AbstractsMathematics

In this thesis, we advance efficient methods to solve parameter estimation problems constrained by partial differential equations (PDEs). If PDE constrained parameter estimation problems are solved by derivative based methods, here, the generalized Gauss – Newton method, and multiple shooting, the numerical effort growths drastically with the number of states. The reduced approach couples the computation of the Jacobians and the subsequent block Gaussian elimination using directional derivatives by exploiting the special structure of the constraints which arises from the shooting formulation. Thus, the computational effort is reduced to the one of single shooting. The advantages of the new method in comparison to the common approach are illustrated by means of two academic examples. Furthermore, we are the first to adapt methods of optimum experimental design for parameter estimation to processes of microbial enhanced oil recovery. We consider a nonlinear coupled PDE model which consists of two parts. The first part, the black oil model, describes two phase flow through porous media and a model of convection – diffusion – reaction type depicts the transport and growth effects of bacteria, nutrients, gas and other metabolites in the two phases. A mixed discontinuous Galerkin finite element discretization is applied in space. The discretized model is solved in time by the extended IMPES method. Under the assumption of rotational symmetry, we examine a one dimensional model formulation for parameter estimation and optimum experimental design. We follow the principles of internal numerical differentiation and algorithmic differentiation to evaluate the required derivatives, i.e., the derivatives of the model functions are computed by software tools and we solve the tangential problems with respect to the model parameters and the control variables. By optimum experimental design, a new experiment is planned to reduce the uncertainties of the estimated parameters. The designed experiment differs substantially from the experiments which are usually realized in practice. The confidence intervals for the estimated parameters are reduced by a factor of one hundred. The developed methods for parameter estimation are implemented in the software package PAREMERA which is embedded in the optimum experimental design software VPLAN. The model equations for microbial enhanced oil recovery are implemented in a simulation tool which computes not only the nominal equation but also evaluates the derivatives with respect to parameters and controls up to second order.