Explorations in Monte Carlo methods:
Gespeichert in:
| Hauptverfasser: | , |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Dordrecht ; Heidelberg ; London ; New York
Springer
[2009]
|
| Schriftenreihe: | Undergraduate texts in mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | xii, 243 Seiten Illustrationen |
| ISBN: | 9780387878362 |
Internformat
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| 245 | 1 | 0 | |a Explorations in Monte Carlo methods |c Ronald W. Shonkwiler ; Franklin Mendivil |
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Datensatz im Suchindex
| _version_ | 1804138873752649728 |
|---|---|
| adam_text | Titel: Explorations in Monte Carlo methods
Autor: Shonkwiler, Ronald W.
Jahr: 2009
Contents
Preface
v
1 Introduction to Monte Carlo Methods............... 1
1.1 How Can Random Numbers Solve Problems?.......... 1
1.1.1 History of the Monte Carlo Method............. 2
1.1.2 Histogramming Simulation Results.............. 4
1.1.3 Sample Paths................................ 8
1.2 Some Basic Probability............................. 10
1.2.1 Events and Random Variables.................. 10
1.2.2 Discrete and Continuous Random Variables...... 11
1.2.3 The Probability Density Function............... 14
1.2.4 Expected Values.............................. 16
1.2.5 Conditional Probabilities...................... 20
1.2.6 Variance for a Sum of Random Variables-
Joint Probability Densities..................... 24
1.3 Random Number Generation........................ 28
1.3.1 Requirements for a Random Number Generator
(RNG)...................................... 28
1.3.2 Middle-Square and Other Middle-Digit Techniques 31
1.3.3 Linear Congruential Random Number Generators. 32
1.4 Some Applications ................................. 37
Problems: Chapter 1.................................... 43
2 Some Probability Distributions and Their Uses...... 51
2.1 CDF Inversion-Discrete Case: Bernoulli Trials......... 51
2.1.1 Two-Outcome CDF Inversion.................. 52
2.1.2 Multiple-Outcome Distributions................ 53
Contents
2.2 Walker s Alias Method: Roulette Wheel Selection ...... 55
2.3 Probability Simulation: The Binomial Distribution..... 57
2.3.1 Sampling from the Binomial................... 60
2.4 Another Simulation: The Poisson Distribution......... 60
2.4.1 Sampling from the Poisson Distribution by
Simulation................................... 62
2.5 CDF Inversion, Continuous Case: The Exponential
Distribution....................................... 64
2.5.1 Inverting the CDF—The Canonical Method for
the Exponential.............................. 66
2.5.2 Discrete Event Simulation..................... 68
2.5.3 Transforming Random Variables: The Cauchy
Distribution.................................. 70
2.6 The Central Limit Theorem and the Normal Distribution 72
2.6.1 Sampling from the Normal Distribution ......... 75
2.6.2 Approximate Sampling via the Central Limit
Theorem .................................... 77
2.6.3 Error Estimates for Monte Carlo Simulations..... 79
2.7 Rejection Sampling Example: The Beta Distribution .... 81
2.7.1 Sampling from the Beta Distribution............ 85
2.7.2 Sampling from an Unbounded Beta Distribution .. 87
2.8 Composition Example: The Gamma Distribution ....... 88
2.8.1 The Gamma Distribution...................... 89
2.8.2 Sampling from G(a, 1) ........................ 90
Problems: Chapter 2.................................... 93
Markov Chain Monte Carlo..........................101
3.1 Discrete Markov Chains.............................101
3.1.1 Random Walk on a Graph.....................101
3.1.2 Matrix Representation of a Chain...............105
3.2 Markov Chain Monte Carlo Sampling—The Metropolis
Algorithm.........................................107
3.2.1 Some Examples ..............................109
3.2.2 Why Does the Metropolis Algorithm Work?......112
3.3 MCMC Sampling and the Ergodic Theorem...........114
3.4 Statistical Mechanics...............................115
3.5 Ising Model and the Metropolis Algorithm ............121
3.6 Counting..........................................126
3.7 Applications.......................................129
3.7.1 Shuffling with Constraints.....................129
Contents xi
3.7.2 Coupling from the Past........................131
Problems: Chapter 3....................................135
Optimization by Monte Carlo Methods..............139
4.1 Simulated Annealing................................140
4.2 Application of SA to the Traveling Salesman Problem... 143
4.3 Genetic Algorithms.................................147
4.4 An Application of GA to Function Maximization.......149
4.5 An Application of GA to the Permanent Problem......152
Problems: Chapter 4....................................158
Random Walks.......................................165
5.1 Diffusion..........................................165
5.2 Brownian Motion ..................................171
5.3 Random Walk Applications I........................174
