Monte Carlo Methods for Applied Scientists

The following sections are included:

• Preface

• Acknowledgements

• Contents

Monte Carlo methods are a powerful tool in many fields of mathematics, physics and engineering. It is known that the algorithms based on this method give statistical estimates for any linear functional of the solution by performing random sampling of a certain random variable (r.v.) whose mathematical expectation is the desired functional.

In [Binder and Heermann (1998); Bremaud (1999); Berg (2004); Chen and Shao (2000); Dubi (2000); Ermakov and Mikhailov (1982); Evans and Swartz (2000); Fishman (1995); MacKeown (1997); Gilks et al. (1995); Gould and Tobochnik (1989); Hellekalek and Larcher (1998); Jacoboni and Lugli (1989); Landau and Binder (2000); Liu (2001); Manly (1997); Manno (1999); Mikhailov and Sabelfeld (1992); Newman and Barkema (1999); Rubinstein (1992); Robert and Casella (2004); Sabelfeld (1991); Shreider (1966); Sloan and Joe (1994); Sobol (1973)] one can find various definitions and different understanding of what Monte Carlo methods are. Nevertheless, in all above mentioned works there is a common understanding that it's a method which uses r.v. or random processes to find an approximate solution to the problem. We use the following definition.

The following sections are included:

• Convergence and Error Analysis of Monte Carlo Methods

• Integral Evaluation
  • Plain (Crude) Monte Carlo Algorithm
  • Geometric Monte Carlo Algorithm
  • Computational Complexity of Monte Carlo Algorithms

• Monte Carlo Methods with Reduced Error
  • Separation of Principal Part
  • Integration on a Subdomain
  • Symmetrization of the Integrand
  • Importance Sampling Algorithm
  • Weight Functions Approach

• Superconvergent Monte Carlo Algorithms
  • Error Analysis
  • A Simple Example

• Adaptive Monte Carlo Algorithms for Practical Computations
  • Superconvergent Adaptive Monte Carlo Algorithm and Error Estimates
  • Implementation of Adaptive Monte Carlo Algorithms. Numerical Tests
  • Discussion

• Random Interpolation Quadratures

• Some Basic Facts about Quasi-Monte Carlo Methods

• Exercises

The following sections are included:

• Introduction

• Description of the Method and Theoretical Estimates

• Estimates of the Computational Complexity

• Numerical Tests

• Concluding Remarks

The following sections are included:

• Iterative Monte Carlo Algorithms

• Solving Linear Systems and Matrix Inversion

• Convergence and Mapping

• A Highly Convergent Algorithm for Systems of Linear Algebraic Equations

• Balancing of Errors

• Estimators

• A Refined Iterative Monte Carlo Approach for Linear Systems and Matrix Inversion Problem
  • Formulation of the Problem
  • Refined Iterative Monte Carlo Algorithms
  • Discussion of the Numerical Results
  • Conclusion

The following sections are included:

• Formulation of the Problems
  • Bilinear Form of Matrix Powers
  • Eigenvalues of Matrices

• Almost Optimal Markov Chain Monte Carlo
  • MC Algorithm for Computing Bilinear Forms of Matrix Powers (v, A^kh)
  • MC Algorithm for Computing Extremal Eigenvalues
  • Robust MC Algorithms
  • Interpolation MC Algorithms

• Computational Complexity
  • Method for Choosing the Number of Iterations k
  • Method for Choosing the Number of Chains

• Applicability and Acceleration Analysis

• Conclusion

The following sections are included:

• BVP for Elliptic Equations

• Grid Monte Carlo Algorithm

• Grid-Free Monte Carlo Algorithms
  • Local Integral Representation
  • Monte Carlo Algorithms
  • Parallel Implementation of the Grid-Free Algorithm and Numerical Results
  • Concluding Remarks

The following sections are included:

• Problem Formulation

• The Methods

• Error Balancing

• Concluding Remarks

The following sections are included:

• Formulation of the Problems

• A Monte Carlo Method for Solving Non-linear Integral Equations of Fredholm Type

• An Efficient Algorithm

• Numerical Examples

The following sections are included:

• Parallel Efficiency Criterion

• Markov Chain Algorithms for Linear Algebra Problems

• Algorithms for Boundary Value Problems
  • Algorithm A (Grid Algorithm)
  • Algorithm B (Random Jumps on Mesh Points Algorithm)
  • Algorithm C (Grid-Free Algorithm)
  • Discussion
  • Vector Monte Carlo Algorithms

The following sections are included:

• The Boltzmann Transport
  • Numerical Monte Carlo Approach
  • Convergency Proof
  • Error Analysis and Algorithmic Complexity

• The Quantum Kinetic Equation
  • Physical Aspects
  • The Monte Carlo Algorithm
  • Monte Carlo Solution

• The Wigner Quantum-Transport Equation
  • The Integral Form of the Wigner Equation
  • The Monte Carlo Algorithm
  • The Neumann Series Convergency

• A Grid Computing Application to Modeling of Carrier Transport in Nanowires
  • Physical Model
  • The Monte Carlo Method
  • Grid Implementation and Numerical Results

• Conclusion

The following sections are included:

• 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