High-dimensional optimization in the presence of uncertainty: applications in random heterogeneous media (invited talk)

This talk is concerned with the optimization/design/control of complex systems character- ized by high-dimensional uncertainties and design variables. Expected utility maximization involves maximizing, with respect to design/control parameters, an integral expression with respect to the random variables. The problem is difficult because the integration is embed- ded in the maximization and has to be evaluated many times for different design parameters. Furthermore each of these evaluations require several calls to the forward model (e.g. dis- cretized system of PDEs). We are especially concerned with problems relating to random heterogeneous materials where uncertainties arise from the stochastic variability of their properties. In particular, we formulate topology optimization problems to account for the design of truly random composites. Deterministic optimization tools cannot be directly ex- tended to stochastic settings [1]. Traditional stochastic optimization techniques result in exuberant computational cost and fail to identify multiple local optima [2]. In this work we reformulate the problem as one of probabilistic inference [3] and employ sampling tools suitable for high-dimensions [4]. The methodological advances proposed in this paper allow the analyst to identify several local maxima that provide important information with re- gards to the robustness of the design [5]. We further propose statistical learning tools that exploit information from approximate models and can ultimately lead to further reductions in computational cost.      «

This talk is concerned with the optimization/design/control of complex systems character- ized by high-dimensional uncertainties and design variables. Expected utility maximization involves maximizing, with respect to design/control parameters, an integral expression with respect to the random variables. The problem is difficult because the integration is embed- ded in the maximization and has to be evaluated many times for different design parameters. Furthermore each of these evaluations...     »