LisMath Seminar 

27/05/2016, 16:00 — 17:00 — Room V1.07, Civil Engineering Building, IST
Ragaa Ahmed, Universidade de Lisboa

The Monte Carlo method and some applications

There is an interesting technique known as the Monte Carlo method that can be used to solve many types of problems [1]. Monte Carlo method is a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. It is mainly used in three distinct problem classes [2]: optimization, numerical integration, and generating draws from a probability distribution. We can use Monte Carlo algorithm for finding the solution of Dirichlet partial differential equations (DPDE) [3, 4] and to find the solution of Schro ̈dinger equation in quantum mechanics, Brownian motion [5, 6] as well as statistical mechanics. Since the main objective of statistical mechanics is to show how the properties of matter (macroscopic properties such as pressure, heat capacity, entropy, etc [7,8]) can be calculated from the properties of individual molecules (positions, molecular geometry and intermolecular forces, etc), we can use the statistical methods (such as Monte Carlo method, Green’s function [9,10]) to understand the relation between the behavior of atoms, energy and the treatment of huge numbers of distinguishable and indistinguishable particles, for example the particles in crystals net, the particles of gas (different in the particles position). Here, we present a brief overview of the Monte Carlo method and then illustrate its use in PDE and integration. 

Bibliography

1. Stanley, J. Farlow, Partial differential equations for Scientists and Engineers, Dover Publications, INC. NewYork (1982).
2. Kroese, D. P., Brereton, T., Taimre, T. and Botev, Z. I., Why the Monte Carlo method is so important today. WIREs Computational Statistics 386-392 (2014). DOI:10.1002/wics.1314.
3. Vajargah, B. F. and Vajargah, K. F., Monte Carlo Method for Finding the Solution of Dirichlet Partial Differential Equations, Applied Mathematical Sciences, 453-462 (2007).
4. Talay, D., Monte Carlo Methods for PDE’s. In Encyclopaedia of Mathematics, M. Hazewinkel (Ed.). Kluwer Academic Press (1997).
5. Jacod, J.; Lejay, A. and Talay, D., Estimation of the Brownian dimension of a continuous Ito process, Bernoulli, 469-498 (2008) DOI: 10.3150/07-BEJ6190.
6. Maire, S. and Talay, D., On a Monte Carlo method for neutron transport criticality computations, IMA Journal Numerical Analysis, 657-685 (2006).
7. Kraeft, W. D. and Bonitz, M., Thermodynamics of a Correlated Confined Plasma, Journal of Physics: Conference Series, 78 (2006).
8. Reed, T. M. and Keith, E.Gubbins, Applied statistical mechanics (1991).
9. Hussein, N. A., Eisa, D.A., Osman, A.-N. A. and Abbas, R. A., Quantum Binary and Triplet Distribution Functions of Plasma by using Green’s Function Contrib. Plasma Physics, 815 - 826 (2014) / DOI 10.1002/ctpp.201400016.
10. Hussein, N. A., Osman, A.-N. A., Eisa, D. A. and Abbas, R. A., The quantum thermodynamic functions of plasma in terms of the Green’s function, Natural Science, 71-80 (2014) http://dx.doi.org/10.4236/ns.2014.62011. 

See also

LisMath_Ragaa_Ahmed.pdf