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Introduction

Dieter W. Heermann

Monte Carlo Methods

2009

Dieter W. Heermann (Monte Carlo Methods) Introduction 2009 1 / 18

Outline

1 Introduction

2 Poor Man’s Monte Carlo

3 Some History

4 Literature


Introduction

What are Monte Carlo Methods?

In the widest sense of the term, Monte Carlo (MC) simulations meanany simulation (not even necessarily a computer simulation) thatutilizes random numbers in the simulation algorithm.

Monte Carlo simulations are statistical and non-deterministic. Henceeach simulation will give a different result, but the results will berelated via some statistical error.

The Monte Carlo algorithm was named the top algorithm of the 20thcentury by mathematicians and physicists.


Introduction

Why do we need this?

Multidimensional integralsSystems with a large number of degrees of freedom

Many atoms in a gas, liquid, solidMany electrons in an atomGene expressionNetworks...


Poor Man’s Monte Carlo

Problem: Calculate πIdea: Randomly fill a square. Then the ratio between the number ofpoints inside a circle and the overall number of point must beproportional to π



Page 1 of 1main.c

#include <stdio.h>#include <math.h>#include <time.h>#include <stdlib.h>

int main (int argc, const char * argv[]) { int mcs,mcsmax; float x,y,pi; int seed; int count_in; mcsmax = 100; count_in = 0; seed = 4711; srand(seed); for(mcs=1;mcs<=mcsmax;mcs++) { x = ( float) rand() / RAND_MAX; y = ( float) rand() / RAND_MAX; if ( x*x + y*y <= 1) count_in++; } pi = 4* ( float) count_in / (float) mcsmax; printf("%f\n",pi); return 0;}


Some History

Some History

Note on census-taking in Monte Carlo calculationsE. Fermi and R.D. Richtmyer 1948A declassified report by Enrico Fermi. From the Los Alamos Archive

The Monte Carlo Method N. Metropolis and S. Ulam 1949Journal of the American Statistical Association, 44, 335 (1949)

The name apparently originates with Metropolis, who says that heinvented it in 1945 inspired by Stan Ulams gambling uncle who “justhad to go to Monte Carlo”[Metropolis, N. “The Beginning of theMonte Carlo Method”. Los Alamos Science, No. 15, p. 125,http://jackman.stanford.edu/mcmc/metropolis1.pdf ]


Some History


Literature

General literature

K. Binder and D. W. Heermann, Monte Carlo Simulation inStatistical Physics : An Introduction (Springer Series in Solid-StateSciences, 80), Springer-Verlag, 1988, 4th edition , Springer Verlag,Heidelberg, 2002 (5th edition will be coming out soon, 2009)

L. Bauwens, Bayesian Full Information Analysis of SimultaneousEquation Models Using Integration by Monte Carlo (Lecture Notes inEconomics and Mathematical Systems, 232), Springer-Verlag, 1984

P. Bremaud, Markov Chains : Gibbs Fields, Monte Carlo Simulation,and Queues (Texts in Applied Mathematics, 31), Springer-Verlag,1999

M. H. Chen and Q. M. Shao, Monte Carlo Methods in BayesianComputation (Springer Series in Statistics), Springer-Verlag, 2000


Literature

A. Doucet, N.de Freitas and N. Gordon, Sequential Monte CarloMethods in Practice (Statistics for Engineering and InformationScience), Springer-Verlag, 2001

A. Dubi, Monte Carlo Applications in Systems Engineering, Wiley,2000

M. J. Evans and T. Swartz, Approximating Integrals Via Monte Carloand Deterministic Methods (Oxford Statistical Science Series, 20),Oxford University Press, 2000

K. T. Fang and Y. Wang, Number-Theoretic Methods in Statistics,Chapman and Hall, 1993.

K. T. Fang, F. J. Hickernell and H. Niederreiter, Monte Carlo andQuasi-Monte Carlo Methods 2000 : Proceedings of a Conference heldat Hong Kong Baptist University, Hong Kong SAR, China, Nov.27-Dec.1, 2000, Springer-Verlag, 2002

G. S. Fishman, Monte Carlo : Concepts, Algorithms, and Applications(Springer Series in Operations Research), Springer-Verlag, 1996


Literature

B. L. Fox, Strategies for Quasi-Monte Carlo (International Series inOperations Research and Management Science, 22), KluwerAcademic Publishers, 1999