5.3.1 Options Pricing in Finance ....................174
5.3.2 Self-Avoiding Walks...........................180
5.4 Gambler s Ruin....................................181
5.4.1 Gambling Schemes............................185
5.5 Random Walk Applications II—Kelly s Criterion in
Finance...........................................186
5.5.1 The Simple Kelly Game.......................187
5.5.2 The Simple Game with Catastrophic Loss.......188
5.5.3 Option Trading Application I..................191
5.5.4 Option Trading Application II..................192
5.6 Random Walks and Electrical Networks...............196
5.6.1 Markov Chain Solution for Voltages.............200
5.6.2 The Fundamental Matrix and Expected Hitting
Times.......................................203
Problems: Chapter 5....................................208
Generating Uniform Random Numbers..............213
A.I Multiple Stored Value Random Number Generation-----213
A.1.1 Fibonacci Generators .........................213
A.1.2 Finite Field RNG.............................215
A.2 Testing for Nonrandomness..........................218
A.2.1 Chi-Square Test..............................218
A.2.2 Kolmogorov-Smirnov Test.....................219
Problems: Appendix A..................................222
xii Contents
B Perron—Frobenius Theorem..........................225
B.I Proof of Perron-Frobenius...........................225
C Projects .............................................229
References...............................................235
List of Notation.........................................239
Index....................................................241
UNDERGRADUATE TEXTS IN MATHEMATICS
Monte Carlo methods are among the most used and useful computational tools
available today, providing efficient and practical algorithims to solve a wide range
of scientific and engineering problems. Applications covered in this book include
optimization, finance, statistical mechanics, birth and death processes, and gam¬
bling systems.
Explorations in Monte Carlo Methods provides a hands-on approach to learning
this subject. Each new idea is carefully motivated by a realistic problem, thus lead¬
ing from questions to theory via examples and numerical simulations. Program¬
ming exercises are integrated throughout the text as the primary vehicle for
learning the material. Each chapter ends with a large collection of problems illus¬
trating and directing the material.
This book is suitable as a textbook for students of engineering and the sciences,
as well as mathematics. The problem-oriented approach makes it ideal for an ap¬
plied course in basic probability and for a more specialized course in Monte Carlo
methods. Topics include probability distributions, counting combinatorial ob¬
jects, simulated annealing, genetic algorithms, option pricing, gamblers ruin, sta¬
tistical mechanics, sampling, and random number generation.
|
| any_adam_object | 1 |
| author | Shonkwiler, Ronald W. 1942- Mendivil, Franklin |
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| id | DE-604.BV035431505 |
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| indexdate | 2024-07-09T21:35:07Z |
| institution | BVB |
| isbn | 9780387878362 |
| language | English |
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| physical | xii, 243 Seiten Illustrationen |
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| spelling | Shonkwiler, Ronald W. 1942- (DE-588)114726590 aut Explorations in Monte Carlo methods Ronald W. Shonkwiler ; Franklin Mendivil Dordrecht ; Heidelberg ; London ; New York Springer [2009] © 2009 xii, 243 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Undergraduate texts in mathematics Monte Carlo method Monte-Carlo-Simulation (DE-588)4240945-7 gnd rswk-swf Monte-Carlo-Simulation (DE-588)4240945-7 s DE-604 Mendivil, Franklin (DE-588)139391312 aut Erscheint auch als Online-Ausgabe 978-0-387-87837-9 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017351888&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017351888&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Shonkwiler, Ronald W. 1942- Mendivil, Franklin Explorations in Monte Carlo methods Monte Carlo method Monte-Carlo-Simulation (DE-588)4240945-7 gnd |
| subject_GND | (DE-588)4240945-7 |
| title | Explorations in Monte Carlo methods |
| title_auth | Explorations in Monte Carlo methods |
| title_exact_search | Explorations in Monte Carlo methods |
| title_full | Explorations in Monte Carlo methods Ronald W. Shonkwiler ; Franklin Mendivil |
| title_fullStr | Explorations in Monte Carlo methods Ronald W. Shonkwiler ; Franklin Mendivil |
| title_full_unstemmed | Explorations in Monte Carlo methods Ronald W. Shonkwiler ; Franklin Mendivil |
| title_short | Explorations in Monte Carlo methods |
| title_sort | explorations in monte carlo methods |
| topic | Monte Carlo method Monte-Carlo-Simulation (DE-588)4240945-7 gnd |
| topic_facet | Monte Carlo method Monte-Carlo-Simulation |
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