Efficiensy study of solution methods for discrete optimization problems, constrained by partial differential equations
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 problems (OPT-PDE). These problems arise in many important applications. As the name suggests, in the OPT-PDE framework optimization methods are coupled with methods to numerically solve PDEs. The task is to steer the solution of some process in order to fulfill various requirements imposed on it. A classical example is the construction of a machine to meet some desired performance specifications. These problems can also involve finding some unknown parameter function in the equation, which is then estimated based on a control function in the form of measurements, in the interior or on the boundary of the domain of definition of the PDE. In such problems one must solve discretized PDEs many times and, unless properly handled, this can demand an unfeasible amount of computational labor, computer time and memory, even on the presently available high performance computers. Because of the large number of degrees of freedom, much larger than when we only solve discrete PDE problems, iterative methods are the methods of choice and, therefore, constructing efficient preconditioners for this class of problems remains a hot topic of research. Highly parallelizable and numerically efficient preconditioning techniques and their performance on HPC computer platforms is the central topic of this project.