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E-raamat: Monte Carlo Methods For Applied Scientists

(Bulgarian Acad Of Sciences, Bulgaria)
  • Formaat: 308 pages
  • Ilmumisaeg: 21-Dec-2007
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789814500142
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Monte Carlo Methods For Applied ScientistsSuurem pilt
  • Formaat: 308 pages
  • Ilmumisaeg: 21-Dec-2007
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789814500142
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The Monte Carlo method is inherently parallel and the extensive and rapid development in parallel computers, computational clusters and grids has resulted in renewed and increasing interest in this method. At the same time there has been an expansion in the application areas and the method is now widely used in many important areas of science including nuclear and semiconductor physics, statistical mechanics and heat and mass transfer.This book attempts to bridge the gap between theory and practice concentrating on modern algorithmic implementation on parallel architecture machines. Although a suitable text for final year postgraduate mathematicians and computational scientists it is principally aimed at the applied scientists: only a small amount of mathematical knowledge is assumed and theorem proving is kept to a minimum, with the main focus being on parallel algorithms development often to applied industrial problems.A selection of algorithms developed both for serial and parallel machines are provided.
Preface vii
Acknowledgements ix
1. Introduction
1
2. Basic Results of Monte Carlo Integration
11
2.1 Convergence and Error Analysis of Monte Carlo Methods
11
2.2 Integral Evaluation
13
2.2.1 Plain (Crude) Monte Carlo Algorithm
13
2.2.2 Geometric Monte Carlo Algorithm
14
2.2.3 Computational Complexity of Monte Carlo Algorithms
15
2.3 Monte Carlo Methods with Reduced Error
16
2.3.1 Separation of Principal Part
16
2.3.2 Integration on a Subdomain
17
2.3.3 Symmetrization of the Integrand
18
2.3.4 Importance Sampling Algorithm
20
2.3.5 Weight Functions Approach
21
2.4 Superconvergent Monte Carlo Algorithms
22
2.4.1 Error Analysis
23
2.4.2 A Simple Example
26
2.5 Adaptive Monte Carlo Algorithms for Practical Computations
29
2.5.1 Superconvergent Adaptive Monte Carlo Algorithm and Error Estimates
30
2.5.2 Implementation of Adaptive Monte Carlo Algorithms. Numerical Tests
34
2.5.3 Discussion
37
2.6 Random Interpolation Quadratures
39
2.7 Some Basic Facts about Quasi-Monte Carlo Methods
43
2.8 Exercises
46
3. Optimal Monte Carlo Method for Multidimensional Integrals of Smooth Functions
49
3.1 Introduction
49
3.2 Description of the Method and Theoretical Estimates
52
3.3 Estimates of the Computational Complexity
55
3.4 Numerical Tests
60
3.5 Concluding Remarks
63
4. Iterative Monte Carlo Methods for Linear Equations
67
4.1 Iterative Monte Carlo Algorithms
68
4.2 Solving Linear Systems and Matrix Inversion
74
4.3 Convergence and Mapping
77
4.4 A Highly Convergent Algorithm for Systems of Linear Algebraic Equations
81
4.5 Balancing of Errors
84
4.6 Estimators
86
4.7 A Refined Iterative Monte Carlo Approach for Linear Systems and Matrix Inversion Problem
88
4.7.1 Formulation of the Problem
88
4.7.2 Refined Iterative Monte Carlo Algorithms
89
4.7.3 Discussion of the Numerical Results
94
4.7.4 Conclusion
99
5. Markov Chain Monte Carlo Methods for Eigenvalue Problems
101
5.1 Formulation of the Problems
103
5.1.1 Bilinear Form of Matrix Powers
104
5.1.2 Eigenvalues of Matrices
104
5.2 Almost Optimal Markov Chain Monte Carlo
106
5.2.1 MC Algorithm for Computing Bilinear Forms of Matrix Powers (v. Akh)
107
5.2.2 MC Algorithm for Computing Extrema' Eigenvalues
109
5.2.3 Robust MC Algorithms
111
5.2.4 Interpolation MC Algorithms
112
5.3 Computational Complexity
115
5.3.1 Method for Choosing the Number of Iterations k
116
5.3.2 Method for Choosing the Number of Chains
117
5.4 Applicability and Acceleration Analysis
118
5.5 Conclusion
131
6. Monte Carlo Methods for Boundary-Value Problems (BVP)
133
6.1 BVP for Elliptic Equations
133
6.2 Grid Monte Carlo Algorithm
134
6.3 Grid-Free Monte Carlo Algorithms
135
6.3.1 Local Integral Representation
136
6.3.2 Monte Carlo Algorithms
144
6.3.3 Parallel Implementation of the Grid-Free Algorithm and Numerical Results
154
6.3.4 Concluding Remarks
159
7. Superconvergent Monte Carlo for Density Function Simulation by B-Splines
161
7.1 Problem Formulation
162
7.2 The Methods
163
7.3 Error Balancing
169
7.4 Concluding Remarks
170
8. Solving Non-Linear Equations
171
8.1 Formulation of the Problems
171
8.2 A Monte Carlo Method For Solving Non-linear Integral Equations of Fredholin 'Pyle
173
8.3 An Efficient Algorithm
179
8.4 Numerical Exami)les
191
9. Algorithmic Efficiency for Different Computer Models
195
9.1 Parallel Efficiency Criterion
195
9.2 Markov Chain Algorithms for Linear Algebra Problems
197
9.3 Algorithms for Boundary Value Problems
204
9.3.1 Algorithm A (Grid Algorithm)
205
9.3.2 Algorithm B (Random Jumps on Mesh Points Algorithm)
208
9.3.3 Algorithm C (Grid-Free Algorithm)
211
9.3.4 Discussion
213
9.3.5 Vector Monte Carlo Algorithms
214
10. Applications for Transport Modeling in Semiconductors and Nanowires 219
10.1 The Boltzmann Transport
219
10.1.1 Numerical Monte Carlo Approach
222
10.1.2 Convergency Proof
224
10.1.3 Error Analysis and Algorithmic Complexity
225
10.2 The Quantum Kinetic Equation
227
10.2.1 Physical Aspects
230
10.2.2 The Monte Carlo Algorithm
233
10.2.3 Monte Carlo Solution
234
10.3 The Wigner Quantum-Transport Equation.
237
10.3.1 The Integral Form of the Wigner Equation
242
10.3.2 The Monte Carlo Algorithm
243
10.3.3 The Neumann Series Convergency
245
10.4 A Grid Computing Application to Modeling of Carrier Transport in Nanowires
247
10.4.1 Physical Model
247
10.4.2 The Monte Carlo Method
249
10.4.3 Grid Implementation and Numerical Resuits
251
10.5 Conclusion
254
Appendix A Jumps on Mesh Octahedra Monte Carlo 257
Appendix B Performance Analysis for Different Monte Carlo Algorithms 263
Appendix C Sample Answers of Exercises 265
Appendix D Symbol Table 273
Bibliography 275
Subject Index 285
Author Index 289


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