Optimization and Computational Fluid Dynamics - Department of ...

70 H.G. Bock and V. Schulz
Next, the solution of optimization step ∆p = −B −1 g is distributed to all
approximate linearized forward problems
Ai∆yi + ci,p∆p = −ci
which can then again be solved in parallel.
3.2.4 Multigrid Optimization
The main motivation for multigrid (MG) methods is derived from the observation
that the convergence properties of most iterative methods for the
solution of systems of equations, or of optimization problems, deteriorate
when the discretization mesh is refined. This means that when approaching
the continuous problem by the use of successively finer meshes, the effort
required for the solution of the resulting discretized (non-)linear systems increases
more than linearly with the number of variables.
The promise of MG methods is to provide grid refinement independent
convergence rates by performing more work on coarser grids than on the
finer ones. This goal of optimal numerical complexity (which grows then only
linearly with respect to the number of unknowns) can be achieved by MG
methods in many cases, particularly if the problem possesses the feature that
coarser grids are able to provide improvements for finer grids. This is the
reason why theoretical convergence results always assume that the coarsest
grid is already sufficiently fine. The highest advantage can be gained from MG
methods, which are optimally adapted to the problem under investigation,
in particular with respect to the spatial distribution of variables in so-called
geometric MG methods. With general-purpose MG method solvers, e.g., of
algebraic type, one can not expect to achieve optimal complexity, but can
however expect to obtain high flexibility.
Besides the huge progress in MG methods for simulation problems, the
field of multigrid optimization (MG/OPT) has also come to a certain degree
of maturity. The review [12] gives an up-to-date survey on the diverse
MG/OPT approaches in the literature. In the following, we want to focus
on the most simply implementable MG/OPT approach, which is based on
an MG structure only with respect to the free variables p in the reduced
formulation (3.20).
Numerical experiments, e.g. [28], demonstrate that MG/OPT greatly
improves the efficiency of the underlying optimization scheme used as a
“smoother”, suggesting that the MG/OPT scheme may be beneficial in combination
with well known optimization algorithms. This claim appears to be
true as long as a line search along the coarse-grid correction is performed.
Also in [28], it is reported that MG/OPT without a line search diverges in
some cases. Therefore, a line search appears to be necessary for convergence.