Optimization and Computational Fluid Dynamics - Department of ...

76 H.G. Bock and V. Schulz
computed in advance, and have a potential to speed up the final solution
process.
In the special case of NMPC, a sequence of neighboring optimization problems
on a receding time horizon has to be solved, which differ only by (an
estimate of) the initial value of the states y(t0) and possibly by a change in
the set-point, i.e., parameters and scenarios. Minimal response delays can be
achieved by the “real–time iteration” (RTI) in combination with a “perturbation
embedding” as developed in [7, 13, 14]. Here, huge savings in computing
time are gained by a kind of hot start property because already the first
QP solution in a full step inexact SQP method provides a nearly tangential
approximation of the exact solution if the problem is initialized with the solution
of the previous problem, even in the presence of a change of the active
constraints. Furthermore, the calculations can be divided into a preparation
phase that can be performed without knowledge of y0, and a much shorter
feedback phase that allows to make the delay even shorter than the sampling
time. This remaining delay is typically orders of magnitude smaller than the
sampling time and thus we can consider the feedback to be instantaneous. So
the sampling time is practically only needed to prepare the following real–
time iteration. Additionally, cheap feasibility and/or optimality improving
subiterations (FOI) can be performed that reuse Jacobians and Hessians of
the previous step and require only one forward and additionally one adjoint
solution, respectively.
For a more detailed description of the real-time iteration scheme and its
convergence and nominal stability properties, please refer to [14].
3.3.4 Sensitivity Driven Multiple Shooting
In parameter estimation problems for CFD models, it is possible to use a
multiple shooting time-domain decomposition and profit from its superior
convergence properties, and on the other hand reduce the optimization space
only to the parameters to be estimated. In these problems, one uses a generalized
Gauss-Newton approach which achieves rather good convergence already
with information of first order only. The linearized least-squares subproblem
is condensed to the unknown parameters, which can be achieved while only
marching forward in time with a conventional time-integration scheme. This
idea was originally developed in [33], and has been successfully applied to
multiphase problems, e.g., in [19]. Extensions to Newton-type methods for
general objectives, e.g., in control problems were developed in [31, 32].