# APMA 2811L- Topics in Homogenization: Theory and Computation

### Instructor: Konstantinos Spiliopoulos

Office: 37 Manning St, room 101

Office Hours: by appoitment

Email: kspiliop_at_dam.brown.edu

*to send me an email replace _at_ by @.*

**Meets: ** Monday 12:30-3:00 at 102A at 180 George Street

** Organizational meeting: ** Tuesday Sep 6 2011, 2 - 3 pm, Room 110, 182 George Street

**Text:** G. A. Pavliotis and A. M. Stuart, Multiscale Methods: Averaging and Homogenization, Springer, 2007

**Recommended textbooks: **

- For multiscale methods and perturbation theory:
- G. A. Pavliotis and A. M. Stuart, Multiscale Methods: Averaging and Homogenization, Springer, 2007
- A. Bensoussan, J. -L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures (Studies in mathematics and its applications), Elsevier, 1978
- V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of Differential Operators and Integral Functions, Springer-Verlang, 1991
- H. Cheng, Advanced Analytic Methods in Applied Mathematics, Science, and Engineering,
Luban Press, 2005.
- M. H. Holmes, Introduction to Perturbation Methods, Springer, 1998.

- For stochastic calculus and the interplay between PDE's and stochastic processes:
- M. Freidlin, Functional Integration and Partial Differential Equations, Princeton University Press, 1985
- B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, 2007 (6th edition)
- I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, 2nd edition

**Course Description:**
Concepts and analytic and probabilistic tools used in various scientific disciplines. Emphasis will be placed on

- Review of probability theory, introduction to stochastic calculus (Brownian motion, stochastic differential equations, It\^{o} formula, Fokker-Planck eqs, Feynman-Kac formula, relation to PDE's)
- Homogenization for PDE's and stochastic processes using various deterministic and probabilistic tools. Homogenization for diffusion in periodic and random media. Averaging for SDE's. Multiple scales method and two-scale convergence.
- Backward SDE's and their application to homogenization of semi-linear parabolic PDE's.
- Numerical methods for multiscale equations.

The course material will be based on theory, methods and examples from various scientific disciplines.

**Course Objective:**

- To learn various analytic and probabilistc techniques which are useful in the analysis of ODE's, PDE's and SDE's that depend on small (or large) parameters and have rapidly oscillating coefficients. To apply techniques from perturbation theory to study homogenization problems for PDE's and SDE's. To derive rigorous proofs of the formal calculations.