POD for Parameter Estimation in Differential Equations

The numerical treatment of parameter identification problems for partial differential equations (PDEs) arising in diverse areas of science and (industrial or economic) applications has received an increasing amount of attention in the recent past. Parameter estimation of engineering components or systems often requires repeated, reliable, and real-time prediction of the parameters such as forces, critical stresses, flow rates or heat fluxes. In particular, parameter estimation in elliptic, but also parabolic PDEs is an important issue. If these problems can be formulated as constrained optimal control problems, efficient, fast and reliable solvers for these problems should be available. The goal of this project is to develop numerical methods that permits accurate and rapid estimation of parameters in PDEs. We propose to combine fast solution techniques in optimization (i.e., methods with locally superlinear or even quadratic rate of convergence) with a model reduction based on proper orthogonal decomposition (POD). POD is a method for deriving low order models for systems of differential equations. It is based on projecting the system onto subspaces consisting of basis elements that contain characteristics of the expected solution. This is in contrast to, e.g., finite element techniques, where the elements of the subspaces are uncorrelated to the physical properties of the system that they approximate. In our applications the POD basis is derived from solutions to the underlying PDE for different parameter values. Within the numerical optimization methods parameter dependent PDEs have to be solved. Here we apply a POD Galerkin approximation for the spatial discretoization. The used POD basis contains information about the solution of the PDEs for different parameter values. Most POD approximations focus only on temporal variations. Therefore, the development of reduced-order models for parameter dependent PDEs is much less common. The goal of our project is to derive efficient and reliable strategies for the computation of the POD basis and its use within the parameter estimation.

If phenomena in nature, economics and medicine can be sufficiently well described by mathematical models, the question of optimization of such models often arise (e.g., optimal consumption or optimal design). These problems can be solved by strategies from mathematical optimization. In this way one yields numerical algorithms, so that the problems can be solved by computers. Hence, the unknown solution variables (degrees of freedom) can be determined. In this project partial differential equations (PDEs) serve as mathematical models. As an application we focuss - beside others - on an example arising in vehicle acoustics. PDE constrained optimization leads to problems with a huge number of degrees of freedom. Therefore, model reduction techniques are utilized to reduce significantly the degrees of freedom. This implis a reduction of memory requirements and of CPU times. In the project strategies are developped to control the model reduction. This is done in such a way that the obtained solution to the reduced problem is sufficiently close to the solution of the original one. To stimate the quality of the solution to the reduced problem the computation of the original one is not required.