This book provides practical demonstrations of how to carry out definite integrals with Monte Carlo methods using Mathematica. Random variates are sampled by the inverse transform method and the acceptance-rejection method using uniform, linear, Gaussian, and exponential probability distribution functions. A chapter on the application of the Variational Quantum Monte Carlo method to a simple harmonic oscillator is included. These topics are all essential for students of mathematics and physics. The author includes thorough background on each topic covered within the book in order to help readers understand the subject. The book also contains many examples to show how the methods can be applied.
Author(s): Sujaul Chowdhury
Series: Synthesis Lectures on Mathematics & Statistics
Publisher: Springer
Year: 2023
Language: English
Pages: 150
City: Cham
1
Preface
Contents
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Part I Monte Carlo Methods Using Inverse Transform Sampling Utilizing Mathematica
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1 Introduction for Part I
1.1 Random Variable
1.2 Continuous Random Variable
1.3 Uniform Random Variable
1.4 Normal or Gaussian Random Variable
1.5 Generating Random Variates by Inverse Transform Method
1.6 Variance Reduction and Importance Sampling
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2 Evaluation of Definite Integrals Using Inverse Transform Sampling Utilizing Mathematica
2.1 Evaluation of Definite Integrals Using Inverse Transform Sampling: Example I
2.2 Evaluation of Definite Integrals Using Inverse Transform Sampling: Example II
2.3 Evaluation of Definite Integrals Using Inverse Transform Sampling: Example III
2.4 Evaluation of Definite Integrals Using Inverse Transform Sampling: Example IV
2.5 Evaluation of Definite Integrals Using Inverse Transform Sampling: Example V
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Part II Monte Carlo Methods Using Acceptance-Rejection Sampling Utilizing Mathematica
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3 Introduction for Part II
3.1 Random Variable
3.2 Continuous Random Variable
3.3 Uniform Random Variable
3.4 Normal or Gaussian Random Variable
3.5 Generating Random Variates by Acceptance-Rejection Method
3.6 Variance Reduction and Importance Sampling
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4 Evaluation of Definite Integrals Using Acceptance-Rejection Sampling Utilizing Mathematica
4.1 Evaluation of Definite Integrals Using Acceptance-Rejection Sampling: Example I
4.2 Evaluation of Definite Integrals Using Acceptance-Rejection Sampling: Example II
4.3 Evaluation of Definite Integrals Using Acceptance-Rejection Sampling: Example III
4.4 Evaluation of Definite Integrals Using Acceptance-Rejection Sampling: Example IV
4.5 Evaluation of Definite Integrals Using Acceptance-Rejection Sampling: Example V
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5 Variational Monte Carlo Method Applied to Ground State of Simple Harmonic Oscillator Using Acceptance-Rejection Sampling Utilizing Mathematica
5.1 The Variational Method of Quantum Mechanics Applied to Ground State of Any Quantum Mechanical System
5.2 The Variational Method of Quantum Mechanics Applied to Ground State of Simple Harmonic Oscillator
5.3 Ground State Energy of Simple Harmonic Oscillator Using Variational Quantum Monte Carlo Method and Acceptance-Rejection Sampling
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Concluding Remarks
Appendix: Handouts for Computational Lab and Automated Collection of Accepted Random Variates
A.1 Experiment I: Monte Carlo Integration Using Inverse Transform Sampling Using (a) Mathematica and (b) MS Excel
A.2 Experiment II: Monte Carlo Integration Using Acceptance-Rejection Sampling, Gaussian Pdf and Mathematica
A.3 Automated Collection of Accepted Random Variates
References
