This work focuses on model order reduction for parabolic partial differential equations with parametrized random input data. The input data enter the model via model coefficients, external sources or boundary conditions, for instance. The outcome of the model problem is not only the solution, but also a quantity of interest (or output). The output is determined by a functional which maps the solution to a real number. Random data cause randomness of the outcomes of the model problem and, hence, statistical quantities are of interest. In particular, this work studies the expected value of the solution and the output. In order to approximate the expectation, a Monte Carlo estimator is utilized. For high accuracy Monte Carlo requires many evaluations of the underlying problem and, hence, it can become computationally infeasible. In order to overcome this computational issue, a reduced basis method (RBM) is considered. The RBM is a Galerkin projection onto a low-dimensional space (reduced basis space). The construction of the reduced basis space combines a proper orthogonal decomposition (POD) with a greedy approach, called POD-greedy algorithm, which is state of the art for the RBM for parabolic problems. The POD-greedy uses computationally cheap error estimators in order to build a reduced basis.
This thesis proposes efficient reduced order models regarding the expected value of the errors resulting from the model order reduction. To this end, the probability density function of the random input data is used as a weight for the reduced space construction of the RBM, called weighted RBM. In the past, a weighted RBM has been successfully applied to elliptic partial differential equations with parametrized random input data. This work combines the ideas of a RBM for parabolic partial differential equations Grepl and Patera (2005) and a weighted RBM for elliptic problems Chen et al. (2013) in order to extend the weighted approach also for the RBM for parabolic problems.
The performance of a non-weighted and a weighted approach are compared
numerically with respect to the expected solution error and the expected output error. Furthermore, this work provides a numerical comparison of a non-weighted RBM and a weighted RBM regarding an optimum reference. The reference is obtained by an orthogonal projection onto a POD space, which minimizes the expected squared solution error
