Many physical processes are described with partial differential equations (PDE), with important examples such as thermal conduction, Maxwell laws for electrodynamics and Navier-Stokes equations for viscous fluids.

In many applications, we do not want to find just any solution for differential equation, but an optimal solution in a specific sense: We have one or more parameters in the PDE model that we can tune, and we want to find the parameter values that are most beneficial. A classic example is to determine the shape of an airfoil, so that the airflow around the wing results in the best possible flight characteristics.

A related problem is the reconstruction of physical processes based on incomplete measurements. In such a problem, we seek the solution to the differential equation that best fits the measurement data.

Both examples above can be formulated as optimization problems with PDE constraints. In almost all applications, we can not calculate the exact solution for a PDE or a PDE-constrained optimization problem. Instead, we must compute a numerical approximate solution. For PDE-constrained optimization, this usually involves solving large sparse saddle point systems.

In his thesis work Magne has investigated methods to solve such problems in an efficient way. A key technique is operator preconditioning. This involves identifying a mathematical structure in the problem that can be exploited to derive effective solution algorithms. The thesis consists of three papers. The first two paper concerns parameter-robust preconditioning techniques for saddle point systems. In the third paper, a PDE-constrained optimization method is applied to solve a data assimilation problem for cerebral blood flow.