Quasi-Monte Carlo methods and pseudo-random numbers

The Art of Computer Programming

1968 • 16,727 citations

The Monte Carlo Method

1949 • 5,519 citations

Monte Carlo methods

1964 • 3,128 citations

Methods of Numerical Integration.

1986 • 2,870 citations

Uniform distribution of sequences

1974 • 2,581 citations

On the distribution of points in a cube and the approximate evaluation of integrals

1967 • 2,478 citations

On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals

1960 • 1,843 citations

Approximate Calculation of Multiple Integrals

1973 • 1,544 citations

The definition of random sequences

1966 • 1,230 citations

On the Length of Programs for Computing Finite Binary Sequences

1969 • 937 citations

Logik der Forschung

1935 • 859 citations

Approximate Calculation of Integrals

1964 • 856 citations

Approximate Calculation of Integrals.

1965 • 765 citations

Randomness and Mathematical Proof

1975 • 663 citations

Computer Simulation Techniques.

1967 • 639 citations

Pure and Applied Mathematics

1973 • 553 citations

Algorithm 247: Radical-inverse quasi-random point sequence

1964 • 534 citations

RANDOM NUMBERS FALL MAINLY IN THE PLANES

1968 • 524 citations

MONTE CARLO METHODS FOR SOLVING MULTIVARIABLE PROBLEMS

1960 • 508 citations

On irregularities of distribution

1954 • 491 citations

Monte Carlo Principles and Neutron Transport Problems

1971 • 459 citations

Mathematical Methods in Large-scale Computing Units

1951 • 443 citations

Linear Recurring Sequences

1959 • 421 citations

Random numbers generated by linear recurrence modulo two

1965 • 419 citations

Wahrscheinlichkeit Statistik und Wahrheit

1928 • 415 citations

On the concept of a random sequence

1940 • 400 citations

Molecular Schrödinger Equation. VIII. A New Method for the Evaluation of Multidimensional Integrals

1967 • 397 citations

A pseudo-random number generator for the System/360

1969 • 375 citations

Concepts and Methods in Discrete Event Digital Simulation.

1975 • 356 citations

Étude critique de la notion de collectif

1939 • 346 citations

A Retrospective and Prospective Survey of the Monte Carlo Method

1970 • 346 citations

Random Number Generators

1962 • 337 citations

Grundlagen der Wahrscheinlichkeitsrechnung

1919 • 307 citations

Randomization of Number Theoretic Methods for Multiple Integration

1976 • 303 citations

Fourier Analysis of Uniform Random Number Generators

1967 • 295 citations

Irregularities of distribution, VII

1972 • 272 citations

The Logic of Chance.

1964 • 270 citations

On large deviations of the empiric D. F. of vector chance variables and a law of the iterated logarithm

1961 • 266 citations

Logik der Forschung: Zur Erkenntnistheorie der Modernen Naturwissenschaft.

1936 • 264 citations

Numerical Evaluation of Multiple Integrals

1970 • 262 citations

Zufälligkeit und Wahrscheinlichkeit

1971 • 260 citations

Uniform Random Number Generators

1965 • 247 citations

Good lattice points, discrepancy, and numerical integration

1966 • 242 citations

Funktionen von beschränkter Variatiou in der Theorie der Gleichverteilung

1961 • 234 citations

A method for numerical integration

1961 • 200 citations

The Structure of Linear Congruential Sequences

1972 • 164 citations

Zur angen�herten Berechnung mehrfacher Integrale

1962 • 163 citations

Simulation: Statistical Foundations and Methodology.

1974 • 151 citations

Metrical theorems on fractional parts of sequences

1964 • 149 citations

An estimate concerning the Kolmogoroff limit distribution

1949 • 147 citations

Simultaneous approximation to algebraic numbers by rationals

1970 • 143 citations

Some applications of multidimensional integration by parts

1968 • 131 citations

Computer Modeling and Simulation

1968 • 129 citations

Note on irregularities of distribution

1956 • 125 citations

Coding the Lehmer pseudo-random number generator

1969 • 122 citations

Monte Carlo Methods.

1965 • 119 citations

Computational Investigations of Low-Discrepancy Point Sets**This work received financial support from National Science Foundation Grants GB-16665 and GJ171 and National Aeronautics and Space Administration Grant NGL 50-002-001.

