简介
Monte Carlo方法是数学、物理及工程技术领域有效的计算手段。本书论述Monte Carlo方法在自然科学中的应用,为应用领域科技人员应用Monte Carlo数值方法给出系统的指导。与同类专著相比,本书更为注重算法的性能分析,并用实例表明Monte Carlo方法在半导体器件模型等实际问题中有着广阔的应用前景。本书还融入了作者在英国Reading大学为计算机科学专业开设的“随机方法和算法”课程的内容,使本书在理论上也有一定深度。
全收由10章和4个附录组成。1.是引论,给出基本概念和定义;2~3.论述Monte Carlo积分,研究函数空间中的最优算法。其中第2章给出基本结果,第3章给出光滑函数的高维积分析最优Monte Carlo方法(包括积分误差的Bakhvalov下界估计及两个算法实现);4~8.分别论述线性方程组的Monte Carlo迭代方法、特征值问题的Markov链Monte Carlo方法、边值问题的Monte Carlo方法、密度函数B样条模拟的超收敛Monte Carlo方法,以及Fredholm非线性积分方程的Monte Carlo方法;9.研究上述不同方法的算法有效性;10.是Monte Carlo方法在半导体和纤导传输模型中的应用。附录给出与正文有关的一些结果的理论证明及公式汇编等。
本书数学预备知识限于工科大学基础数学,与本书主要读者对象相适应。书中包含一些理论证明,但有些重要结果只叙述不证明。本书可供计算机科学及其它自然科学领域有关专业大学生、研究生用作教材,也可供有关科技人员阅读。
朱尧辰,研究员
(中国科学院应用数学研究所)
Zhu Yaochen, Professor
(Institute of Applied Mathematics,CAS
目录
Contents 12
Preface 8
Acknowledgements 10
1. Introduction 18
2. Basic Results of Monte Carlo Integration 28
2.1 Convergence and Error Analysis of Monte Carlo Methods 28
2.2 Integral Evaluation 30
2.2.1 Plain (Crude) Monte Carlo Algorithm 30
2.2.2 Geometric Monte Carlo Algorithm 31
2.2.3 Computational Complexity of Monte Carlo Algorithms 32
2.3 Monte Carlo Methods with Reduced Error 33
2.3.1 Separation of Principal Part 33
2.3.2 Integration on a Subdomain 34
2.3.3 Symmetrization of the Integrand 35
2.3.4 Importance Sampling Algorithm 37
2.3.5 Weight Functions Approach 38
2.4 Superconvergent Monte Carlo Algorithms 39
2.4.1 Error Analysis 40
2.4.2 A Simple Example 43
2.5 Adaptive Monte Carlo Algorithms for Practical Computations 46
2.5.1 Superconvergent Adaptive Monte Carlo Algorithm and Error Estimates 47
2.5.2 Implementation of Adaptive Monte Carlo Algorithms. Numerical Tests 51
2.5.3 Discussion 54
2.6 Random Interpolation Quadratures 56
2.7 Some Basic Facts about Quasi-Monte Carlo Methods 60
2.8 Exercises 63
3. Optimal Monte Carlo Method for Multidimensional Integrals of Smooth Functions 66
3.1 Introduction 66
3.2 Description of the Method and Theoretical Estimates 69
3.3 Estimates of the Computational Complexity 72
3.4 Numerical Tests 77
3.5 Concluding Remarks 80
4. Iterative Monte Carlo Methods for Linear Equations 84
4.1 Iterative Monte Carlo Algorithms 85
4.2 Solving Linear Systems and Matrix Inversion 91
4.3 Convergence and Mapping 94
4.4 A Highly Convergent Algorithm for Systems of Linear Algebraic Equations 98
4.5 Balancing of Errors 101
4.6 Estimators 103
4.7 A Re ned Iterative Monte Carlo Approach for Linear Systems and Matrix Inversion Problem 105
4.7.1 Formulation of the Problem 105
4.7.2 Re ned Iterative Monte Carlo Algorithms 106
4.7.3 Discussion of the Numerical Results 111
4.7.4 Conclusion 116
5. Markov Chain Monte Carlo Methods for Eigenvalue Problems 118
5.1 Formulation of the Problems 120
5.1.1 Bilinear Form of Matrix Powers 121
5.1.2 Eigenvalues of Matrices 121
5.2 Almost Optimal Markov Chain Monte Carlo 123
5.2.1 MC Algorithm for Computing Bilinear Forms of Matrix Powers (v;Akh) 124
5.2.2 MC Algorithm for Computing Extremal Eigenvalues 126
5.2.3 Robust MC Algorithms 128
5.2.4 Interpolation MC Algorithms 129
5.3 Computational Complexity 132
5.3.1 Method for Choosing the Number of Iterations k 133
5.3.2 Method for Choosing the Number of Chains 134
5.4 Applicability and Acceleration Analysis 135
5.5 Conclusion 148
6. Monte Carlo Methods for Boundary-Value Problems (BVP) 150
6.1 BVP for Elliptic Equations 150
6.2 Grid Monte Carlo Algorithm 151
