IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

PDE compression — asymptotic preserving, numerical homogenization and randomized solvers

  • Feb. 26, 2018
  • 2:15 p.m.
  • LeConte 312

Abstract

Multi scale multi physics problems all have small parameters in the equations. In kinetic theory, the Boltzmann type equations converge to the fluid limits in the small Knudsen number regime. Asymptotic Preserving solvers are then designed to capture such fluid limits without fine discretization. Numerical homogenization is heavily used in equations with porous media where the small parameters are used to present the oscillations. Many numerical solvers are then designed to capture the homogenized limits with grids independent of the scale of the oscillations.

Despite that the equations are from different backgrounds, they share a common trait: the solution space is approximately low rank. There should be a general framework that captures the solution space automatically without analytical input. We build such a framework in this talk. Specifically, a randomized PDE solver will be designed that compresses PDE solution spaces with no analytical knowledge required. The solver is thus general and automatic, and unifies AP and numerical homogenization. We will demonstrate numerical examples: the compression rate for porous media equation is 0.5, and the compression rate for the Boltzmann equation is 0.05.

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