1972 • 113 citations

Pseudo-random numbers and optimal coefficients

1977 • 105 citations

On irregularities of distribution, III

1979 • 100 citations

Zufälligkeit und Wahrscheinlichkeit : Eine algorithmische Begründung der Wahrscheinlichkeitstheorie

1970 • 100 citations

La Méthode des “Bons Treillis” pour le Calcul des Intégrales Multiples

1972 • 100 citations

Uniformity in weak convergence

1967 • 95 citations

The Mathematical Basis of Monte Carlo and Quasi-Monte Carlo Methods

1968 • 95 citations

An Asymptotically Random Tausworthe Sequence

1973 • 94 citations

Deterministic simulation of random processes

1963 • 91 citations

The extreme and L2 discrepancies of some plane sets

1969 • 84 citations

Serial Correlation in the Generation of Pseudo-Random Numbers

1960 • 81 citations

Existence of good lattice points in the sense of Hlawka

1978 • 78 citations

ON THE DISTRIBUTION OF DIGITS IN PERIODIC FRACTIONS

1972 • 77 citations

Subsequences of normal sequences

1973 • 72 citations

One-line random number generators and their use in combinations

1968 • 70 citations

A modified Monte-Carlo quadrature

1966 • 69 citations

An a priori determination of serial correlation in computer generated random numbers

1961 • 69 citations

Pseudo-random numbers

1970 • 69 citations

The Runs Up-and-Down Performance of Tausworthe Pseudo-Random Number Generators

1971 • 67 citations

Numerical Simulation of Stationary and Non-Stationary Gaussian Random Processes

1965 • 65 citations

The Monte Carlo Method: The Method of Statistical Trials

1967 • 64 citations

The Generation of Random Variates.

1971 • 63 citations

On irregularities of distribution, IV

1980 • 62 citations

Analysis of the subtractive algorithm for greatest common divisors

1975 • 62 citations

A comparison of the correlational behavior of random number generators for the IBM 360

1968 • 60 citations

On Some Diophantine Inequalities Involving the Exponential Function

1965 • 60 citations

Notes on generalized Dedekind sums

1977 • 59 citations

Multiplicative congruential pseudo-random number generators

1967 • 59 citations

Discrepancy and convex programming

1972 • 59 citations

Random Numbers Generated by Linear Recurrence Modulo Two

1965 • 54 citations

Die Monte-Carlo-Methode und verwandte Fragen

1975 • 52 citations

Empirical Tests of an Additive Random Number Generator

1959 • 51 citations

Mixing Sequences of Random Variables and Probabilistic Number Theory

1971 • 51 citations

Recherche et Utilisation des “Bons Treillis.” Programmation et Résultats Numériques

1972 • 44 citations

Methods for Estimating Discrepancy

1972 • 44 citations

The Literature on von Mises' Kollektivs Revisited

1969 • 44 citations

Über eine Transformation von gleichverteilten Folgen II

1972 • 44 citations

The lattice structure of multiplicative congruential pseudo-random vectors

1971 • 44 citations

Random Number Generators

1967 • 43 citations

A new uniform pseudorandom number generator

1966 • 43 citations

Diskrepanz und Distanz von Ma�en bez�glich konvexer und Jordanscher Mengen

1975 • 42 citations

Distribution Sampling for Computer Simulation

1975 • 42 citations

A History of Distribution Sampling Prior to the Era of the Computer and its Relevance to Simulation

1965 • 41 citations

Die Approximation der Lösung von Integro-Differentialgleichungen durch endliche Punktmengen

1974 • 40 citations

A quasi-Monte Carlo method for the approximate computation of the extreme values of a function

1983 • 40 citations

Notes on the theory of series (XXIV): a curious power-series

1946 • 39 citations

Concerning rates of convergence of Riemann sums

1971 • 39 citations

Ergodic problems in the theory of congruences and of diophantine approximations

1967 • 39 citations

Random number generator

1956 • 38 citations

Some theorems on diophantine inequalities

1950 • 38 citations

Berry-Esseen bounds and a theorem of Erdös and Turàn on uniform distribution mod1

