Monte Carlo Methods

Many quantitative problems in science, engineering, and economics are nowadays solved via statistical sampling on a computer. Such Monte Carlo methods can be used in three different ways: (1) to generate random objects and processes in order to observe their behavior, (2) to estimate numerical quantities by repeated sampling, and (3) to solve complicated optimization problems through randomized algorithms.

[1]  Takuji Nishimura,et al.  Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator , 1998, TOMC.

[2]  Ted G. Lewis,et al.  Generalized Feedback Shift Register Pseudorandom Number Algorithm , 1973, JACM.

[3]  James C. Spall,et al.  Introduction to stochastic search and optimization - estimation, simulation, and control , 2003, Wiley-Interscience series in discrete mathematics and optimization.

[4]  Pierre L'Ecuyer,et al.  TestU01: A C library for empirical testing of random number generators , 2006, TOMS.

[5]  Dirk P. Kroese,et al.  Handbook of Monte Carlo Methods , 2011 .

[6]  Dirk P. Kroese,et al.  The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation and Machine Learning , 2004 .

[7]  Pierre L'Ecuyer,et al.  Good Parameters and Implementations for Combined Multiple Recursive Random Number Generators , 1999, Oper. Res..

[8]  G. Pflug.  Stochastic Approximation Methods for Constrained and Unconstrained Systems - Kushner, HJ.; Clark, D.S. , 1980 .

[9]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[10]  Dirk P. Kroese,et al.  The Cross-Entropy Method for Continuous Multi-Extremal Optimization , 2006 .

[11]  Robert Tibshirani,et al.  An Introduction to the Bootstrap , 1994 .

[12]  Andrew Gelman,et al.  The No-U-turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo , 2011, J. Mach. Learn. Res..

[13]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[14]  B. Leimkuhler,et al.  Simulating Hamiltonian Dynamics: Hamiltonian PDEs , 2005 .

[15]  Dirk P. Kroese,et al.  Chapter 3 – The Cross-Entropy Method for Optimization , 2013 .

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

[17]  Robert L. Smith,et al.  An analytically derived cooling schedule for simulated annealing , 2007, J. Glob. Optim..

[18]  J. Hammersley.  SIMULATION AND THE MONTE CARLO METHOD , 1982 .

[19]  Christian P. Robert,et al.  Monte Carlo Statistical Methods , 2005, Springer Texts in Statistics.

[20]  Shie Mannor,et al.  A Tutorial on the Cross-Entropy Method , 2005, Ann. Oper. Res..