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

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

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

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

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

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

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

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

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

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 

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

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

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

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

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

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

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

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)

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

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

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

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

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

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

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

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