Richard von Mises Lecture 2021

## Timeline

 14:15-14:20 Opening of the 2021 Richard-von-Mises-Lecture 14:20-15:10 Historical Talk: John von Neumann in Berlin: Ein ungarisch-jüdischer Mathematiker an der Friedrich-Wilhelms-Universität (1921-1933) (Dr. Ulf Hashagen) 15:10-15:45 Coffee Break 15:45-16:45 Richard-von-Mises-Lecture: The Representative Volume Element method in homogenization: how to minimize the statistical bias (Prof. Felix Otto) 16:45-17:00 Open discussion

## Abstracts

The Representative Volume Element method in homogenization: how to minimize the statistical bias
Prof. Felix Otto (MPI for Mathematics in the Sciences (Leipzig))

We are interested in the elastic or conductive behavior of a composite material, as described by a non-constant coefficient field $$a=a(x)$$ and the corresponding linear elliptic operator $$-\nabla\cdot a\nabla$$. Suppose that while the details of the heterogeneity are unknown, its statistical behavior, in form of a probability distribution/ensemble of $$a$$'s, is known. We study the Representative Volume Element (RVE) method, which is an engineering method to approximately infer the effective behavior $$a_{\rm hom}$$ of such a heterogeneous medium, which we think of being statistically homogeneous/stationary. The latter puts us into the context of (quantitative) stochastic homogenization. In line with the theory of homogenization, the RVE method proceeds by computing $$d=3$$ ($$d$$ denoting the space dimension) correctors, however on a ''representative'' volume element, i.e. box with, say, periodic boundary conditions. One message of the talk is: periodize the ensemble instead of its realizations!

By this we mean that it is better to sample from a suitably periodized ensemble than to periodically extend the restriction of a realization $$a(x)$$ from the whole-space ensemble $$\langle\cdot\rangle$$. We make this point by investigating not just the fluctuations (or random error) but also the bias (or systematic error), i.e.the difference between $$a_{\rm hom}$$ and the expected value of the RVE method, in terms of its scaling w.r.t.the lateral size $$L$$ of the box. In case of periodizing $$a$$, we heuristically argue that this error is generically $$O(L^{-1})$$. In case of a suitable periodization of $$\langle\cdot\rangle$$, we rigorously show that it is $$O(L^{-d})$$. In fact, we give a characterization of the leading-order error term for both strategies.

This relies on joint work with N.Clozeau, M.Josien, and Q. Xu; and with numerical simulations by M.Schneider as well as B. & V.Khoromskij.

John von Neumann in Berlin: Ein ungarisch-jüdischer Mathematiker an der Friedrich-Wilhelms-Universität (1921-1933)
Dr. Ulf Hashagen (Deutsches Museum in Munich)

John von Neumann (*1903 Budapest; † 1957 Washington D.C.) wird heute zu den bedeutendsten Wissenschaftlern des 20. Jahrhunderts gezählt und ist für seine grundlegenden Beiträge zur reinen Mathematik (Mathematischen Logik, Maßtheorie, Spektraltheorie, Operatoren-Algebren), theoretischen Physik (Grundlagen der Quantenmechanik, Ergodentheorie), Wirtschaftswissenschaften (Spieltheorie) und Informatik (von-Neumann-Architektur, Scientific Computing) berühmt. Der Vortrag untersucht die frühe wissenschaftliche Karriere Neumanns als Student und Hochschullehrer an der Friedrich-Wilhelms-Universität in Berlin bis zu seiner Emigration in die USA im Jahr 1933 und diskutiert Neumanns Beziehungen zu den Berliner Mathematikprofessoren Erhard Schmid, Issai Schur und Richard von Mises. Weiterhin werden die Karrierechancen von ausländischen und jüdischen Mathematikern im deutschen Wissenschaftssystem der Weimarer Republik und die Gründe für Neumanns Emigration diskutiert.