Optimization and Computational Fluid Dynamics - Department of ...

72 H.G. Bock and V. Schulz
Itisshownin[12]thatthereforethecoarse grid correction is indeed a secant
direction for the optimization problems on the higher refinement levels.
A typical area of application for this multigrid strategy is the inverse problems
for distributed parameters and control or shape optimization problems,
where the optimization variable is of function type.
3.3 Unsteady Problems
In principle, unsteady optimization problems of the form (3.1, 3.2, 3.3) can
be treated by the methods discussed above after a discretization in space
and time. However, the adjoint variables to the CFD equation always have
the same dimensionality as the solution of the CFD equation, i.e., if y is a
function of space and time, then so is λ. When performing simulations of
CFD problems, one typically provides only a space discretization of the flow
variables and marches forward in time so that during the whole simulation,
the main memory is only populated by one (or two, but at most six, depending
on the time marching scheme) spatially discretized state variable pertaining
to the solution at the current time-step. Unfortunately, the time direction is
reversed for the corresponding adjoint problems and, in nonlinear problems,
the solution of the (discretized) adjoint problem needs information from the
(discretized) primal CFD solution y. That means, a direct application of the
methodology sketched above to time-dependent problems requires storage for
at least the size of the whole space-time history of the solution y of the CFD
problem. If this amount of storage easily fits into the available memory, one
can just skip the following comments and continue with the next section,
where we discuss special features of the resulting SQP methods. Otherwise,
there are mainly four options:
• It is possible to counteract the lack of storage by investing more computing
time. In the situation of time-dependent optimization problems, this can
be facilitated by the use of so-called check-pointing strategies as described
in [18]. The algorithmic complexity then grows only logarithmically in the
number of time-steps to be thus virtually stored and so does the actual
storage space required. Note that the multiple shooting approach outlined
below provides a natural kind of checkpoints.
• Another interesting option is model reduction. In a trivial form this could
just mean choosing a coarser space-grid, which is however not feasible in
many cases. But the same line of thought leads to proper orthogonal decomposition
(POD), where low dimensional ansatz spaces are constructed
by the use of snapshots, which mirror already the major properties of the
solution of the CFD problem. The result is a low dimensional system of
Ordinary Differential Equations (ODE) or Differential Algebraic Equations
(DAE), for which the optimization problems can be solved in the
fashion of the methods discussed below. A description of the use of POD