Many physical processes in micro-heterogeneous media such as modern composite and functional materials are described by partial differential equations with coefficients that represent complicated material microstructures. The coefficients are often the result of measurements combined with inverse modelling and, hence, underlie errors and uncertainty. Given the complexity of these processes, the key to efficiently and reliably simulate some relevant classes of such problems involves the construction of appropriate macroscopic computational models with significantly reduced oscillations and randomness. Homogenization is a multiscale method for the derivation of such effective models and this series of lectures aims to promote the recent methodological progress in this context.
This series of lectures will start with a brief survey of constructive approaches in the mathematical theory of deterministic and stochastic homogenization, their quantitative analysis under the usual periodicity, ergodicity and scale separation assumptions. The main part will then be on novel computational approaches based on subspace decomposition that remain valid beyond such strong structural assumptions. In the end, we will bridge the analytical and computational approaches to homogenization and illustrate some surprising connections to seemingly unrelated areas such as the theory of iterative solvers and domain decomposition, multiresolution analysis, classical stabilization techniques in the theory of finite elements, principal component analysis , and information games.
The lectures require basic knowledge of partial differential equations and its numerical treatment (the finite element method).
References to the underlying research articles may be found in the survey papers:
Daniel Peterseim holds the chair of Computational Mathematics at the University of Augsburg, Germany. His research covers all aspects of computational partial differential equations with applications in engineering and physics. He is most well known for his contributions in the area of computational multiscale methods.
Professor Peterseim gained his doctorate in Mathematics at the University of Zurich in 2007. Before he moved to Augsburg in 2017, he was appointed as a junior research group leader at the Humboldt-Universität zu Berlin (within the Research Center Matheon) in 2009 and as a professor for numerical simulation at the University of Bonn in 2013.