CMX Student/Postdoc Seminar
Learning Homogenization for Elliptic Operators with Discontinuous Parameters
Elliptic partial differential equations describe important physical phenomena such as elastic material deformation and fluid flow through porous media. When the coefficient in this equation is discontinuous, involves corner interfaces, or changes across multiple scales, numerically solving the equation can be computationally expensive. In this talk, we will discuss in depth the challenges associated with learning homogenized solutions to equations with such parameterizations. A full analysis in one dimension gives insight into addressing the problem in higher dimensions. PDE theory on solutions at corner interfaces in two dimensions suggests learning methods. An approximation theorem for the homogenized map is given, and different approaches to learning are discussed.