Abstract: Karen Willcox

MIT

Surrogate and Multifidelity Modeling for Optimization Under Uncertainty of Large-scale Complex Systems

Numerical tools to support decision under uncertainty are essential in many settings, from supporting policy decisions, to the design, development and operation of complex systems. These tools are especially important in the early stages of the decision process, when decisions have the greatest influence on subsequent development efforts and final system performance. However, to support effective decision-making, it is essential to address the many different sources and types of uncertainty, including those stemming from the fidelity of the underlying mathematical models.

A first challenge is to characterize uncertainties and formulate the decision task as an optimization under uncertainty problem, with objective functions and constraints taking the form of expectations, variances, probabilities, etc. A second challenge is then to solve the resulting stochastic optimization problem, which is particularly difficult for systems governed by partial differential equations. The discretized forward models describing such systems typically are of very high dimension and are expensive to solve, and the number of parameters representing the optimization decision variables may be large. The computational resources required to solve a stochastic optimization problem therefore quickly become prohibitive.

Surrogate modeling has an important role to play in addressing this second challenge, by producing low-order approximate models that retain the essential system dynamics but that are fast to solve. This talk will discuss two aspects of surrogate modeling in optimization under uncertainty for complex systems. First, we discuss formulation and derivation of projection-based reduced-order models for optimization under uncertainty. Our methods use state approximations through the proper orthogonal decomposition, reductions in parameter dimensionality through parameter basis approximations, and the discrete empirical interpolation method for efficient evaluation of nonlinear terms. Second, we present a multifidelity optimization framework that combines a derivative-free trust-region model management method with a Bayesian calibration strategy. The method iteratively calibrates lower-fidelity information, provided through surrogate models, to the high-fidelity function, and is provably convergent to an optimum of the high-fidelity optimization problem. Gradients of high-fidelity functions are not required, which is important for problems with non-smooth objectives or constraints; instead, convergence is achieved using sensitivity information from the calibrated surrogate models.


 

Aachen Institute for Advanced Study in Computational Engineering Science (AICES)
at RWTH Aachen University, Germany. Email: acces11@aices.rwth-aachen.de
Tel. +49 (0)241 80 99130, Fax +49 (0)241 80 628498