** Project highlights: **

Optimal design, optimal control and parameter estimation of systems governed by
partial differential equations (PDE) give rise to a class of problems referred
to as *PDE-constrained optimization* (OPT-PDE).
OPT-PDE pursues the idea to influence phenomena and processes, governed by PDEs.
As PDEs describe almost every aspect of physics, chemistry, engineering, biology,
finance etc., that fit into a continuum framework, OPT-PDE can
be regarded as the ultimate/farthermost
goal of any application problem to steer the underlying systems in a desired way.

As a starting point, we consider the OPT-PDE setting with a stationary or time-dependent reaction-diffusion equation as constraint, combined with pointwise (box) constraints of the state and the control variable, which latter is in addition sparse (localized).

The aim of the project is two-fold. On the *modelling and discretization side* we will

- study appropriate formulations of the cost functional and regularization techniques;
- by proper stabilization handle constraints and problem features applied on subregions of the control domain, not aligned with any discretization mesh;
- analyse the interplay between regularization, stabilization and discretization parameters in order to formalize their mutual dependence and how to balance the corresponding errors, the ill-conditioning of the arising algebraic systems of equations and the attainable accuracy of the computed optimal solution.

On the *solution method side* we will

- develop efficient preconditioners to speed up the linear solvers, used in the nonlinear solution procedure, inevitably required for the target OPT-PDE problem;
- develop two- and multilevel mixed nonlinear-linear methods to reduce the computational cost of the nonlinear solver on the finest discretization level, required to guarantee a desired level of accuracy;
- on all stages of development and implementation of the solution procedures and algorithms, enhance their performance on HPC resources.

First results:

State | Control | State | Sparse control |

Article in review: I. Dravins, M. Neytcheva,
*PDE-Constrained Optimization: Matrix Structures and Preconditioners *

As a far goal, the constraint could be extended to a system of equations, to model coupled heating-cooling physics processes.

Maya Neytcheva Gunilla Kreiss Ivo Dravins |

Scientific Computing, Uppsala University |