Optimization of Parabolic Systems: Iterative Methods, Suboptimal Controls, and Preconditioning

William Marsh Rice University, Houston TX

Investigators

Abstract

Optimization of linear or nonlinear parabolic differential equations in the context of optimal control, optimal design, or parameter estimation plays an important role in science and engineering. Algorithms for the solution of parabolic equations often involve marching in time, starting from an initial condition. In optimization, however, the values of the solution of the parabolic equation at later times feed into the optimization at early times. This coupling in time makes the practical solution of these very large-scale optimization problems challenging. To cope with storage and computer time demands required by an exact optimization of parabolic systems, so-called suboptimal control techniques, such as reduced bases techniques and instantaneous control have recently been proposed. The analysis of these techniques is still incomplete and the limits of their applicability are not clearly described. This research integrates selected suboptimal control techniques into an optimization framework, where they are interpreted as truncated iterative methods or are used as preconditioners in optimization methods. This improves our theoretical understanding of these techniques and broadens their applicability. The resulting methods are applied to specific optimal control problems in or related to fluid mechanics. Optimal control attempts to determine system parameters or inputs to increase the performance of the system. For example, micro electromechanical systems may be used to alter the flow characteristics on an aircraft wing to reduce drag, or heaters may be adjusted to achieve a desired temperature profile in a furnace while minimizing energy consumption. Many systems can be modeled by mathematical equations. In this case mathematical techniques can be used, at least in principle, to determine the optimal system inputs. For systems that can be adequately modeled by moderately complex mathematical equations this is done routinely and successfully. Detailed mathematical descriptions of other systems, including flow over an aircraft wing or the temperature distribution in a furnace, however, are so complex that present mathematical techniques for the determination of optimal control strategies require such large computer resources that render them impractical. The goal of this research is to develop and analyze computational mathematics tools for the determination of optimal control strategies for a class of complex systems and the demonstration of the practicability of these tools using selected applications in fluid mechanics.