Contents -- Preface -- Acknowledgements -- 1. Introduction -- 2. Basic Results of Monte Carlo Integration -- 2.1 Convergence and Error Analysis of Monte Carlo Methods -- 2.2 Integral Evaluation -- 2.2.1 Plain (Crude) Monte Carlo Algorithm -- 2.2.2 Geometric Monte Carlo Algorithm -- 2.2.3 Computational Complexity of Monte Carlo Algorithms -- 2.3 Monte Carlo Methods with Reduced Error -- 2.3.1 Separation of Principal Part -- 2.3.2 Integration on a Subdomain -- 2.3.3 Symmetrization of the Integrand -- 2.3.4 Importance Sampling Algorithm -- 2.3.5 Weight Functions Approach -- 2.4 Superconvergent Monte Carlo Algorithms -- 2.4.1 Error Analysis -- 2.4.2 A Simple Example -- 2.5 Adaptive Monte Carlo Algorithms for Practical Computations -- 2.5.1 Superconvergent Adaptive Monte Carlo Algorithm and Error Estimates -- 2.5.2 Implementation of Adaptive Monte Carlo Algorithms. Numerical Tests -- 2.5.3 Discussion -- 2.6 Random Interpolation Quadratures -- 2.7 Some Basic Facts about Quasi-Monte Carlo Methods -- 2.8 Exercises -- 3. Optimal Monte Carlo Method for Multidimensional Integrals of Smooth Functions -- 3.1 Introduction -- 3.2 Description of the Method and Theoretical Estimates -- 3.3 Estimates of the Computational Complexity -- 3.4 Numerical Tests -- 3.5 Concluding Remarks -- 4. Iterative Monte Carlo Methods for Linear Equations -- 4.1 Iterative Monte Carlo Algorithms -- 4.2 Solving Linear Systems and Matrix Inversion -- 4.3 Convergence and Mapping -- 4.4 A Highly Convergent Algorithm for Systems of Linear Algebraic Equations -- 4.5 Balancing of Errors -- 4.6 Estimators -- 4.7 A Re ned Iterative Monte Carlo Approach for Linear Systems and Matrix Inversion Problem -- 4.7.1 Formulation of the Problem -- 4.7.2 Re ned Iterative Monte Carlo Algorithms -- 4.7.3 Discussion of the Numerical Results -- 4.7.4 Conclusion.

5. Markov Chain Monte Carlo Methods for Eigenvalue Problems -- 5.1 Formulation of the Problems -- 5.1.1 Bilinear Form of Matrix Powers -- 5.1.2 Eigenvalues of Matrices -- 5.2 Almost Optimal Markov Chain Monte Carlo -- 5.2.1 MC Algorithm for Computing Bilinear Forms of Matrix Powers (v -- Akh) -- 5.2.2 MC Algorithm for Computing Extremal Eigenvalues -- 5.2.3 Robust MC Algorithms -- 5.2.4 Interpolation MC Algorithms -- 5.3 Computational Complexity -- 5.3.1 Method for Choosing the Number of Iterations k -- 5.3.2 Method for Choosing the Number of Chains -- 5.4 Applicability and Acceleration Analysis -- 5.5 Conclusion -- 6. Monte Carlo Methods for Boundary-Value Problems (BVP) -- 6.1 BVP for Elliptic Equations -- 6.2 Grid Monte Carlo Algorithm -- 6.3 Grid-Free Monte Carlo Algorithms -- 6.3.1 Local Integral Representation -- 6.3.2 Monte Carlo Algorithms -- 6.3.3 Parallel Implementation of the Grid-Free Algorithm and Numerical Results -- 6.3.4 Concluding Remarks -- 7. Superconvergent Monte Carlo for Density Function Simulation by B-Splines -- 7.1 Problem Formulation -- 7.2 The Methods -- 7.3 Error Balancing -- 7.4 Concluding Remarks -- 8. Solving Non-Linear Equations -- 8.1 Formulation of the Problems -- 8.2 A Monte Carlo Method for Solving Non-linear Integral Equations of Fredholm Type -- 8.3 An Efficient Algorithm -- 8.4 Numerical Examples -- 9. Algorithmic Efficiency for Different Computer Models -- 9.1 Parallel Efficiency Criterion -- 9.2 Markov Chain Algorithms for Linear Algebra Problems -- 9.3 Algorithms for Boundary Value Problems -- 9.3.1 Algorithm A (Grid Algorithm) -- 9.3.2 Algorithm B (Random Jumps on Mesh Points Algorithm) -- 9.3.3 Algorithm C (Grid-Free Algorithm) -- 9.3.4 Discussion -- 9.3.5 Vector Monte Carlo Algorithms -- 10. Applications for Transport Modeling in Semiconductors and Nanowires -- 10.1 The Boltzmann Transport.

10.1.1 Numerical Monte Carlo Approach -- 10.1.2 Convergency Proof -- 10.1.3 Error Analysis and Algorithmic Complexity -- 10.2 The Quantum Kinetic Equation -- 10.2.1 Physical Aspects -- 10.2.2 The Monte Carlo Algorithm -- 10.2.3 Monte Carlo Solution -- 10.3 The Wigner Quantum-Transport Equation -- 10.3.1 The Integral Form of the Wigner Equation -- 10.3.2 The Monte Carlo Algorithm -- 10.3.3 The Neumann Series Convergency -- 10.4 A Grid Computing Application to Modeling of Carrier Transport in Nanowires -- 10.4.1 Physical Model -- 10.4.2 The Monte Carlo Method -- 10.4.3 Grid Implementation and Numerical Results -- 10.5 Conclusion -- Appendix A Jumps on Mesh Octahedra Monte Carlo -- Appendix B Performance Analysis for Different Monte Carlo Algorithms -- Appendix C Sample Answers of Exercises -- Appendix D Symbol Table -- Bibliography -- Subject Index -- Author Index.