter, H. (1992), Random Number Generation and Quasi-Monte Carlo Methods, Vol. 63 of CBMS-NSF regional conference series in applied mathematics, SIAM, Philadelphia.
Noskov, M. V. (1988 a), ‘Cubature formulae for the approximate integration of functions of three variables’, USSR Comput. Math. and Math. Phys. 28, 200202.
Noskov, M. V. (1988 b), ‘Formulas for the approximate integration of periodic functions’, Metody Vychisl. 15, 1922. Russian.
Piessens, R. and Haegemans, A. (1975), ‘Cubature formulas of degree nine for symmetric planar regions’, Math. Comp. 29, 810815.
Rabinowitz, P. and Richter, N. (1969), ‘Perfectly symmetric two-dimensional integration formulas with minimal number of points’, Math. Comp. 23, 765799.
Radon, J. (1948), ‘Zur mechanischen Kubatur’, Monatsh. Math. 52, 286300.
Rasputin, G. G. (1983 a), ‘On the construction of cubature formulas containing prespecified knots’, Metody Vychisl. 13, 122128. Russian.
Rasputin, G. G. (1983 b), ‘On the question of numerical characteristics for orthogonal polynomials of two variables’, Metody Vychisl. 13, 145154. Russian.
Rasputin, G. G. (1986), ‘Construction of cubature formulas containing preassigned nodes’, Soviet Math. (Iz. VUZ) 30, 5867.
Richardson, L. F. (1927), ‘The deferred approach to the limit’, Philos. Trans. Royal Soc. London 226, 261299.
Richtmyer, R. D. (1952), The evaluation of definite integrals, and a quasi-Monte-Carlo method based on the properties of algebraic numbers, Report LA-1342, Los Alamos Scientific Laboratory.
Schmid, H. J. (1978), ‘On cubature formulae with a minimal number of knots’, Numer. Math. 31, 281297.
Schmid, H. J. (1980 a), ‘Interpolatorische Kubaturformeln und reelle Ideale’, Math. Z. 170, 267282.
Schmid, H. J. (1980 b), Interpolatory cubature formulae and real ideals, in Quantitative Approximation (Vore, R. D. and Scherer, K., eds), Academic, New York, pp. 245254.
Schmid, H. J. (1983), Interpolatorische Kubaturformeln, Vol. CCXX of Dissertationes Math., Polish Scientific Publishers, Warsaw.
Schmid, H. J. (1995), ‘Two-dimensional minimal cubature formulas and matrix equations’, SIAM J. Matrix Anal. 16(3), 898921.
Schmid, H. J. and Xu, Y. (1994), ‘On bivariate Gaussian cubature formulae’, Proc. Amer. Math. Soc. 122, 833842.
Sloan, I. H. and Joe, S. (1994), Lattice Methods for Multiple Integration, Oxford University Press.
Sloan, I. H. and Kachoyan, P. J. (1987), ‘Lattice mathods for multiple integration: theory, error analysis and examples’, SIAM J. Numer. Anal. 24, 116128.
Sobolev, S. L. (1962), ‘The formulas of mechanical cubature on the surface of a sphere’, Sibirsk. Mat. Zž. 3, 769796. Russian.
Stroud, A. H. (1960), ‘Quadrature methods for functions of more than one variable’, New York Acad. Sci. 86, 776791.
Stroud, A. H. (1971), Approximate calculation of multiple integrals, Prentice-Hall, Englewood Cliffs, NJ.
Sturmfels, B. (1996), Gröbner Bases and Convex Polytopes, Vol. 8 of University Lecture Series, AMS, Providence, RI.
Taylor, M. (1995), ‘Cubature for the sphere and the discrete spherical harmonic transform’, SIAM J. Numer. Anal. 32(2), 667670.
Tchakaloff, V. (1957), ‘Formules de cubatures mécaniques à coefficients non négatifs’, Bull. des Sciences Math., 2e série 81, 123134.
Verlinden, P. and Cools, R. (1992), ‘On cubature formulae of degree 4k + 1 attaining Möller's lower bound for integrals with circular symmetry’, Numer. Math. 61, 395407.
Verlinden, P. and Haegemans, A. (1993), ‘An error expansion for cubature with an integrand with homogeneous boundary singularities’, Numer. Math. 65, 383406.
Wei, S.ß (1991), Über Kubaturformeln vom Grad 2k – 2, Master's thesis, Universität Erlangen.
Wissman, J. W. and Becker, T. (1986), ‘Partially symmetric cubature formulas for even degrees of exactness’, SIAM J. Numer. Anal. 23, 676685.
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