1973 • 36 citations

Irregularities for distribution IX

1975 • 36 citations

Axiomatik der Wahrscheinlichkeitsrechnung

1932 • 36 citations

Regularities in congruential random number generators

1970 • 35 citations

Mehrfach rekursiv‐erzeugte Pseudo‐Zufallszahlen

1973 • 34 citations

Pseudo-random numbers. The exact distribution of pairs

1971 • 33 citations

Normal numbers and selection rules

1975 • 33 citations

An Estimate Concerning the Kolmogroff Limit Distribution

1949 • 32 citations

Method in randomness

1965 • 32 citations

The Application of Automatic Computers to Sampling Experiments

1954 • 31 citations

Continued fraction expansions of algebraic numbers

1962 • 31 citations

On the sum ∑⟨nα⟩^−tand numerical integration

1969 • 31 citations

Admissible Numbers in the Theory of Probability

1928 • 30 citations

Metrical Theorems on Fractional Parts of Sequences

1964 • 30 citations

On the distribution of pseudo-random numbers generated by the linear congruential method. II

1974 • 30 citations

Die Theorie der asymptotischen Verteilung und die numerische L�sung von Integrodifferentialgleichungen

1973 • 30 citations

An exact determination of serial correlations of pseudo-random numbers

1971 • 29 citations

Good lattice points modulo composite numbers

1974 • 29 citations

Some Monte Carlo Experiments in Computing Multiple Integrals

1956 • 26 citations

Weights of cyclic codes

1977 • 26 citations

�ber die Diskrepanz mehrdimensionaler Folgen mod. 1

1961 • 26 citations

Some Monte Carlo experiments in computing multiple integrals

1956 • 26 citations

La discrépance isotrope et l'intégration numérique

1970 • 25 citations

Discrepancy and Riemann Integration

1990 • 25 citations

Convergence Rates for the Isotrope Discrepancy

1977 • 25 citations

A Uniform Random Number Generator Based on the Combination of Two Congruential Generators

1967 • 24 citations

A table of good lattice points in three dimensions

1974 • 24 citations

On a number-theoretical integration method

1972 • 24 citations

On the cycle structure of linear recurring sequences.

1976 • 23 citations

On a sequence of points of interest for numerical quadrature

1966 • 22 citations

A Modified Monte-Carlo Quadrature

1966 • 21 citations

A Remark on Uniformly Distributed Sequences and Riemann Integrability

1968 • 21 citations

An extension of the law of the iterated logarithm

1951 • 21 citations

Some New Results in Pseudo-Random Number Generation

1967 • 21 citations

Deterministic Simulation of Random Processes

1963 • 20 citations

On the uniform ε-distribution of residues within the periods of rational fractions with applications to normal numbers

1973 • 20 citations

Uniform distribution modulo 1 and numerical analysis

1990 • 20 citations

A Transformation of Equidistributed Sequences**Translated from German by H. Niederreiter.

1972 • 20 citations

Lattice Structure and Reduced Bases of Random Vectors Generated by Linear Recurrences**Work performed under the auspices of the U.S. Atomic Energy Commission.

1972 • 19 citations

Construction of a random sequence

1965 • 19 citations

The approximate solution of integral equations

1935 • 18 citations

SOME PROBLEMS IN THE THEORY OF DIOPHANTINE APPROXIMATION

1967 • 18 citations

The Monte Carlo Method in Quantum Statistics

1968 • 18 citations

Statistical Interdependence of Pseudo-Random Numbers Generated by the Linear Congruential Method

1972 • 18 citations

Good lattice points in the sense of Hlawka and Monte-Carlo integration

1968 • 18 citations

Generating and Testing Pseudo Random Numbers on the IBM Type 701

1956 • 18 citations

Simulationstechnik

1976 • 17 citations

On the Nature of Certain Quadrature Formulas and Their Errors

1968 • 16 citations

Numerical integration of functions of very many variables

1972 • 16 citations

A Fast Method of Generating Digital Random Numbers

1970 • 16 citations

Autocorrelations between pseudo-random numbers

1964 • 16 citations

On optimal extreme-discrepancy point sets in the square

1977 • 15 citations

On irregularities of distribution, II

1976 • 15 citations

An A Priori Determination of Serial Correlation in Computer Generated Random Numbers

1961 • 15 citations

On uniform distribution of algebraic numbers

1953 • 14 citations

Stationary functions and their applications to turbulence II. Turbulent solutions of the Navier-Stokes equations

