CIMMS Lunchtime Series: Metric Based Up-Scaling and Universality at Small Scales

Dr. Houman Owhadi
Applied & Computational Mathematics and Control & Dynamical Systems, Caltech

Heterogeneous multi scale structures can be found everywhere in nature. Can these structures be accurately simulated at a coarse level? Homogenization theory allows us to do so by transferring bulk (averaged) information from sub-grid scales to computational scales. This theory has been initiated long time ago by the work of Poisson (1826), Faraday (1839), Maxwell (1873), Rayleigh (1892) on the macroscopic behavior of two phase conductors and the work of Einstein (1905-1906) on effective viscosity. Its justification (Spaniolo, Tartar, Lions, Papanicolaou, Oleinik, Jikov...) in the 70's has been grounded on two key hypotheses: scale-separation and ergodicity at small scales.

Can we get rid of these assumptions? Can we upscale a partial differential equation when its coefficients are arbitrary? What do we mean by up-scaling? Can we simulate a physical system in an energy landscape by a Markov chain on a coarse graph when the energy landscape is arbitrary?

Can we compress the operator describing a physical system at different arbitrary scales of resolution? If yes, then what minimal quantity of information should be kept from the small scales in order to so? What orthogonal information should be added to the compressed operator in order to decompress it? Can an operator become singular after it has been up-scaled? What does this mean for the physical system?

We will address these issues in this talk. Such an up-scaling can be done rigorously for divergence form linear operators by transferring a new metric and a new measure of volumes from small scales in addition to traditional bulk quantities and error bounds can be given.