Interplay of Theory and Numerics for Deterministic and Stochastic Homogenization
The workshop has brought together experts in the broad field of partial differential equations with highly heterogeneous coefficients. Analysts and computational and applied mathematicians have shared results and ideas on a topic of considerable interest both from the theoretical and applied viewpoints. A characteristic feature of the workshop has been to encourage discussions on the theoretical as well as numerical challenges in the field, both from the point of view of deterministic as well as stochastic modeling of the heterogeneities. Mathematics Subject Classification (2010) 35A15, 35A35, 35J60, 49S05, 65C05, 35B27, 74Q15, 65N99, 74Q20, 37A25. Introduction by the Organisers Multiscale problems are ubiquitous in modern science and engineering. Applications are many: materials science, underground flows, wave propagation, etc. Indeed, materials that seem to be a continuum are made up of atoms. Geological media have many scales ranging from very small (atomistic) to an order of magnitude larger (pore and crack sizes) and to even much larger sizes of tectonic plates. Composite materials that are critical for modern technology are typically multiscale (e.g. small filling particles of different sizes in a sample of much larger size or layers of various thickness in a laminate composite) Many, though not all, of the models involve random modeling. Mathematics evidently plays a major role in this area. It covers the whole spectrum from theory 802 Oberwolfach Report 14/2013 (with homogenization theory, variational calculus, etc) to scientific computing, via numerical analysis (with the development of adequate, multiscale, finite element methods and related approaches) Statistics based methods are also employed. While modeling at each specific fundamental scale (e.g. molecular dynamics at atomistic scales, continuum PDEs at much larger scales) has been quite successful, the coupling between different scales is not well understood mathematically. The workshop has been an opportunity to make a state of the art review of the mathematical knowledge in a broad sense, to draw up a list of the challenges to overcome in the near future. It was a unique opportunity to bring together experts from all the areas involved. Approximately 45 scientists with very different backgrounds attended. Many delivered a talk, presenting their recent contributions in the light of their own perspectives. The quite compact schedule we had did not prevent many informal interactions to take place over coffee breaks, meals and lively evenings. On the theoretical side, we heard about recent theoretical achievements on quantitative rates of convergence in stochastic homogenization, on the Einstein relation, or on derivations of models for rough boundaries. Several contributions presented novel results on homogenization in the presence of large random potentials and on the derivation of macroscopic models to understand the energy density and phase information of waves propagating in highly heterogeneous media. Several discussions focused on the relatively uncharted territory of understanding the propagation of uncertainty from the coefficients in an equation to the solution of said equation. Many numerical analysis talks were devoted to novel numerical methods for approximating the solution space of PDEs with rough coefficients and possibly with non separated scales. This generated intense discussion among the participants. Common features and differences between the approaches were discussed in terms of robustness, cost, assumptions made, generality, ease of implementation, mathematical rigor and optimality. While there is a variety of heuristic numerical schemes, the mathematical understanding of such problems remains relatively underdeveloped. The workshop helped fill that gap through such discussions. Techniques using polynomial chaos expansion were also a topic of choice. Variance reduction issues, and more general aspects of Monte Carlo methods, were the topic of several talks. Applications as varied as transport processes through membranes, wave propagation phenomena, underground flows in porous media, nanotechnologies, colloid dynamics, optimal design of trussed materials, modeling of composite materials used in the aerospace industry, were addressed. Based on the feedback already received from participants, we consider the organization of such a workshop equally beneficial for scientists interested in theoretical issues, applied mathematicians developing numerical techniques, and mechanical engineers in contact with practical problems. The mixing of experts at deterministic and stochastic techniques was extremely rewarding for both camps. We hope other workshops in the same spirit will be organized in the near future. Theory and Numerics for Deterministic and Stochastic Homogenization 803 Workshop: Interplay of Theory and Numerics for Deterministic and Stochastic Homogenization