Mini-Workshop: Numerical Methods for Instationary Control Problems
Authors: Kunisch Karl , Kunoth Angela , Rannacher Rolf 
Published: December 2009

The topic for the current Mini-Workshop organized by Karl Kunisch (Univ. Graz), Angela Kunoth (Univ. Bonn) and Rolf Rannacher (Univ. Heidelberg) emerged from the Oberwolfach Workshop “Numerical Techniques for Optimization Problems with PDE Constraints” which was held February 16–22, 2003. It was realized that numerically solving control problems which are constrained by time-dependent nonlinear PDEs (Partial Differential Equations) are particularly challenging with respect to the complexity of the problem.
Mathematically, one has to minimize a functional under PDE constraints and possibly additional constraints on the state and the control. Standard discretizations on uniform grids in space and time will only yield solutions where the inherent structures of the problem (nonlinearity, constraints) are not sufficiently captured. Certain optimization problems for large coupled systems of partial differential equations are currently not yet numerically treatable or do not satisfy the time constraints required in practice. Overcoming this barrier can only be achieved by designing new mathematically founded algorithmic approaches. The road towards this goal leads to many interesting problems in optimization, linear algebra, numerics, analysis, and approximation theory.
The conference had 21 participants which represented continuous optimization, numerical analysis and scientific computing. 18 talks were given. The 10 longer, overview-style talks were on optimization with PDEs, focussing on
These talks were intended to bridge the gap between the different research fields Optimization and Numerics. They were complemented by 8 shorter talks on more specialized research topics, ranging from efficiency indices for optimization over iterative methods for the coupled systems and multigrid acceleration to modelling issues in optimization, and data compression by means of proper orthogonal decompositions and central Voronoi tesselations.
Different modern approaches to overcome the complexity issues in numerical simulations for PDE-constrained optimization have been presented and discussed. One of the approaches is to employ fast iterative solvers like multigrid on uniform grids. The methodology which conceptually provides the largest potential is to introduce adaptivity. This drastically reduces complexity but depending on the context may require solving an additional problem. Wavelet approaches particularly allow to resolve each of the variables separately and in addition provide a built-in preconditioning. Yet another approach uses compressed information in order to efficiently solve the primal-dual system.
All these issues are currently very active research areas. The extensive discussions held during this workshop have produced a number of new ideas and connections. It was agreed upon that a mere black-box-style matching of efficient PDE codes with optimization tool boxes would on one hand remain much below its potential and on the other hand not help overcoming complexity barriers. Some concepts from automatic differentiation seem to carry over to adaptive methods. Even combining adaptivity with proper orthogonal decompositions may be a very promising direction. The many new ideas discussed during this workshop will have to be further elaborated in future.