MA725 Advanced Numerical Analysis


Instructor: Nikolay S. Strigul
E-mail: nstrigul@stevens.edu
Office Hours: by appointment

Lectures: Pierce 220, Thursday 6:15 pm -8:45 pm
Assignments, Exams, Quizzes: There will be regular homework assignments and a final project.

Grading:

Course program Course program (PDF)

General comments:

This course is designed for graduate students majoring in computational sciences; in particular, this course may be of interest to students majoring in applied and financial mathematics, computer sciences, physics and engineering. The goal of this course is to study stochastic simulation methods such as Monte Carlo methods. Methods of stochastic simulations have been intensively developed since 1940s when they were first applied for simulation of physical systems; in particular, these methods were broadly employed in the Manhattan project. Nowadays Monte Carlo methods are widely used in modern science including financial mathematics, physics, biology and engineering. Monte Carlo simulations are used when deterministic algorithms are not efficient, or simply do not exist. The idea of the Monte Carlo simulations is to consider a given problem as a probabilistic problem, even, if initially it was established as a deterministic problem. These methods employ repeated random sampling from some parametric domain. In this course, we will consider the stochastic simulation methods based on random walk, in particular Markov Chain Monte Carlo (MCMC) methods. We will review random walk methods and discuss different approaches to improve or avoid random walk. The Metropolis-Hastings algorithm will be considered in depth including theoretical background, computer realisations and practical applications. A substantial part of this course will be devoted to computer simulations; in particular we will apply MCMC to the forest inventory (FIA) data. For practical applications we will employ the OPENBUGS package and original programs in C++.

Textbooks

1) Monte Carlo Statistical Methods by C.P. Robert and G. Casella, ISBN 0387212396
2) Bayesian Statistical Modelling by P. Congdon, ISBN-13: 9780470018750
3) Bayesian Data Analysis by A. Gelman, J.B. Carlin, H.S. Stern and D.B. Rubin, ISBN: 158488388X

Course program:



Lecture 1. - Probability theory

1. Elementary definition of probability
2. The sample space, events, random variables
3. Probability theory and measure theory
4. Probability space and probability measure
5. Random variables, probability distribution functions
6. Moments

Lecture 2. - Introduction to Bayesian methods

7. Bayes formula
8. Likelihood methods
9. Prior and posterior distributions
10. Introduction to OPENBUGS

Lecture 3. - Discrete probability distributions

11. Discrete uniform distribution
12. Binomial and negative binomial distributions
13. Hypergeometric distribution
14. Poisson distribution
15. Applications in biology

Lecture 4. - Continuous probability distributions

16. The normal distribution.
17. Lognormal distribution
18. Gamma distribution
19. Beta distribution
20. Student's t distribution
21. F distribution

Lecture 5. - Markov Chains 1

22. Random processes in discrete and continuous time
23. Markov property
24. Random walks
25. The Chapman-Kolmogorov equations
26. Simulated Annealing

Lecture 6. - Markov Chains 2

27. Irreducible chains
28. Reccurent chains
29. Stationary chains
30. Kac's theorem

Lecture 7. - Markov Chains 3.

31. Ergodicity
32. Convergence
33. Ergodic theorems
34. Central limit theorems

Lecture 8. - The Metropolis-Hastings algorithm

35. The MCMC principle
55. Definition of the Metropolis-Hastings algorithm
56. Convergence
57. Random walks
58. Examples

Lecture 8. - Gibbs sampler

59. Slice and Gibbs sampling
60. Gibbs algorithm
61. Program in C++
62. Examples

Lecture 9. - Practical applications

Lecture 10. - Practical applications

Lecture 11. - Practical applications

Lecture 12. - Practical applications

Lecture 13. - Practical applications

Lecture 14. - Practical applications

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