Quasi-Monte Carlo methods and pseudo-random numbers

CONTENTS

[1]  R. Mises.  Grundlagen der Wahrscheinlichkeitsrechnung , 1919 .

[2]  Karl Pearson,et al.  Random sampling numbers , 2022 .

[3]  R. Mises,et al.  Wahrscheinlichkeit, Statistik und Wahrheit. , 1936 .

[4]  Arthur H. Copeland,et al.  Admissible Numbers in the Theory of Probability , 1928 .

[5]  Hans Reichenbach,et al.  Axiomatik der Wahrscheinlichkeitsrechnung , 1932 .

[6]  R. Iglisch.  Zum Aufbau der Wahrscheinlichkeitsrechnung , 1933 .

[7]  E. N. Oberg.  The approximate solution of integral equations , 1935 .

[8]  Jean-Luc Ville.  Étude critique de la notion de collectif , 1939 .

[9]  A. Church.  On the concept of a random sequence , 1940 .

[10]  J. Littlewood,et al.  Notes on the theory of series (XXIV): a curious power-series , 1946, Mathematical Proceedings of the Cambridge Philosophical Society.

[11]  K. Chung,et al.  An estimate concerning the Kolmogoroff limit distribution , 1949 .

[12]  N. Metropolis,et al.  The Monte Carlo method. , 1949 .

[13]  Some theorems on diophantine inequalities , 1950 .

[14]  J. Cassels.  An extension of the law of the iterated logarithm , 1951, Mathematical Proceedings of the Cambridge Philosophical Society.

[15]  Gilbert W. King.  The Monte Carlo Method as a Natural Mode of Expression in Operations Research , 1953, Oper. Res..

[16]  Wouter Peremans,et al.  Reduced sequences of integers and pseudo-random numbers , 1953 .

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

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

[19]  O. Taussky,et al.  Generation and testing of pseudo-random numbers , 1954 .

[20]  K. D. Tocher.  The Application of Automatic Computers to Sampling Experiments , 1954 .

[21]  P. Davis,et al.  Some Monte Carlo experiments in computing multiple integrals , 1956 .

[22]  D. L. Johnson.  Generating and testing pseudo random numbers on the IBM Type 701 , 1956 .

[24]  J. Franklin.  On the equidistribution of pseudo-random numbers , 1958 .

[25]  I. M. Sobol.  Pseudo-Random Numbers for the Machine “Strela” , 1958 .

[26]  N. Zierler.  Linear Recurring Sequences , 1959 .

[27]  Bert F. Green,et al.  Empirical Tests of an Additive Random Number Generator , 1959, JACM.

[28]  R. R. Coveyou.  Serial Correlation in the Generation of Pseudo-Random Numbers , 1960, JACM.

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

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

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

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

[33]  W. Liniger.  On a method by D. H. Lehmer for the generation of pseudo random numbers , 1961 .

[34]  M. Greenberger.  An a priori determination of serial correlation in computer generated random numbers , 1961 .

[35]  J. Kiefer.  On large deviations of the empiric D. F. of vector chance variables and a law of the iterated logarithm. , 1961 .

[36]  Note on the numerical integration of periodic functions and of partially periodic functions , 1961 .

[37]  Edmund Hlawka.  Über die Diskrepanz mehrdimensionaler Folgen mod. 1 , 1961 .

[38]  R. D. Richtmyer,et al.  Continued fraction expansions of algebraic numbers , 1962 .

[39]  I. M Sobol.  An exact estimate of the error in multidimensional quadrature formulae for functions of the classes W̃1 and H̃1 , 1962 .

[40]  On quadrature formulae for functions of an infinitely large number of variables , 1962 .

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

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

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

[44]  L. Arnold,et al.  Numerische Experimente zur mehrdimensionalen Quadratur , 1963 .

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

[46]  J. Halton,et al.  Algorithm 247: Radical-inverse quasi-random point sequence , 1964, CACM.

[47]  Edmund Hlawka.  Uniform distribution modulo $1$ and numerical analysis , 1964 .

[48]  E. Hlawka.  Trigonometrische Interpolation bei Funktionen von mehreren Variablen , 1964 .

[49]  G. Helmberg.  Gleichverteilte Folgen in lokal kompakten Räumen , 1964 .

[50]  Birger Jansson,et al.  Autocorrelations between pseudo-random numbers , 1964 .

[51]  J. Hammersley,et al.  Monte Carlo Methods , 1965 .

[52]  W. Schmidt.  Metrical theorems on fractional parts of sequences , 1964 .

[53]  D. L. Jagerman.  Some theorems concerning pseudo-random numbers , 1965 .

[54]  J. Franklin,et al.  Numerical Simulation of Stationary and Non-Stationary Gaussian Random Processes , 1965 .