1974 • 14 citations

A quasi-Monte Carlo method for computing double and other multiple integrals

1970 • 14 citations

Multiplicative Pseudo-Random Number Generators with Prime Modulus

1971 • 14 citations

Interpolation Analytischer Funktionen Auf Dem Einheitskreis

1969 • 14 citations

A Method for Numerical Integration

1961 • 14 citations

Mean-square discrepancies of the Hammersley and Zaremba sequences for arbitrary radix

1975 • 13 citations

Remarks Concerning Numerical Integration

2005 • 13 citations

Letters to the editor: on the multidimensional uniformity of pseudorandom generators

1969 • 13 citations

Convergence of certain quadrature processes

1973 • 13 citations

The Monte Carlo method

1977 • 13 citations

On the discrepancy of certain sequences mod. 1

1963 • 13 citations

Pseudo-Random Numbers: The Exact Distribution of Pairs

1971 • 12 citations

The serial test for linear congruential pseudo-random numbers

1978 • 12 citations

Reduced sequences of integers and pseudo-random numbers

1953 • 12 citations

Pseudo-random numbers for comparative Monte Carlo calculations

1970 • 12 citations

The Discrepancy of a G-Adic Sequence

1968 • 12 citations

Reviews and Bibliography

1962 • 12 citations

The Monte Carlo Method as a Natural Mode of Expression in Operations Research

1953 • 12 citations

Plane nets of integration

1967 • 12 citations

Herstellung von Zufallszahlen auf Rechenautomaten

1957 • 11 citations

Fortran Tausworthe pseudorandom number generator

1970 • 11 citations

On the equidistribution of pseudo-random numbers

1958 • 11 citations

Randomness Statistics and Emergence

1970 • 11 citations

A Convergence Theorem for Certain Riemann Sums

1969 • 11 citations

Zur Definition der Diskrepanz

1971 • 11 citations

Das Gesetz vom iterierten Logarithmus mit Anwendungen auf die Zahlentheorie

1971 • 10 citations

Generating and testing pseudo random numbers on the IBM Type 701

1956 • 10 citations

Pseudo-Random Numbers for the Machine “Strela”

1958 • 10 citations

Letters to the editor: an implementation of the Tausworthe generator

1969 • 10 citations

Statistical independence of linear congruential pseudo-random numbers

1976 • 10 citations

Gleichverteilte Folgen in lokal kompakten R�umen

1964 • 9 citations

Diskrepanz in kompakten abelschen Gruppen I

1969 • 9 citations

Simulation von Produktionsprozessen

1971 • 9 citations

Diskrepanz in kompakten abelschen Gruppen II

1969 • 8 citations

ON THE UNIFORM DISTRIBUTION OF THE SEQUENCE $ \{\alpha\lambda^x\}$

1975 • 8 citations

The Lattice Structure of Multiplicative Congruential Pseudo-Random Vectors

1971 • 8 citations

ON UNIFORM DISTRIBUTION AND NUMERICAL ANALYSIS (III): NUMBER-THEORETIC METHOD

2005 • 8 citations

On a method by D. H. Lehmer for the generation of pseudo random numbers

1961 • 8 citations

On Periods of Pseudo-Random Sequences

1964 • 8 citations

A fast random number generator with good statistical properties

1970 • 8 citations

Some theorems concerning pseudo-random numbers

1965 • 8 citations

Distances of probability measures and uniform distribution mod 1

1975 • 7 citations

Experiments on Optimal Coefficients**This work was supported in part by NASA Research Grant NSG-398 to the Computer Science Center, University of Maryland.

1972 • 7 citations

Das Gesetz vom iterierten Logarithmus mit Anwendungen auf die Zahlentheorie

1969 • 7 citations

Trigonometrische Interpolation bei Funktionen von mehreren Variablen

1964 • 7 citations

On distribution functions (mod 1): Quantitative Fourier inversion

1972 • 7 citations

Approximate solution of integral equations

1964 • 6 citations

Concepts of Randomness

1973 • 6 citations

Note on the numerical integration of periodic functions and of partially periodic functions

1961 • 6 citations

Fonctions entières et répartition modulo un. II

1973 • 6 citations

Concepts of randomness

1972 • 6 citations

Simulation déterministe du hasard

1975 • 6 citations

On the discrepancy of certain sequences $\mathrm{mod}\,1$

1967 • 6 citations

Estimating the Accuracy of Quasi-Monte Carlo Integration**The work described in this paper was done with the support of the National Science Foundation, under Grants GJ-171 and GJ-29986X, for which the author wishes to express his gratitude.

