Sep 24, 2014
Martin Gander, University of Geneva, Section of Mathematics, Geneva, Switzerland The history of constrained optimization spans nearly three centuries. It goes back to a letter Johann Bernoulli sent in 1715 to Varignon, announcing a very simple rule with which the many hundreds of different problems in fluid and solid mechanics considered in detail by Varignon can be solved in the blink of an eye. Varignon then explains this rule at the end of his book, but unfortunately cites the letter of Johann Bernoulli with an incorrect date. Bernoulli’s rule, based on virtual velocities, was later carefully explained by Lagrange, and led to the discovery of the famous multiplier method of Lagrange, with which many optimization problems can be easily treated. Using so called Lagrange multipliers is however a much more far reaching concept, and we will see that one can, armed only with Lagrange multipliers, discover the important primal and dual equations in optimal control and the famous maximum principle of Pontryagin. Pontryagin himself however did not discover his maximum principle using Lagrange multipliers, he used a more geometric argument. We will finally give the complete formulation of PDE constrained optimization based on duality introduced by Lions.
MOdelling REvisited + MOdel REduction Modeling, analysis and computing in nonlinear PDEs. September 21-26, 2014, Chateau Liblice, Czech Republic. Recently developed implicit constitutive theory allows one to describe nonlinear response of complex materials in complicated processes and to model phenomena in both fluid and solid mechanics that have hitherto remained unexplained. The theory also provides a thermodynamically consistent framework for technologically important but so far only ad hoc engineering models without sound footing. The overall goal of the project is to develop accurate, efficient and robust numerical methods that allow one to perform large-scale simulations for the models arising from the new theoretical framework. A natural part of the goal is rigorous mathematical analysis of the models. The nonstandard structure of constitutive relations arising from the new framework requires the reconsideration of many existing approaches in the mathematical theory of partial differential equations and the development of new ones. In particular, basic notions such as the concept of the solution and its well-posedness need to be reconsidered. Further, the complexity of the constitutive relations calls for rigorous investigation on model reduction - the identification of simplified models that capture the chosen (practically relevant) information about the behaviour of the system and disregard irrelevant information. Reliable numerical simulations require the derivation of sharp a posteriori error estimates to control all possible sources of errors, including rarely studied but important algebraic errors. We believe that in solving difficult problems in mathematical modelling the individual aspects discussed above - physics, mathematical analysis and numerical analysis - are so closely interrelated that no breakthrough can be achieved without emphasising the holistic approach as the main principle. Our vision is to follow this principle: the entire process of modelling of complex materials will be revisited in an innovative manner.
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