[55]  P. Prescott,et al.  Monte Carlo Methods , 1964, Computational Statistical Physics.

[56]  A. Baker,et al.  On Some Diophantine Inequalities Involving the Exponential Function , 1965, Canadian Journal of Mathematics.

[57]  An inequality connected with Weyl''s criterion for uniform distribution , 1965 .

[58]  R. Tausworthe.  Random Numbers Generated by Linear Recurrence Modulo Two , 1965 .

[59]  Martin Greenberger,et al.  Method in randomness , 1965, CACM.

[60]  Donald E. Knuth,et al.  Construction of a random sequence , 1965 .

[61]  D. Teichroew.  A History of Distribution Sampling Prior to the Era of the Computer and its Relevance to Simulation , 1965 .

[62]  Octav Onicescu.  Nombres et systèmes aléatoires , 1966 .

[63]  Per Martin-Löf,et al.  The Definition of Random Sequences , 1966, Inf. Control..

[64]  David W. Hutchinson.  A new uniform pseudorandom number generator , 1966, CACM.

[65]  Seymour Haber.  On a sequence of points of interest for numerical quadrature , 1966 .

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

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

[68]  Gregory J. Chaitin,et al.  On the Length of Programs for Computing Finite Binary Sequences , 1966, JACM.

[69]  Birger Jansson,et al.  Random Number Generators , 1967 .

[70]  I. V. Vilenkin.  Plane nets of integration , 1967 .

[71]  Wilfred J. Westlake,et al.  A Uniform Random Number Generator Based on the Combination of Two Congruential Generators , 1967, JACM.

[72]  Allen Van Gelder,et al.  Some New Results in Pseudo-Random Number Generation , 1967, JACM.

[73]  I. Sobol.  On the distribution of points in a cube and the approximate evaluation of integrals , 1967 .

[74]  D. Y. Downham,et al.  Multiplicative congruential pseudo-random number generators , 1967, Comput. J..

[75]  R. R. Coveyou,et al.  Fourier Analysis of Uniform Random Number Generators , 1967, JACM.

[76]  Représentations arithmétiques de grandeurs aléatoires , 1967 .

[77]  Philip Rabinowitz,et al.  Methods of Numerical Integration , 1985 .

[78]  Garry J. Tee,et al.  The Monte Carlo Method: The Method of Statistical Trials , 1967 .

[79]  Patrick Billingsley,et al.  Uniformity in weak convergence , 1967 .

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

[81]  A. Postnikov,et al.  Ergodic Problems in the Theory of Congruences and of Diophantine Approximations , 1967 .

[82]  A remark on uniformly distributed sequences and Riemann integrability , 1968 .

[83]  Good lattice points in the sense of Hlawka and Monte-Carlo integration , 1968 .

[84]  John R. B. Whittlesey,et al.  A comparison of the correlational behavior of random number generators for the IBM 360 , 1968, Commun. ACM.

[85]  S. C. Zaremba.  Some applications of multidimensional integration by parts , 1968 .

[86]  Integration over regions different from a unit cube , 1968 .

[87]  L. Fosdick.  The Monte Carlo Method in Quantum Statistics , 1968 .

[88]  H. G. Meijer,et al.  The Discrepancy of a G-Adic Sequence , 1968 .

[89]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[90]  S. Zaremba.  The Mathematical Basis of Monte Carlo and Quasi-Monte Carlo Methods , 1968 .

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

[92]  George Marsaglia,et al.  One-line random number generators and their use in combinations , 1968, Commun. ACM.

[93]  C. Baker.  On the Nature of Certain Quadrature Formulas and Their Errors , 1968 .

[94]  Bryna M. Fellen.  Letters to the editor: an implementation of the Tausworthe generator , 1969, CACM.

[95]  S. K. Zaremba.  A quasi-Monte Carlo method for computing double and other multiple integrals , 1969 .

[96]  Das Gesetz vom iterierten Logarithmus mit Anwendungen auf die Zahlentheorie , 1969 .

[97]  I. P. Mysovskih.  Approximate Calculation of Integrals , 1969 .

[98]  E. Hlawka,et al.  Diskrepanz in kompakten abelschen Gruppen I , 1969 .

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

[100]  On a Theorem of Piatetsky-Shapiro and Approximation of Multiple Integrals , 1969 .

[101]  S. K. Zaremba,et al.  The extreme and L2 discrepancies of some plane sets , 1969 .

[102]  C. Chui.  A Convergence Theorem for Certain Riemann Sums , 1969, Canadian Mathematical Bulletin.

[104]  Edmund Hlawka.  Interpolation Analytischer Funktionen Auf Dem Einheitskreis , 1969 .

[105]  P. A. W. Lewis,et al.  A Pseudo-Random Number Generator for the System/360 , 1969, IBM Syst. J..