1972 • 6 citations

Relations within sequences of congruential pseudo-random numbers

1969 • 6 citations

Über eine Methode von E. Hecke in der Theorie der Gleichverteilung

1973 • 5 citations

Sequences of Numbers That Are Approximately Completely Equidistributed

1970 • 5 citations

Discrepancy and uniform distribution of sequences

1990 • 5 citations

On Cartesian products of good lattices

1976 • 4 citations

A NEW TYPE OF EXPONENTIAL SERIES

1972 • 4 citations

On the Distribution of Pseudo-Random Numbers Generated by the Linear Congruential Method

1972 • 4 citations

Equirepartition et theorie des nombres premiers

1975 • 4 citations

Resultats nouveaux dans la theorie quantitative de l’equirepartition

1975 • 4 citations

Quantitative versions of a result of Hecke in the theory of uniform distribution mod 1

1975 • 4 citations

Application of the method of optimal coefficients to numerical integration

1969 • 4 citations

On the Distribution of Pseudo-Random Numbers Generated by the Linear Congruential Method. II

1974 • 4 citations

Some Theorems Concerning Pseudo-Random Numbers

1965 • 4 citations

Nombres et Systemes aleatoires.

1966 • 4 citations

On the exponential sum

1972 • 3 citations

A note on interpolation of a certain class of functions

1994 • 3 citations

Solution of the Cauchy problem for a parabolic equation by a quasi-Monte-Carlo method

1973 • 3 citations

On the evaluation of infinite-dimensional integrals

1962 • 3 citations

Sur la discr�pance des suites al�atoires

1971 • 3 citations

Zum Aufbau der Wahrscheinlichkeitsrechnung

1933 • 3 citations

�ber die C-Gleichverteilung von Ma�en

1971 • 3 citations

Numerische Experimente zur mehrdimensionalen Quadratur

1963 • 3 citations

MULTIDIMENSIONAL INTEGRALS AND THE MONTECARLO METHOD

1957 • 3 citations

Über Quadraturformeln auf Tω

1970 • 2 citations

2929. On the error in numerical integration

1960 • 2 citations

On quadrature formulae for functions of an infinitely large number of variables

1962 • 2 citations

On a theorem of Piatetsky-Shapiro and approximation of multiple integrals

1969 • 2 citations

Tests de générateurs pseudo-aléatoires

1973 • 2 citations

NOMBRES ET SYSTÈMES ALÉATOIRES

1966 • 2 citations

Quasi-random sequences by power residues

1972 • 2 citations

ON NUMERICAL INTEGRATION OF PERIODIC FUNCTIONS OF SEVERAL VARIABLES

2005 • 2 citations

Tables of Random Sampling Numbers. By M. G. Kendall and B. Babington Smith. Pp. x, 60. 3s. 9d. Tracts for Computers No. xxiv. (Department of Statistics, University College, London)

1940 • 2 citations

Computing the isotropic discrepancy of point sets in two dimensions

1975 • 2 citations

On the Evaluation of Multiple Integrals

1994 • 2 citations

A Remarkable Lattice Generated by Fibonacci Numbers

1970 • 2 citations

On Cartesian Products of Good Lattices

1976 • 2 citations

Multiply schemes and shuffling

1975 • 2 citations

Evaluation of Multiple Integrals

1974 • 2 citations

On Uniform Distribution of Algebraic Numbers

1953 • 2 citations

L'ERREUR DANS LE CALCUL DES INTÉGRALES DOUBLES PAR LA METHODE DES BONS TREILLIS

1975 • 2 citations

On a Theorem of Piatetsky-Shapiro and Approximation of Multiple Integrals

1969 • 1 citations

Anwendung zahlentheoretischer Methoden auf Probleme der numerischen Mathematik I

1990 • 1 citations

On a new class of multiplicative pseudo-random number generators

1970 • 1 citations

On the Periods of Certain Pseudorandom Sequences

1974 • 1 citations

Représentations arithmétiques de grandeurs aléatoires

1967 • 1 citations

Deleted Work

1955 • 0 citations