Space Mapping for Optimal Control of Partial Differential Equations

Solving optimal control problems for nonlinear partial differential equations represents a significant numerical challenge due to the tremendous size and possible model difficulties (e.g., nonlinearities) of the discretized problems. In this paper, a novel space-mapping technique for solving the aforementioned problem class is introduced, analyzed, and tested. The advantage of the space-mapping approach compared to classical multigrid techniques lies in the flexibility of not only using grid coarsening as a model reduction but also employing (perhaps less nonlinear) surrogates. The space mapping is based on a regularization approach which, in contrast to other space-mapping techniques, results in a smooth mapping and, thus, avoids certain irregular situations at kinks. A new Broyden’s update formula for the sensitivities of the space map is also introduced. This quasi-Newton update is motivated by the usual secant condition combined with a secant condition resulting from differentiating the space-mapping surrogate. The overall algorithm employs a trust-region framework for global convergence. Issues involved in the computations are highlighted, and a report on a few illustrative numerical tests is given.