Monte Carlo Methods Utilizing Mathematica®: Applications in Inverse Transform and Acceptance-Rejection Sampling

150 Pages·2023·1.165 MB·English

by  Sujaul Chowdhury

Synthesis Lectures on Mathematics
& Statistics
SeriesEditor
StevenG.Krantz,DepartmentofMathematics,WashingtonUniversity,SaintLouis,MO,
USA
This series includes titles in applied mathematics and statistics for cross-disciplinary
STEMprofessionals,educators,researchers,andstudents.Theseriesfocusesonnewand
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Sujaul Chowdhury
Monte Carlo Methods
Utilizing Mathematica®
Applications in Inverse Transform
and Acceptance-Rejection Sampling
SujaulChowdhury
DepartmentofPhysics
ShahjalalUniversityofScience
andTechnology
Sylhet,Bangladesh
ISSN1938-1743 ISSN1938-1751 (electronic)
SynthesisLecturesonMathematics &Statistics
ISBN978-3-031-23293-0 ISBN978-3-031-23294-7 (eBook)
https://doi.org/10.1007/978-3-031-23294-7
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Preface
This book on Monte Carlo Methods Using Inverse Transform and Acceptance-Rejection
Sampling Utilizing Mathematica is intended for undergraduate students of Mathemat-
ics, Statistics, and Physics who wish to know how Monte Carlo methods work with
calculations done using a user-friendly program Mathematica.
All treatments have been done as less manually as practicable. All calculations except
for gathering accepted values of random variate in acceptance-rejection sampling have
been done using Mathematica.
Definite integrals of a number of functions F(x) have been evaluated using uniform,
linear,Gaussianandexponentialprobabilitydensityfunctionsp(x).Itisshownthatresults
agree with known exact values better if p(x) is proportional to F(x). Deviation from the
proportionality has been shown to result in worse agreement.
Allnecessarybackgroundmaterialsarecovered.Aseparatechapterhasbeendedicated
to variational Monte Carlo method applied to ground state of simple harmonic oscillator.
Necessary background on variational method has been covered so that readers need not
consult Quantum Mechanics textbooks.
ThebookisonMonteCarlomethods.ThesearenumericalmethodsforComputational
Physics. These are parts of the syllabus for undergraduate students of Mathematics and
Physics for the course titled “Computational Physics”.
Besides the four referenced books, this is the only book to teach how basic Monte
Carlo methods work. This is possibly the 1st Mathematica-based book on the topic.
Pedagogical features: the book contains practical demonstrations of how to carry out
definite integrals using Monte Carlo methods using Mathematica and using random vari-
ates sampled by inverse transform method and acceptance-rejection method. Moreover,
the book contains a chapter on Variational Quantum Monte Carlo method applied to
simple harmonic oscillator.
v
vi Preface
Key words for the book are: Monte Carlo methods, Mathematica, basic Monte Carlo
integration,VariationalQuantumMonteCarlomethod,simpleharmonicoscillator,inverse
transform sampling, and acceptance-rejection sampling.
Sylhet, Bangladesh Sujaul Chowdhury
2022
Contents
PartI Monte Carlo Methods Using Inverse Transform Sampling
UtilizingMathematica
1 IntroductionforPartI ................................................ 3
1.1 Random Variable ................................................. 3
1.2 Continuous Random Variable ...................................... 3
1.3 Uniform Random Variable ......................................... 4
1.4 Normal or Gaussian Random Variable .............................. 4
1.5 Generating Random Variates by Inverse Transform Method ............ 6
1.6 Variance Reduction and Importance Sampling ....................... 9
2 EvaluationofDefiniteIntegralsUsingInverseTransformSampling
UtilizingMathematica ................................................ 13
2.1 Evaluation of Definite Integrals Using Inverse Transform Sampling:
Example I ....................................................... 13
2.2 Evaluation of Definite Integrals Using Inverse Transform Sampling:
Example II ...................................................... 17
2.3 Evaluation of Definite Integrals Using Inverse Transform Sampling:
Example III ..................................................... 23
2.4 Evaluation of Definite Integrals Using Inverse Transform Sampling:
Example IV ..................................................... 27
2.5 Evaluation of Definite Integrals Using Inverse Transform Sampling:
Example V ...................................................... 31
PartII MonteCarloMethodsUsingAcceptance-RejectionSampling
UtilizingMathematica
3 IntroductionforPartII ............................................... 45
3.1 Random Variable ................................................. 45
3.2 Continuous Random Variable ...................................... 45
3.3 Uniform Random Variable ......................................... 46
3.4 Normal or Gaussian Random Variable .............................. 46
vii
viii Contents
3.5 Generating Random Variates by Acceptance-Rejection Method ......... 49
3.6 Variance Reduction and Importance Sampling ....................... 51
4 EvaluationofDefiniteIntegralsUsingAcceptance-RejectionSampling
UtilizingMathematica ................................................ 53
4.1 Evaluation of Definite Integrals Using Acceptance-Rejection
Sampling: Example I ............................................. 53
4.2 Evaluation of Definite Integrals Using Acceptance-Rejection
Sampling: Example II ............................................ 67
4.3 Evaluation of Definite Integrals Using Acceptance-Rejection
Sampling: Example III ............................................ 81
4.4 Evaluation of Definite Integrals Using Acceptance-Rejection
Sampling: Example IV ............................................ 96
4.5 Evaluation of Definite Integrals Using Acceptance-Rejection
Sampling: Example V ............................................ 102
5 VariationalMonteCarloMethodAppliedtoGroundStateofSimple
HarmonicOscillatorUsingAcceptance-RejectionSamplingUtilizing
Mathematica ......................................................... 111
5.1 The Variational Method of Quantum Mechanics Applied to Ground
State of Any Quantum Mechanical System .......................... 111
5.2 The Variational Method of Quantum Mechanics Applied to Ground
State of Simple Harmonic Oscillator ................................ 113
5.3 Ground State Energy of Simple Harmonic Oscillator
Using Variational Quantum Monte Carlo Method
and Acceptance-Rejection Sampling ................................ 116
ConcludingRemarks .................................................... 133
Appendix:HandoutsforComputationalLabandAutomatedCollection
ofAcceptedRandomVariates .................................. 135
References .............................................................. 149
Part I
Monte Carlo Methods Using Inverse Transform
Sampling Utilizing Mathematica
1
Introduction for Part I
This chapter provides background for Chap. 2. It introduces readers to continuous, uni-
form and Gaussian random variables. Inverse transform method of generating random
variate for given probability distribution function, and concept of importance sampling
are also introduced.
1.1 Random Variable
InthecontextofMonteCarlomethods,todefinearandomvariable,wemustindicatethe
values that the variable can assume and probabilities of occurrence of these values. We
do not know value of the variable in any given case; but we do know the values that the
variable can assume and the probabilities of occurrence of these values.
1.2 Continuous Random Variable
Arandomvariablex iscalledcontinuousifitcanassumeanyfractionalvalueinacertain
interval, say a to b. Besides specifying the interval containing all its possible values, we
needtostateafunctionp(x)calledprobabilitydensityfunctionorprobabilitydistribution
function.
We have
(1) p(x)≥0 for a≤x≤b,
(2) the product p(x)dx is probability that x≤x≤x + dx,
(cid:2)
(3) b(cid:4) p(x)dx is probability that a(cid:4)≤x≤b(cid:4) where a < a(cid:4) and b(cid:4) < b,
(cid:2)a(cid:4)
(4) b p(x)dx =1 indicating the surety that x lies in the interval a to b.
a
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S.Chowdhury,MonteCarloMethodsUtilizingMathematica®,SynthesisLectures
onMathematics&Statistics,https://doi.org/10.1007/978-3-031-23294-7_1