[106]  Wolfgang M. Schmidt,et al.  Simultaneous approximation to algebraic numbers by rationals , 1970 .

[107]  G. Cenacchi,et al.  Pseudo-random numbers for comparative Monte Carlo calculations , 1970 .

[108]  George Marsaglia,et al.  Regularities in congruential random number generators , 1970 .

[109]  W. H. Payne,et al.  Fortran Tausworthe pseudorandom number generator , 1970, CACM.

[110]  Seymour Haber,et al.  Numerical Evaluation of Multiple Integrals , 1970 .

[111]  Seymour Haber.  Sequences of Numbers That Are Approximately Completely Equidistributed , 1970, JACM.

[113]  J. Halton.  A Retrospective and Prospective Survey of the Monte Carlo Method , 1970 .

[114]  J. P. R. Tootill,et al.  The Runs Up-and-Down Performance of Tausworthe Pseudo-Random Number Generators , 1971, JACM.

[115]  S. Zaremba.  Sur la discrépance des suites aléatoires , 1971 .

[116]  W. A. Beyer,et al.  The lattice structure of multiplicative congruential pseudo-random vectors , 1971 .

[117]  U. Dieter,et al.  Pseudo-random numbers. The exact distribution of pairs , 1971 .

[118]  Simulation von Produktionsprozessen , 1971 .

[119]  C. S. Smith,et al.  Multiplicative Pseudo-Random Number Generators with Prime Modulus , 1971, JACM.

[120]  Claus-Peter Schnorr,et al.  Zufälligkeit und Wahrscheinlichkeit - Eine algorithmische Begründung der Wahrscheinlichkeitstheorie , 1971, Lecture Notes in Mathematics.

[121]  Charles K. Chui,et al.  Concerning rates of convergence of Riemann sums , 1971 .

[122]  U. Dieter,et al.  An exact determination of serial correlations of pseudo-random numbers , 1971 .

[123]  Edmund Hlawka.  Zur Definition der Diskrepanz , 1971 .

[124]  Walter Philipp.  Mixing Sequences of Random Variables and Probabilistic Number Theory , 1971 .

[125]  Über die C-Gleichverteilung von Maßen , 1971 .

[126]  N M Korobov.  ON THE DISTRIBUTION OF DIGITS IN PERIODIC FRACTIONS , 1972 .

[127]  Harald Niederreiter,et al.  Methods for Estimating Discrepancy , 1972 .

[128]  Quasi-random sequences by power residues , 1972 .

[129]  Harald Niederreiter,et al.  Discrepancy and convex programming , 1972 .

[130]  Seymour Haber.  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 .

[131]  G. Marsaglia.  The Structure of Linear Congruential Sequences , 1972 .

[132]  Edmund Hlawka,et al.  A Transformation of Equidistributed Sequences , 1972 .

[133]  On the exponential sum , 1972 .

[134]  Tony Warnock,et al.  Computational investigations of low-discrepancy point-sets. , 1972 .

[135]  On distribution functions (mod 1): Quantitative Fourier inversion , 1972 .

[136]  Takao Tsuda.  Numerical integration of functions of very many variables , 1972 .

[137]  Dominique Maisonneuve.  Recherche et Utilisation des “Bons Treillis.” Programmation et Résultats Numériques , 1972 .

[138]  U. Dieter,et al.  Statistical Interdependence of Pseudo-Random Numbers Generated by the Linear Congruential Method , 1972 .

[139]  Harald Niederreiter,et al.  On a number-theoretical integration method , 1972 .

[140]  A NEW TYPE OF EXPONENTIAL SERIES , 1972 .

[141]  W. A. Beyer.  Lattice Structure and Reduced Bases of Random Vectors Generated by Linear Recurrences , 1972 .

[142]  C. Chui.  Convergence of certain quadrature processes , 1972 .

[143]  S. K. Zaremba,et al.  La Méthode des “Bons Treillis” pour le Calcul des Intégrales Multiples , 1972 .

[144]  John H. Halton,et al.  Estimating the Accuracy of Quasi-Monte Carlo Integration , 1972 .

[145]  W. Schmidt.  On irregularities of distribution vii , 1972 .

[146]  Peter Kirschenmann,et al.  Concepts of randomness , 1972, J. Philos. Log..

[147]  Harald Niederreiter,et al.  On the distribution of pseudo-random numbers generated by the linear congruential method. II , 1972 .

[148]  Fonctions entières et répartition modulo un. II , 1973 .

[149]  A. Grube,et al.  Mehrfach rekursiv‐erzeugte Pseudo‐Zufallszahlen , 1973 .

[150]  Solution of the Cauchy problem for a parabolic equation by a quasi-Monte-Carlo method , 1973 .

