Optimization and Computational Fluid Dynamics - Department of ...

3 Mathematical Aspects of CFD-based Optimization 75
ing technique in general and especially in the context of Nonlinear Model
Predictive Control (NMPC), the reader is referred to [13, 14].
Within classical multiple shooting, these blockwise matrices, also called
Wronskians, are built up in a sensitivity approach, which means that the
Jacobian matrix of the constraints are explicitly constructed and completely
stored. Matrix factorization techniques dovetailed to the multiple shooting
QP are applied in order to solve the quadratic programs. Note, however, that
these Jacobians need only be approximated in a coarse manner, when using
formulation (3.16, 3.17) such as by secant update techniques. In case of CFD
optimization problems they should only capture unstable and slowly decaying
modes. In certain cases, the frequent generation of the constraint Jacobian
can also be avoided, so that only the very first Jacobian is constructed and
factorized by the use of formulation (3.16, 3.17), where the matrix A denotes
the (multiple shooting) Jacobian at the data of the first optimization
iteration.
3.3.2 Parallel Multiple Shooting
Parallelization of multiple shooting is trivially based on the idea that the
initial value solvers on each multiple shooting interval can work completely
independently, provided the initial values si are available. Parallel realizations
of multiple shooting have existed for a long time, cf. [15, 24, 36], and more
recently as parareal in [27]. Ulbrich has described the use of this approach
within an optimal control context [35] in a formulation similar to Eqs. (3.16)-
(3.17). There the initial values are provided by a coarse discretization scheme
(often implicit Euler) which operates only on the multiple shooting nodes.
It is obvious that this approach works only well with CFD problems, which
are dissipative enough so that even coarse time discretizations give some
consistent information.
3.3.3 Real-time Optimization and Nonlinear Model
Predictive Control
Real-time optimization is not just fast optimization, rather it means a shift
in philosophy: in order to minimize the delay time such as for a control
response to a perturbation of the process, one tries to compute as much
information as possible before data about the values of states and parameters
or of scenarios for the real process become available in real-time. In the CFD
context this means in particular that matrix factorizations, which are often
avoided for computational speed, or the pre-computation of sophisticated
pre-conditioners, receive an increased attention, insofar as they can often be