6.3 Grid-Free Monte Carlo Algorithms 152
6.3.1 Local Integral Representation 153
6.3.2 Monte Carlo Algorithms 161
6.3.3 Parallel Implementation of the Grid-Free Algorithm and Numerical Results 171
6.3.4 Concluding Remarks 176
7. Superconvergent Monte Carlo for Density Function Simulation by B-Splines 178
7.1 Problem Formulation 179
7.2 The Methods 180
7.3 Error Balancing 186
7.4 Concluding Remarks 187
8. Solving Non-Linear Equations 188
8.1 Formulation of the Problems 188
8.2 A Monte Carlo Method for Solving Non-linear Integral Equations of Fredholm Type 190
8.3 An Efficient Algorithm 196
8.4 Numerical Examples 208
9. Algorithmic Efficiency for Different Computer Models 212
9.1 Parallel Efficiency Criterion 212
9.2 Markov Chain Algorithms for Linear Algebra Problems 214
9.3 Algorithms for Boundary Value Problems 221
9.3.1 Algorithm A (Grid Algorithm) 222
9.3.2 Algorithm B (Random Jumps on Mesh Points Algorithm) 225
9.3.3 Algorithm C (Grid-Free Algorithm) 228
9.3.4 Discussion 230
9.3.5 Vector Monte Carlo Algorithms 231
10. Applications for Transport Modeling in Semiconductors and Nanowires 236
10.1 The Boltzmann Transport 236
10.1.1 Numerical Monte Carlo Approach 239
10.1.2 Convergency Proof 241
10.1.3 Error Analysis and Algorithmic Complexity 242
10.2 The Quantum Kinetic Equation 244
10.2.1 Physical Aspects 247
10.2.2 The Monte Carlo Algorithm 250
10.2.3 Monte Carlo Solution 251
10.3 The Wigner Quantum-Transport Equation 254
10.3.1 The Integral Form of the Wigner Equation 259
10.3.2 The Monte Carlo Algorithm 260
10.3.3 The Neumann Series Convergency 262
10.4 A Grid Computing Application to Modeling of Carrier Transport in Nanowires 264
10.4.1 Physical Model 264
10.4.2 The Monte Carlo Method 266
10.4.3 Grid Implementation and Numerical Results 268
10.5 Conclusion 271
Appendix A Jumps on Mesh Octahedra Monte Carlo 274
Appendix B Performance Analysis for Different Monte Carlo Algorithms 280
Appendix C Sample Answers of Exercises 282
Appendix D Symbol Table 290
Bibliography 292
Subject Index 302
Author Index 306
Preface 8
Acknowledgements 10
1. Introduction 18
2. Basic Results of Monte Carlo Integration 28
2.1 Convergence and Error Analysis of Monte Carlo Methods 28
2.2 Integral Evaluation 30
2.2.1 Plain (Crude) Monte Carlo Algorithm 30
2.2.2 Geometric Monte Carlo Algorithm 31
2.2.3 Computational Complexity of Monte Carlo Algorithms 32
2.3 Monte Carlo Methods with Reduced Error 33
2.3.1 Separation of Principal Part 33
2.3.2 Integration on a Subdomain 34
2.3.3 Symmetrization of the Integrand 35
2.3.4 Importance Sampling Algorithm 37
2.3.5 Weight Functions Approach 38
2.4 Superconvergent Monte Carlo Algorithms 39
2.4.1 Error Analysis 40
2.4.2 A Simple Example 43
2.5 Adaptive Monte Carlo Algorithms for Practical Computations 46
2.5.1 Superconvergent Adaptive Monte Carlo Algorithm and Error Estimates 47
2.5.2 Implementation of Adaptive Monte Carlo Algorithms. Numerical Tests 51
2.5.3 Discussion 54
2.6 Random Interpolation Quadratures 56
2.7 Some Basic Facts about Quasi-Monte Carlo Methods 60
2.8 Exercises 63
3. Optimal Monte Carlo Method for Multidimensional Integrals of Smooth Functions 66
3.1 Introduction 66
3.2 Description of the Method and Theoretical Estimates 69
3.3 Estimates of the Computational Complexity 72
3.4 Numerical Tests 77
3.5 Concluding Remarks 80
4. Iterative Monte Carlo Methods for Linear Equations 84
