Some recent developments on numerical homogenization
Organizer
Speaker
Lei Zhang
Time
Friday, December 31, 2021 2:00 PM - 3:00 PM
Venue
1110
Online
Zoom 388 528 9728
(BIMSA)
Abstract
Problems with a wide range of coupled temporal and spatial scales are ubiquitous in many phenomena and processes of materials science and biology. Numerical homogenization concerns the approximation of the high dimensional solution space of multiscale PDEs by a low dimensional approximation space with optimal error control, and furthermore, the efficient construction of such an approximation space, e.g., the localization of the basis on a coarse patch. There has been a vast amount of work concerning the design and analysis of numerical homogenization type methods for multiscale problems, such as asymptotic homogenization, numerical upscaling, heterogeneous multi-scale methods, multi-scale finite element methods, variational multi-scale methods, flux norm homogenization, rough polyharmonic splines (RPS), generalized multi-scale finite element methods, localized orthogonal decomposition, etc. Surprisingly, numerical homogenization has deep connections with Bayesian inference, kernel learning and probabilistic numerics. The Bayesian homogenization approach provides a unified framework for the construction of a proper coarse space with desired approximation and localization properties, and practically it corresponds to a variational problem with functional constraints, which is close to in this sense. In this talk, I will give a brief introduction of the recent development in numerical homogenization, its connection with Bayesian inference, fast solvers, and machine learning. I will also discuss some possible further directions.