The general area of homogenization is about finding averaged properties of solutions to inhomogeneous equations depending on small parameters and set in self averaging media. This is a classical area of mathematics which has been studied a lot in the past in periodic settings both from the pde and probabilistic points of view as well as numerically and computationally. The periodic setting is, however, rather restrictive (rigid) for applications. For more realistic modeling, it is necessary to consider random environments and equations/models with random coeffcients. The goal is to obtain effective equations by "averaging out" all of the oscillations and randomness, and to understand in a precise way how microscopic oscillations give rise to the macroscopic behavior of the system. This is the general goal of stochastic homogenization, an area which has undergone substantial growth and development in the last few years. There are many interesting and important questions both theoretical and applied that need to be studied. The program at IML will bring interested researchers up to speed with the most recent developments.
Some of the themes that will be covered by the program are:
- Homogenization for first and second order PDE in oscillatory random environments
- Interfaces and free boundary problems in random media
- Random walks in random environments
- Numerical methods
- Multi-scale numerical methods, dynamical systems.