Optimal Wavelet Solvers for Control Problems Constrained by Evolution PDEs


Abstract: Numerical solvers for PDEs (partial differential equations)
have matured over the past decades in effciency, largely due to the
development of sophisticated algorithms based on closely intertwining
theory with numerical analysis. Consequently, systems of PDEs as they
arise from optimization problems with PDE constraints also have become
more and more tractable. For constraints in form of a time-dependent
PDE, the first-order necessary conditions for optimality
require to repeatedly solve a system of PDEs for the unknowns (state,
costate and control) which is coupled globally in space and time. For
these, conventional time-marching methods quickly reach computational
limitations due to the massive requirement of storage. To overcome
this problem, adaptive methods which judiciously distribute degrees of
freedom with respect to both space and time during the computations to
resolve singularities in the data and/or domain appear to be most
promising.

Specifying the situation to control problems constrained by a linear
evolution PDE of parabolic type, I start out by formulating the
constraint in a weak space-time form. This is the basis for the design
of an adaptive wavelet method for which convergence and optimal
convergence rates (when compared to wavelet-best N -term
approximations, a concept from nonlinear approximation theory) can be
shown. In addition, wavelets as discretization ingredients allow for
fast solvers of optimal linear complexity. Furthermore, the even more
difficult and recently increasingly more important class of
parameter-dependent control problems depending on even possibly
countable many parameters can be attacked in this framework.

I would like to conclude with some remarks concerning corresponding
finite element discretizations. The results were obtained
in collaboration with Max Gunzburger (Florida State University) and
with Christoph Schwab (ETH Zürich).