Speaker：Lei Zhang, Shanghai Jiaotong University

Title：From Numerical Homogenization to Fast Solver for PDEs with Rough Coefficients

Time：2016-2-25, 15:00-16:00pm

Venue: Room 316 of Environment Building, Renmin University

Abstract: Numerical homogenization concerns the finite dimensional approximation of the solution space of PDEs such as the divergence form elliptic equation with arbitraily rough $L^\infty$ coefficients, which allows for nonseparable scales and high contrast. Standard methods such as finite element methods with piecewise polynomial elements can perform arbitrarily badly for such problems. In this talk, I will introduce a framework for numerical homogenization which precomputes $H^{-d}$ localized bases on patches of size $H\log(1/H)$. The localization is due to the exponential decay of the corresponding fine scale solutions with Lagrange type constraints. Interestingly, this approach can be reformulated as a Bayesian inference problem with partial information or decision theory problem. Furthermore, the numerical homogenization method can be used to construct efficient and robust fine scale multigrid solver or domain decomposition preconditioner. This is a joint work with Houman Owhadi (Caltech) and Leonid Berlyand (PSU).