1. Monte Carlo Methods and Grid Computing
Yaohang Li
Department of Computer Science
North Carolina A&T State University

2. Textbook Monte Carlo Methods Markov Chain Monte Carlo in Practice
By J. M. Hammersley and D. C. Handscomb
Markov Chain Monte Carlo in Practice
By Gilks, Richardson, and Spiegelhalter

3. Course Goals
Let the students master the theory of the Monte Carlo methods in Computational Science
Enable the students to apply the Monte Carlo methods to simulate and solve real-life problems
Allow the students to use the parallel and distributed platform such as the grid to perform large-scale Monte Carlo simulation

4. Course Design Methods Simulation Applications Applications on the Grid
Basic Monte Carlo Methods
Analysis of Monte Carlo Methods
Simulation Applications
Biology
Nuclear Physics
Materials
Engineering
Chemistry
Applications on the Grid
Course Projects
Understand the theory behind Monte Carlo methods
Apply Monte Carlo methods to solve real-life problems
Using the Grid as the computational platform
Term Papers
Survey of the development of Monte Carlo methods

5. Course Outline Simulation Applications Monte Carlo Methods
Nuclear Physics
Radiation Shielding and Reactor Criticality
Polymer Science
Long Polymer Molecules
Statistical Mechanics
Markov Chain
Computational Biology
Protein Folding
Protein Docking
Material Science
Ab initio Monte Carlo
Grid Computing
Characteristics of Grid Computing
Grid-based Monte Carlo Applications
Monte Carlo Methods
The General Nature of Monte Carlo Methods
Short Resume of Statistical Terms
Random Number Generation
Pseudorandom Number Generation
Quasirandom Number Generation
General Principles of the Monte Carlo Methods
Conditional Monte Carlo
Direct Simulation
Solution of Linear Operator Equations
Metropolis Method

6. Introduction to Monte Carlo Methods
What is Monte Carlo method?
Experimental mathematics
Experiments on random numbers
General Nature of Monte Carlo Methods
Understand sophisticated problems
Extensive use in Simulation
Operational research
Nuclear physics
Chemistry
Biology
Medicine
Engineering
According to word of mouth, about 50% of CPU time of Department of Energy is spent on Monte Carlo Computation

7. Short Resume of Statistical Terms
Recall the basic terms and theories in statistics and probability
Random events and probability
Random variables
Discrete random variable
Continuous random variable
Distribution
Probability density function
Cumulative distribution function
Various Distributions
Uniform distribution
Rectangular distribution
Binomial distribution
Poisson distribution
Normal Distribution
Expectation

8. Random Number Generation (I)
Generating high-quality random numbers is critical in Monte Carlo applications
How random numbers will affect the accuracy of Monte Carlo applications
Generating Pseudorandom Numbers
Requirement of Pseudorandom Numbers
Randomness
Reproducibility
Efficiency
Uniformity
Pseudorandom Number Generator
LCG
LFG
ICG
Test of Pseudorandom Number Generator
Quasirandom Number Generator
What are the advantages of quasirandom number generator?
Types of Quasirandom Number Generators
Van der Corput Generator
Halton Generator
Faure Generator
Sobol’ Generator

9. Random Number Generator (II)
Sampling from non-rectangular distributions
Using the inverse function
Using the acceptance-rejection method
Class Project 1
Build a Pseudorandom Number Generator
Test the Randomness of this Random Number Generator
Generating Gaussian Samples using the Pseudorandom Number Generator
Build a Quasirandom Number Generator

10. Direct Simulation Direct Simulation Methods
Base on a probabilistic problem
Build a Monte Carlo model
Simple general structure
Miscellaneous examples of Direct Simulations
Operational research problem
Comparative Simulation
Course Project II
Develop a Buffon’s Needle Applet to calculate 

11. General Principles of the Monte Carlo Method
Crude Monte Carlo Method
Integral Estimation
Expectation and variance
Convergence rate O(cN-1)
Error estimation
Avoiding the curse of dimentionality
Compare with the deterministic numerical methods
Compare with hit-or-miss Monte Carlo

