• Course Description
  

  This course will cover basic mathematical and numerical aspects of Monte Carlo simulations, molecular dynamics simulations, and models of stochastic dynamics. Techniques of coarse graining and passage from discrete to continuum will be discussed. Different computational methods for tracking interface motion will also be introduced, if time permits. Areas of application include but are not limited to mechanical engineering, biological physics, materials science, and statistical learning.

  Some background of numerical computation and stochastic process will be helpful but not absolutely necessary for taking this course. No knowledge of the related application will be assumed.
  

• Office hours: By appointment.
  

• Class Project: Due 2:30 pm, Thursday, June 13, 2019.

  Detailed Instructions on Course Project

  Suggested projects

  1. Monte Carlo integrtion with variance reduction.
  2. Monte Carlo optimization with the simulated annealing.
  3. Monte Carlo simulation of simple liquid with Lenard-Jone interation potential.
  4. Monte Carlo simulation of an ionic system to study the ionic size effect.
  5. Monte Carlo simulation of an ionic system to study the possible like-charge attraction phenomenon.
  6. Convergence analysis of Markov chain Monte Carlo simulation for a simple system.
  7. Molecular dynamics simulations with GROMACS of the solvation of a spherical charged molecule in water with some ions.
  8. Free energy calculation with molecular dynamics simulations of a simple molecular system. using either Monte Carlo or molecular dynamics simulations.
  9. Hybrid Brownian dynamics simulations and finite element modeling of charged systems.
  10. Combined Monte Carlo simulation and continuum solvation model for the binding of a ligand-receptor system.
  11. Numerical investigation of the convergence of a hybrid computational method combining the agent-based discrete simulations with continuum descritions of background fields.
  12. Study the coarsening dynamics: from discrete to continuum descriptions.
  13. Prove the continuum electrostatic energy is the limit of a sequence of discrete elecrostatic energies.
  14. Design your own project and get the instructor's approval.

• Grades: Based on the class participation and possible homework/project.

Lecture Notes on Monte Carlo Methods

• Chapter 1. Introduction (17 pages)
• Chapter 2. Sampling Random Variables (26 pages)
• Chapter 3. Variance Reduction (23 pages)
• Chapter 4. Markov Chain Monte Carlo (26 pages)
• Chapter 5. Statistical Analysis of Simulation Data (27 pages)
• Chapter 6. Monte Carlo Optimization
• Chapter 7. Rare-Event Simulation
• Chapter 8. Application in Statistical Physics (15 pages)
• Appendix: Review of Probability Theory
  • Events and Probabilities (5 pages)
  • Random Variables and Their Distributions (20 pages)
  • Convergence of Random Variables (6 pages)
  • Discrete-State Markov Chains (26 pages)
• Course Project

References on Probability and Stochastic Processes

1. G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, Oxford University Press, Oxford, 3rd ed., 2001.
2. P. G. Hoel, S. C. Port, and C. J. Stone, Introduction to Probability Theory, Houghton Mifflin Company, 1971.
3. P. G. Hoel, S. C. Port, and C. J. Stone, Introduction to Stochastic Processes, Houghton Mifflin Company, 1972.
4. G. F. Lawler, Introduction to Stochastic Processes, 2nd ed., Chapman & Hall/CRC, 2006.

1. P. Baldi, Stochastic Calculus, Universitext, Springer, 2017.
2. E. Cinlar, Probability and Stochastics, Graduate Texts in Mathematics 261, Springer, 2011.
3. J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus, Graduate Texts in Mathematics, Springer, 2016.
4. A. Gut, Probability: A Graduate Course, Springer Texts in Statistics, 2nd ed., Springer, 2013.
5. N. Krylov, Introduction to the Theory of Random Processes, Amer. Math. Soc., 2002.
6. S. Resnick, A Probability Path, Birkhauser, 2005ed.
7. J. B. Walsh, Knowing the Odds: An Introduction to Probability, Graduate Studies in Mathematics, Amer. Math. Soc., 2012.

1. L. Breiman, Probability, Addison-Wesley, 1968.
2. K. L. Chung, A Course in Probability Theory, 3rd ed., Academic Press, 2001.
3. W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed., Wiley, 1968.
4. W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd ed., Wiley, 1971.
5. I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics 113, 2nd ed., Springer, 1991.
6. D. W. Strook, Probability Theory, Cambridge University Press, 1993.
7. S. R. S. Varadhan, Probability Theory, Courant Lecture Notes 7, Amer. Math. Soc., 2001.
8. S. R. S. Varadhan, Stochastic Processes, Courant Lecture Notes 16, Amer. Math. Soc., 2007.

References on Monte Carlo Methods

1. S. Asmussen and P. W. Glynn, Stochastic Simulation, Springer, 2007.
2. G. S. Fishman, Monte Carlo: Concepts, Algorithms and Applications. Springer-Verlag, New York, 1996.
3. M. H. Kalos and and P. A. Whitlock, Monte Carlo Methods, 2nd ed., Wiley, 2008.
4. D. P. Kroese, T. Taimre, and Z. I. Botev, Handbook of Monte Carlo Methods, John Wiley & Sons, 2011.
5. J. S. Liu, Monte Carlo Strategies in Scientific Computing, Springer, 2008.
6. N. Madras, Lectures on Monte Carlo Methods, Amer. Math. Soc., 2002.
7. C. P. Robert and G. Casella, Monte Carlo Statistical Methods, 2nd ed., Springer, New York, 2004.
8. R. Y. Rubinstein and D. P. Kroese, Simulation and the Monte Carlo Method, 3rd ed., John Wiley & Sons, New York, 2017.
9. P. J. M. van Laarhoven and E. H. L. Aarts, Simulated Annealing: Theory and Applications, Springer, 1989.

1. M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, Oxford University Press, 1987.
2. K. Binder and D. W. Heermann, Monte Carlo Simulation in Statistical Physics. An Introduction, 5th ed., Springer, 2010.
3. D. Frenkel and B. Smit, Understanding Molecular Simulation, 2nd ed., Academic Press, 2002.
4. D. P. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics, 3rd ed., Cambridge University Press, 2009.

1. C. Graham and D. Talay, Stochastic Simulation and Monte Carlo Methods, Springer, 2013.

1. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller. Equations of state calculations by fast computing machines, J. Chem. Phys., 21(6), 1087-1092, 1953.
2. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, Optimization by simulated annealing, Science, 220:4598, 671-680, 1983.