电子科技大学：《材料设计与计算 Materials Design and Computation》课程教学资源（课件讲稿）MDC 12 An introduction of Monte Carlo method

Review of last class Design materials using computation ·Multiscale coupling Materials Genome and materials informatics Mathematical models in material science Modeling Modeling Modeling Model preparation assumption construction solving F T Model Model Model application testing analysis

Review of last class • Design materials using computation • Multiscale coupling • Materials Genome and materials informatics • Mathematical models in material science 建模准备 建模假设 构造模型 模型求解 模型应用 模型检验 模型分析 F T T F Modeling preparation Modeling assumption Modeling construction Model solving Model analysis Model testing Model application

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An introduction of Monte Carlo method

Outline ·Introduction Basics of Monte Carlo method 0 Statistical Uncertainty Improving Efficiency Techniques An Application in Surface Diffusion

Outline • Introduction • Basics of Monte Carlo method • Statistical Uncertainty • Improving Efficiency Techniques • An Application in Surface Diffusion

INTRODUCTION

INTRODUCTION

Hierarchy of time and space scale of computational simulation Continuum TIME (s) Based on SDSC Blue Horizon(SP3) Methods 512-1024 processors 1.728 Tflops peak performance 100 CPU time =1 week/processor Atomistic Mesoscale methods (ms)103 Simulation Methods Finite elements methods (us)10-6 Semi-empirical (ns)109 methods Monte Carlo Molecular dynamics (ps)1012 Ab initio tight-binding (fs)10-15 10-10 109 108 107 10-6 105 10-4 (nm) (um) LENGTH(m)

Hierarchy of time and space scale of computational simulation 6

MD&MC There are two dominant methods of simulation for complex many particle systems 1)Molecular Dynamics Solve the classical equations of motion from mechanics. Particles interact via a given interaction potential. ● Deterministic behaviour(within numerical precision). Find temporal evolution. 2)Monte Carlo Simulation ● Find mean values (expectation values)of some system components. Random behaviour from given probability distribution laws. The Monte Carlo technique is a very far spread technique, because it is not limited to systems of particles

There are two dominant methods of simulation for complex many particle systems 1) Molecular Dynamics • Solve the classical equations of motion from mechanics. • Particles interact via a given interaction potential. • Deterministic behaviour (within numerical precision). • Find temporal evolution. 2) Monte Carlo Simulation • Find mean values (expectation values) of some system components. • Random behaviour from given probability distribution laws. The Monte Carlo technique is a very far spread technique, because it is not limited to systems of particles. MD & MC

The origin of the name The name refers to the grand casino in the Principality of Monaco at Monte Carlo,which is well-known around the world as an icon of gambling AN E VENI N G IN M ON TE C ARL O

The origin of the name The name refers to the grand casino in the Principality of Monaco at Monte Carlo, which is well-known around the world as an icon of gambling

Application of Monte Carlo method CAD MeCad MCNP CAD-Modell von ITER Konversion in Monte Carlo-Geometrie Monte-Carlo-Modell (Vertikalschnitt) Nuclear reactor design Diffusion Monte Carlo 8P5 Sico24▣-130 Econometrics Radiation cancer therapy and more Oil well exploration Traffic flow

Application of Monte Carlo method Monte Carlo and more Nuclear reactor design Radiation cancer therapy Traffic flow Econometrics Oil well exploration Diffusion

Monte Carlo method This is a kind of "experimental statistics".In other branches of science,for example physics,the relationship between theory and experiment can be depicted in this way: Equipment, Experimental Theoretical Observation Physics Physics In statistics,theory developed from simple observations in card and dice games in the 17th century and later.The fully-fledged"experimental"approach,now known as Monte Carlo and thus acknowledging the origins of statistics in gambling,had to await the development of fast personal computers and random-number generators: Gambling: Cards,Dice Then Experimental Theoretical Very recent Statistics Statistics Random- Monte- Fast PCs number Carlo generators methods

Theoretical Physics Experimental Physics Equipment, Observation Gambling: Cards, Dice Fast PCs Random￾number generators Monte￾Carlo methods Experimental Statistics Theoretical Statistics Then Very recent Monte Carlo method This is a kind of “experimental statistics”. In other branches of science, for example physics, the relationship between theory and experiment can be depicted in this way: In statistics, theory developed from simple observations in card and dice games in the 17th century and later. The fully-fledged “experimental” approach, now known as Monte Carlo and thus acknowledging the origins of statistics in gambling, had to await the development of fast personal computers and random-number generators:

Monte Carlo method Monte Carlo methods are stochastic techniques. > The basic concept is that games of chance can be played to approximate solutions to real world problems. >Monte Carlo methods solve non-probabilistic problems using probabilistic methods. > The Monte Carlo method provides approximate solutions to a variety of mathematical problems by performing statistical sampling experiments on a computer. > The method applies to problems with no probabilistic content as well as to those with inherent probabilistic structure

 Monte Carlo methods are stochastic techniques.  The basic concept is that games of chance can be played to approximate solutions to real world problems.  Monte Carlo methods solve non-probabilistic problems using probabilistic methods.  The Monte Carlo method provides approximate solutions to a variety of mathematical problems by performing statistical sampling experiments on a computer.  The method applies to problems with no probabilistic content as well as to those with inherent probabilistic structure. Monte Carlo method