Optimization and Computational Fluid Dynamics - Department of ...

3 Mathematical Aspects of CFD-based Optimization 73
for optimization problems together with strategies for the adaptation of
the POD-bases can be found in [1, 22, 25].
• If the dimension of the optimization variables is comparatively low, it is
possible to apply the boundary value problem techniques described below
in a forward sensitivity driven manner as described in [19, 31, 32, 33]. The
resulting storage space required is then equivalent to the dimension of the
optimization variables times the dimension of one space discretization of y.
• The direct usage of the reduced formulation (3.20) can be considered as an
ultima ratio, if the available main memory is not large enough to exploit all
other options. Nevertheless, one has to be careful in evaluating derivatives
and also be aware that the resulting unconstrained optimization problem is
typically much more nonlinear than the original, constrained formulation
(3.18,3.19).
Furthermore, an inconsistent adjoint time integration scheme becomes a
major and frequent pitfall in CFD-optimization. It is important that the adjoint
discretization scheme is indeed adjoint to the discretized primal CFD
time integration scheme, according to the principle of internal numerical differentiation
(IND). A deviation from this requirement may lead to gradient
approximations which are not descent directions and would thus lead to
premature stopping of gradient-based algorithms. The requirement of consistency
of the adjoint scheme with the forward scheme leads typically to
adjoint schemes which cannot be interpreted as integration schemes for the
(infinite) adjoint CFD problem. More details can be found in [11].
3.3.1 Time-domain Decomposition by Multiple
Shooting
The direct multiple shooting method for optimization of unsteady processes
as described below was introduced by Bock and Plitt for optimal control in
[10, 30], and for parameter estimation problems in [5]. The basic idea is to
decompose the time history of the problem into subdomains, in which the
unsteady PDE solutions are uniquely parameterized by their initial data and
by the unknown parameters. For this purpose, one divides the time interval
[0,T] into subintervals [ti,ti+1]with
0=t0