Many important problems in computational engineering involve the design or control of systems that require several hours of computer time to produce accurate simulations. The expense of this runtime often precludes applying an optimization procedure directly to a full physics simulation to determine decision variables. Engineers are approaching this problem by applying optimization to the objective function evaluated using less expensive system simulations. These surrogate simulations are built by several different classes of techniques, from simplifications of the physics underlying the more expensive models, to response surface methods, in which a surface is fitted to carefully chosen sample runs of the expensive simulation.
Research in this area by CRPC Optimization Group leader John Dennis and collaborators focuses on the fact that a decision variable vector obtained from optimizing an approximation model can be evaluated for progress in optimizing the expensive problem by using a single expensive simulation.
The corresponding change in the objective function, whether or not it has been improved, can be compared to the change predicted by the change in the objective function using the surrogate simulation as a means of assessing the suitability of the surrogate problem for improving the decision variables. The group is using this idea to build algorithms for optimizing the expensive problem using steps furnished from the optimization of the inexpensive problem, with only periodic checks to verify accuracy. The algorithms use tried and true constructs borrowed from the highly successful trust-region methods of nonlinear programming. The group predicts that rigorous convergence theorems for these methods will be easy to prove once their form is finalized by computational testing.
This work is being applied in two projects: in the first project, which has the helicopter rotor design problem as its motivation, the design objective is not differentiable, so to optimize the objective, the group employs the parallel direct search algorithm developed by Dennis and Virginia Torczon and the DFO algorithm developed by Andy Conn of IBM and Philippe Toint of Leuven.
The second project now getting underway is based on the class of surrogates that match not only the object values but the gradients at points where the expensive simulation has been carried out. This work is being conducted by Dennis and Torczon as well as Natalia Alexandrov of NASA Langley and Robert Michael Lewis of ICASE. They are talking to engineers at NASA about obtaining the simulation software for a motivating example for this project in order to ensure its eventual applicability.