| Description |
1 online resource |
| 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 |
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2.4.1 Error Analysis2.4.2 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 |
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3.4 Numerical Tests3.5 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 |
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5. Markov Chain Monte Carlo Methods for Eigenvalue Problems5.1 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 |
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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 |
| Summary |
The Monte Carlo method is inherently parallel and the extensive and rapid development in vector and parallel computers 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 work 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 or postgraduate mathematicians 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 algorithm development often to applied industrial problems. Algorithms are developed both for MIMD machines with distributed memory and SIMD machines; a selection of programs are provided |
| Subject |
Monte Carlo method.
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Monte Carlo method
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| Form |
Electronic book
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| Author |
McKee, Sean.
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| ISBN |
128193819X |
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9781281938190 |
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9789812779892 |
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9812779892 |
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