4.1 Iterative Monte Carlo Algorithms 85
4.2 Solving Linear Systems and Matrix Inversion 91
4.3 Convergence and Mapping 94
4.4 A Highly Convergent Algorithm for Systems of Linear Algebraic Equations 98
4.5 Balancing of Errors 101
4.6 Estimators 103
4.7 A Re ned Iterative Monte Carlo Approach for Linear Systems and Matrix Inversion Problem 105
4.7.1 Formulation of the Problem 105
4.7.2 Re ned Iterative Monte Carlo Algorithms 106
4.7.3 Discussion of the Numerical Results 111
4.7.4 Conclusion 116
5. Markov Chain Monte Carlo Methods for Eigenvalue Problems 118
5.1 Formulation of the Problems 120
5.1.1 Bilinear Form of Matrix Powers 121
5.1.2 Eigenvalues of Matrices 121
5.2 Almost Optimal Markov Chain Monte Carlo 123
5.2.1 MC Algorithm for Computing Bilinear Forms of Matrix Powers (v;Akh) 124
5.2.2 MC Algorithm for Computing Extremal Eigenvalues 126
5.2.3 Robust MC Algorithms 128
5.2.4 Interpolation MC Algorithms 129
5.3 Computational Complexity 132
5.3.1 Method for Choosing the Number of Iterations k 133
5.3.2 Method for Choosing the Number of Chains 134
5.4 Applicability and Acceleration Analysis 135
5.5 Conclusion 148
6. Monte Carlo Methods for Boundary-Value Problems (BVP) 150
6.1 BVP for Elliptic Equations 150
6.2 Grid Monte Carlo Algorithm 151
6.3 Grid-Free Monte Carlo Algorithms 152
6.3.1 Local Integral Representation 153
6.3.2 Monte Carlo Algorithms 161
6.3.3 Parallel Implementation of the Grid-Free Algorithm and Numerical Results 171
6.3.4 Concluding Remarks 176
7. Superconvergent Monte Carlo for Density Function Simulation by B-Splines 178
7.1 Problem Formulation 179
7.2 The Methods 180
7.3 Error Balancing 186
7.4 Concluding Remarks 187
8. Solving Non-Linear Equations 188
8.1 Formulation of the Problems 188
8.2 A Monte Carlo Method for Solving Non-linear Integral Equations of Fredholm Type 190
8.3 An Efficient Algorithm 196
8.4 Numerical Examples 208
9. Algorithmic Efficiency for Different Computer Models 212
9.1 Parallel Efficiency Criterion 212
9.2 Markov Chain Algorithms for Linear Algebra Problems 214
9.3 Algorithms for Boundary Value Problems 221
9.3.1 Algorithm A (Grid Algorithm) 222
9.3.2 Algorithm B (Random Jumps on Mesh Points Algorithm) 225
9.3.3 Algorithm C (Grid-Free Algorithm) 228
9.3.4 Discussion 230
9.3.5 Vector Monte Carlo Algorithms 231
10. Applications for Transport Modeling in Semiconductors and Nanowires 236
10.1 The Boltzmann Transport 236
10.1.1 Numerical Monte Carlo Approach 239
10.1.2 Convergency Proof 241
10.1.3 Error Analysis and Algorithmic Complexity 242
10.2 The Quantum Kinetic Equation 244
10.2.1 Physical Aspects 247
10.2.2 The Monte Carlo Algorithm 250
10.2.3 Monte Carlo Solution 251
10.3 The Wigner Quantum-Transport Equation 254
10.3.1 The Integral Form of the Wigner Equation 259
10.3.2 The Monte Carlo Algorithm 260
10.3.3 The Neumann Series Convergency 262
10.4 A Grid Computing Application to Modeling of Carrier Transport in Nanowires 264
10.4.1 Physical Model 264
10.4.2 The Monte Carlo Method 266
10.4.3 Grid Implementation and Numerical Results 268
10.5 Conclusion 271
Appendix A Jumps on Mesh Octahedra Monte Carlo 274
Appendix B Performance Analysis for Different Monte Carlo Algorithms 280
Appendix C Sample Answers of Exercises 282
Appendix D Symbol Table 290
Bibliography 292
Subject Index 302
Author Index 306
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