12. Variance Reduction Techniques
Theory Behind Variance Reduction Techniques
Reduce the constant c of the convergence rate O(cN-1) of crude Monte Carlo
Variance Reduction Techniques
Importance Sampling
Control Variates
Antithetic Variates
Regression Methods
Use of Orthonormal Functions

13. Course Project III
Develop a program using Crude Monte Carlo to estimate an integral
Compare with deterministic methods as dimension increases
Apply a variance reduction technique to compare the convergence rate

14. Conditional Monte Carlo
Theory of Conditional Monte Carlo
A special case of the foregoing theory
Conditional Monte Carlo
Compare Conditional Monte Carlo with direct simulation

15. Solution of Linear Operation Equation
Simultaneous Linear Equations
x = a + Hx
Condition of convergence
von Neumann and Ulam’s random walk method
Sequential Monte Carlo
Fredholm integral equations of the second kind
The Dirichlet Problem
Eigenvalue problems
Course Project IV
Use the random walk method to develop a program to simulate the Google search for finding the eigenvector corresponding to largest eigenvalue

16. Radiation Shielding and Reactor Criticality
The Shielding Problem
When a thick shield of absorbing material is exposed to -radiation (photons), of specified energy and angle of incidence, what is the intesity and energy-distribution of the radiation that penetrates the shield?
Special Handling
The Criticality Problem
When a pulse of neutrons is injected into a reactor assembly, will it cause a multiplying chain reaction or will it be absorbed, and, in particular, what is the size of the assembly at which the reaction is just able to sustain itself?
Eigenvalue Problem
Matrix Method

17. Problems in Statistical Mechanics
Markov Chains
Definition of Markov Chains
Characteristics of Markov Chains
Categorizing Markov Chains
Problems in equilibrium statistical mechanics
Thermal equilibrium
Statistical mechanics system
Order-disorder phenomena
Quantum statistics

18. Exploring the Equilibrium States
Global Minimum and Local Minimum
Equilibrium States
Markov Chain Monte Carlo (MCMC)
Metropolis-Hastings Method
Simulated Annealing
Simulated Tempering
Accelerated Simulated Tempering (AST)
Developed by Yaohang Li et al.
Parallel Tempering
Accelerated Parallel Tempering (APT)

19. Course Project V
Develop a Program using Metropolis-Hastings Method to Explore a rough 1-dimensional energy landscape
Extend the program using a simulated annealing method
Extend the program using a simulated tempering method
Analyze the algorithms when dimension increases
Analyze the algorithms when the landscape becomes rougher

20. Long Polymer Molecules
Self-avoiding Random Walks
Walks on a mesh
Walks in continuous space
Crude Sampling
Generation of very long walks
Recent development
Protein Folding and Docking

21. Monte Carlo Method in Material Science
Monte Carlo Process resembles a Physical Process
Ab initio Monte Carlo
Ising Model
Complex System

22. High Performance Monte Carlo Simulation
Characteristics of Monte Carlo Simulation
Parallel Random Number Generation
Natural Parallel Monte Carlo Simulation
High Performance Monte Carlo Simulation

23. Monte Carlo Applications and Grid Computing
Widely distributed
Large-scale resource sharing
Issues
Performance
Trustworthiness
Monte Carlo Applications
Approximate solutions to a variety of mathematical problems
Performing statistical sampling experiments
Slow convergence rate, approximately O(N-1/2)
Computationally intensive and naturally parallel

24. Grid-based Monte Carlo Applications
Issues in Grid Computing
From application point of view
Performance
Trustworthiness
Our Approaches
Address issues from application level
Analyze characteristics of Monte Carlo applications
Statistical nature
Cryptographic aspect of underlying random number generator
Develop strategies and tools

25. Techniques for Grid-based Monte Carlo Applications
Improve Performance
N-out-of-M Strategy
Bio-inspired Scheduling
Lightweight Checkpoint
Improve Trustworthiness
Partial Result Validation Scheme
Intermediate Value Checking

26. Conclusion Stimulate the students’ interest in computational science
Allow the students to master the methods and techniques of high performance statistical simulation
Develop interdisciplinary research and educational areas
Encourage the students to apply the computational techniques to real-life applications