D. Gamerman, Markov Chain Monte Carlo : Stochastic Simulation forBayesian Inference, Chapman and Hall, 1997

J. E. Gentle, Random Number Generation and Monte Carlo Methods(Statistics and Computing), Springer-Verlag, 1998

W. R. Gilks, S. Richardson and D.J. Spiegelhalter, Markov ChainMonte Carlo in Practice, Chapman and Hall, 1995

P. Grassberger, W. Nadler and G. T. Barkema, Monte Carlo Approachto Biopolymers and Protein Folding, World Scientific, 1998

J. M. Hammersley and D.C. Handscomb, Monte Carlo Methods,Methuen, London, 1964.

P. Hellekalek and G. Larcher, Random and Quasi-Random Point Sets(Lecture Notes in Statistics, 138), Springer-Verlag, 1998

L. K. Hua and Y. Wang, Applications of Number Theory toNumerical Analysis, Springer-Verlag, 1981.


Literature

C. Jacoboni and P. Lugli, The Monte Carlo Method forSemiconductor Device Simulation (Computational Microelectronics),Springer-Verlag, 1989

M. H. Kalos and P. A. Whitlock, Monte Carlo Methods, Volume 1,Basics, Wiley, 1986

J. P. C. Kleijnen, Statistical techniques in simulation, Parts I and II,Dekker, 1974

A. Kling, F. J. C. Barao, M. Nakagawa, L. Tavora, and P. Vaz,Advanced Monte Carlo Computer Programs for Radiation Transport:Proceedings of the Monte Carlo 2000 Conference, Lisbon, 23- 26October 2000, Springer-Verlag, 2001

D. E. Knuth, The Art of Computer Programming (Vol. 2),Addison-Wesley, 1997.

D. P. Landau and K. Binder, A Guide to Monte Carlo Simulations inStatistical Physics, Cambridge University Press, 2000

W. A. Lester, Jr., Recent Advances in Quantum Monte CarloMethods (Recent Advances in


Literature

J. S. Liu, Monte Carlo Strategies in Scientific Computing (SpringerSeries in Statistics), Springer- Verlag, 2001

B. F. J. Manly, Randomization, Bootstrap and Monte Carlo Methodsin Biology, Chapman and Hall, 1997

I. Manno, Introduction to the Monte-Carlo Method, Budapest :Akademiai Kiado, 1999

G. A. Mikhailov and K. K. Sabelfeld, Optimization of WeightedMonte Carlo Methods (Sprigner Series in Computational Physics),Springer-Verlag, 1992

C. Moglestue, Monte Carlo Simulation of Semiconductor Devices,Kluwer Academic Pulishers, 1993

M. E. J. Newman and G. T. Barkema, Monte Carlo Methods inStatistical Physics, Oxford University Press, 1999

H. Niederreiter, Random Number Generation and Quasi-Monte CarloMethods (CBMS-NSF Regional Conference Series in AppliedMathematics 63), SIAM, 1992


Literature

H. Niederreiter, P. Hellekalek, G. Larcher and P. Zinterhof, MonteCarlo and Quasi-Monte Carlo Methods 1996 (Lecture Notes inStatistics, 127), Springer-Verlag, 1998.

H. Niederreiter and J. Spanier, Monte-Carlo and Quasi-Monte CarloMethods 1998 : Proceedings of a Conference at the ClaremontGraduate University, Claremont, California, USA, June 22-26, 1998,Springer-Verlag, 2000

H. Niederreiter and P. J. S. Shiue, Monte Carlo and Quasi-MonteCarlo Methods in Scientific Computing (Lecture Notes in Statistics,106), Springer-Verlag, 1995

C. P. Robert and G. Casella Monte Carlo Statistical Methods(Springer Texts in Statistics), Springer-Verlag, 1999

R. Y. Rubinstein, Simulation and the Monte Carlo Method, Wiley,1981

R. Y. Rubinstein, Monte Carlo Optimization, Simulation andSensitivity of Queuing Networks, Krieger Publishing, 1992


Literature

K. K. Sabelfeld, Monte Carlo Methods : in Boundary Value Problems(Springer Series in Computational Physics), Springer-Verlag, 1991

I. H. Sloan and S. Joe, Lattice Methods for Multiple Integration,Oxford University Press, 1994.

I. M. Sobol, A Primer for the Monte Carlo Method, CRC Press, 1994

S. Tezuka, Uniform Random Numbers : Theory and Practice (TheKluwer International Series in Engineering and Computer Science,315), Kluwer Academic Publishers, 1995.

D. Vose, Risk Analysis: A Quantitative Guide, Wiley, 2000

G. Winkler, Image Analysis, Random Fields and Dynamic MonteCarlo Methods : A Mathematical Introduction (Applications ofMathematics, 27), Springer-Verlag, 1995


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