Pseudo-Time Methods for Constrained Optimization Problems Governed by PDE

In this paper we present a novel method for solving optimization problems governed by partial di erential equations. Existing methods use gradient information in marching toward the minimum, where the constrained PDE is solved once (sometimes only approximately) per each optimization step. Such methods can be viewed as a marching techniques on the intersection of the state and costate hypersurfaces while improving the residuals of the design equation per each iteration. In contrast, the method presented here march on the design hypersurface and at each iteration improve the residuals of the state and costate equations. The new method is usually much less expensive per iteration step, since in most problems of practical interest the design equation involves much fewer unknowns than either the state or costate equations. Convergence is shown using energy estimates for the evolution equations governing the iterative process. Numerical tests shows that the new method allows the solution of the optimization problem in cost equivalent to solving the analysis problem just a few times, independent of the number of design parameters. The method can be applied using single grid iterations as well as with multigrid solvers. This research was supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-19480 while the author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23681-0001.