Numerical Evaluation of Multiple Integrals

This paper is an expository survey of the main methods that have been developed for numerical evaluation of multiple integrals. Among the approaches discussed are: the Monte Carlo method and its generalizations; number-theoretical methods, based essentially on the ideas of Diophantine approximation and equidistribution modulo 1; the functional analysis approach, in which the quadrature error is regarded as a linear functional and one attempts to minimize its norm; and the classical approach of designing formulas to be exact for polynomials of high degree while using as few values of the integrand as possible. Most of the research in this field is quite recent.

[1]  P. Rabinowitz,et al.  New Error Coefficients for Estimating Quadrature Errors for Analytic Functions , 1970 .

[2]  On the sum ∑ ⟨nα⟩−t and numerical integration , 1969 .

[3]  G. Marsaglia Random numbers fall mainly in the planes. , 1968, Proceedings of the National Academy of Sciences of the United States of America.

[4]  V. L. N. Sarma Eberlein measure and mechanical quadrature formulae. I. Basic theory. , 1968 .

[5]  S. Haber A combination of Monte Carlo and classical methods for evaluating multiple integrals , 1968 .

[6]  A. Stroud,et al.  Some Extensions of Integration Formulas , 1968 .

[7]  R. Barnhill An error analysis for numerical multiple integration. III , 1968 .

[8]  Harold Conroy,et al.  Molecular Schrödinger Equation. VIII. A New Method for the Evaluation of Multidimensional Integrals , 1967 .

[9]  Lloyd Rosenberg Bernstein Polynomials and Monte Carlo Integration , 1967 .

[10]  D. I. Golenko,et al.  The Monte Carlo Method. , 1967 .

[11]  N M Korobov SOME PROBLEMS IN THE THEORY OF DIOPHANTINE APPROXIMATION , 1967 .

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

[13]  S. K. Zaremba,et al.  Good lattice points, discrepancy, and numerical integration , 1966 .

[14]  M. J. D. Powell,et al.  Weighted Uniform Sampling — a Monte Carlo Technique for Reducing Variance , 1966 .

[15]  S. Haber A modified Monte-Carlo quadrature. II. , 1966 .

[16]  I. P. Mysovskikh Proof of the minimality of the number of nodes in the cubature formula for a hypersphere , 1966 .

[17]  A. Stroud,et al.  Gaussian quadrature formulas , 1966 .

[18]  J. N. Lyness Integration rules of hypercubic symmetry over a certain spherically symmetric space , 1965 .

[19]  J. N. Lyness Symmetric integration rules for hypercubes. I. Error coefficients , 1965 .

[20]  J. N. Lyness Symmetric Integration Rules for Hypercubes III. Construction of Integration Rules Using Null Rules , 1965 .

[21]  I. J. Schoenberg On Monosplines of Least Deviation and Best Quadrature Formulae , 1965 .

[22]  D. Secrest Best approximate integration fuormulas and best error bounds , 1965 .

[23]  Remarks on a Monte Carlo integration method , 1964 .

[24]  Random quadratures of improved accuracy , 1964 .

[25]  A. D. McLaren,et al.  Optimal numerical integration on a sphere , 1963 .

[26]  S. Sobolev Some new problems in the theory of partial differential equations , 1963 .

[27]  I. F. Sharygin,et al.  A lower estimate for the error of quadrature formulae for certain classes of functions , 1963 .

[28]  On the discrepancy of certain sequences mod. 1 , 1963 .

[29]  J. Franklin Deterministic Simulation of Random Processes , 1963 .

[30]  T. E. Hull,et al.  Random Number Generators , 1962 .

[31]  E. Hlawka Zur angenäherten Berechnung mehrfacher Integrale , 1962 .

[32]  A. Stroud,et al.  Approximate Calculation of Integrals , 1962 .

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

[34]  C. Haselgrove,et al.  A method for numerical integration , 1961 .

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

[36]  J. Hammersley MONTE CARLO METHODS FOR SOLVING MULTIVARIABLE PROBLEMS , 1960 .

[37]  J. Miller Numerical Quadrature Over a Rectangular Domain in Two or More Dimensions , 1960 .

[38]  S. M. Ermakov,et al.  Polynomial Approximations and the Monte-Carlo Method , 1960 .

[39]  W. C. Rheinboldt,et al.  The hypercircle in mathematical physics , 1958 .

[40]  H. Thacher Optimum quadrature formulas in dimensions , 1957 .

[41]  A. Stroud Remarks on the disposition of points in numerical integration formulas. , 1957 .

[42]  William H. Peirce,et al.  Numerical integration over the spherical shell , 1957 .

[43]  J. Cassels,et al.  An Introduction to Diophantine Approximation , 1957 .

[44]  F. B. Hildebrand,et al.  Introduction To Numerical Analysis , 1957 .

[45]  O. J. Marlowe,et al.  Numerical integration over simplexes and cones , 1956 .

[46]  A. Stroud,et al.  Numerical integration over simplexes , 1956 .

[47]  K. F. Roth On irregularities of distribution , 1954 .

[48]  P. Davis,et al.  On the estimation of quadrature errors for analytic functions , 1954 .

[49]  On uniform distribution of algebraic numbers , 1953 .

[50]  R. D. Richtmyer THE EVALUATION OF DEFINITE INTEGRALS, AND A QUASI-MONTE-CARLO METHOD BASED ON THE PROPERTIES OF ALGEBRAIC NUMBERS , 1951 .

[51]  Leroy F. Meyers,et al.  Best Approximate Integration Formulas , 1950 .

[52]  N. Metropolis,et al.  The Monte Carlo method. , 1949, Journal of the American Statistical Association.

[53]  J. Radon,et al.  Zur mechanischen Kubatur , 1948 .

[54]  J. L. Coolidge,et al.  A history of geometrical methods , 1947 .

[55]  D. Jackson,et al.  fourier series and orthogonal polynomials , 1943, The Mathematical Gazette.

[56]  J. A. Clarkson,et al.  On definitions of bounded variation for functions of two variables , 1933 .

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