@article{1114.49033,
author="Hinterm\"uller, Michael and Vicente, Lu\'{\i}s N.",
title="{Space mapping for optimal control of partial differential equations.}",
language="English",
journal="SIAM J. Optim. ",
volume="15",
number="4",
pages="1002-1025",
year="2005",
doi={10.1137/S105262340342907X},
abstract="{Summary: 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
    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.}",
keywords="{space mapping; optimal control of PDEs; simulation-based
    optimization; PDE-constrained optimization; trust regions; quasi-Newton
    methods}",
classmath="{*49M15 (Methods of Newton-Raphson, Galerkin and Ritz types)
49K20 (Optimal control problems with PDE (nec./ suff.))
49M25 (Finite difference methods)
90C26 (Nonconvex programming)
90C53 (Methods of quasi-Newton type)
}",
}