[151]  Über eine Methode von E. Hecke in der Theorie der Gleichverteilung , 1973 .

[152]  R. B. Statnikov,et al.  Location of the characteristic roots of a matrix , 1973 .

[153]  Tests de générateurs pseudo-aléatoires , 1973 .

[154]  J. P. R. Tootill,et al.  An Asymptotically Random Tausworthe Sequence , 1973, JACM.

[155]  R. G. Stoneham.  On the uniform ε-distribution of residues within the periods of rational fractions with applications to normal numbers , 1973 .

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

[157]  Berry-Esseen bounds and a theorem of Erdös and Turàn on uniform distribution $\mod 1$ , 1973 .

[158]  H. Neunzert,et al.  Die Theorie der asymptotischen Verteilung und die numerische Lösung von Integrodifferentialgleichungen , 1973 .

[159]  T. Kamae.  Subsequences of normal sequences , 1973 .

[160]  L Hua.  ON UNIFORM DISTRIBUTION AND NUMERICAL ANALYSIS (III) (NUMBER-THEORETIC METHOD) , 1973 .

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

[162]  Evaluation of Multiple Integrals , 1974 .

[163]  S. K. Zaremba.  Good lattice points modulo composite numbers , 1974 .

[164]  G. Kedem,et al.  A table of good lattice points in three dimensions , 1974 .

[166]  G. A. Mihram,et al.  Simulation: Statistical Foundations and Methodology. , 1974 .

[167]  H. Neunzert,et al.  Die Approximation der Lösung von Integro-Differentialgleichungen durch endliche Punktmengen , 1974 .

[168]  G. Chaitin.  Randomness and Mathematical Proof , 1975 .

[169]  S. M. Ermakow.  Die Monte-Carlo-Methode und verwandte Fragen , 1975 .

[170]  The Search for Good Lattice Points in N Dimensions. , 1975 .

[171]  Simulation déterministe du hasard , 1975 .

[172]  A. Yao,et al.  Analysis of the subtractive algorithm for greatest common divisors , 1975 .

[173]  ON THE UNIFORM DISTRIBUTION OF THE SEQUENCE $ \{\alpha\lambda^x\}$ , 1975 .

[174]  L'ERREUR DANS LE CALCUL DES INTÉGRALES DOUBLES PAR LA METHODE DES BONS TREILLIS , 1975 .

[175]  M. Rosenblatt.  Multiply Schemes and Shuffling. , 1975 .

[176]  Distances of probability measures and uniform distribution mod 1 , 1975 .

[177]  S. K. Zaremba.  Computing the isotropic discrepancy of point sets in two dimensions , 1975, Discret. Math..

[178]  H. Niederreiter.  Resultats nouveaux dans la theorie quantitative de l’equirepartition , 1975 .

[179]  H. Niederreiter,et al.  Diskrepanz und Distanz von Maßen bezüglich konvexer und Jordanscher Mengen , 1975 .

[180]  Mean-square discrepancies of the Hammersley and Zaremba sequences for arbitrary radix , 1975 .

[181]  T. Kamae,et al.  Normal numbers and selection rules , 1975 .

[182]  Quantitative versions of a result of Hecke in the theory of uniform distribution mod 1 , 1975 .

[183]  Equirepartition et theorie des nombres premiers , 1975 .

[184]  Statistical independence of linear congruential pseudo-random numbers , 1976 .

[185]  George S. Fishman,et al.  Solution of Large Networks by Matrix Methods , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[186]  R. Cranley,et al.  Randomization of Number Theoretic Methods for Multiple Integration , 1976 .

[187]  K. F. Roth.  On irregularities of distribution, II , 1976 .

[188]  E. S. Page,et al.  Distribution sampling for computer simulation , 1976 .

[189]  H. Niederreiter.  On the cycle structure of linear recurring sequences. , 1976 .

[190]  S. K. Zaremba.  On Cartesian products of good lattices , 1976 .

[191]  W. Stute.  Convergence Rates for the Isotrope Discrepancy , 1977 .

[192]  Harald Niederreiter,et al.  Weights of Cyclic Codes , 1977, Information and Control.

[193]  Brian E. White,et al.  On optimal extreme-discrepancy point sets in the square , 1977 .

[194]  Donald E. Knuth,et al.  Notes on generalized Dedekind sums , 1977 .

[195]  H. Niederreiter.  Pseudo-random numbers and optimal coefficients☆ , 1977 .

[196]  H. Niederreiter.  Existence of good lattice points in the sense of Hlawka , 1978 .

[197]  Harald Niederreiter.  The serial test for linear congruential pseudo-random numbers , 1978 .

[198]  K. F. Roth,et al.  On irregularities of distribution IV , 1979 .