The numerical solution of kinetic partial differential equations (PDEs) usually exhibits high computational costs and memory requirements. This problem can be overcome when using numerical reduction techniques such as dynamical low-rank approximation (DLRA). Its main idea consists in representing and evolving the solution to a given equation on a low-rank manifold, thereby splitting up the solution of one high-dimensional problem into lower-dimensional subproblems. Efficient fully discrete DLRA schemes must be carefully constructed in order to account for the underlying structure of the problem and to ensure numerical stability. The first part of this thesis is devoted to the derivation of stable fully discrete DLRA schemes for different linear PDEs. For the thermal radiative transfer equations (RTEs) with Su-Olson closure, a provably energy stable and mass conservative DLRA algorithm is proposed. For its construction an implicit coupling of particle density and internal energy as well as a rank-adaptive augmented low-rank integrator and a suitable conservative truncation strategy are used. In certain settings, a multiplicative splitting of the kinetic distribution function is advantageous for the construction of an efficient DLRA scheme. We first reconsider the thermal RTEs with Su-Olson closure with a multiplicative splitting of the distribution function, giving rise to additional complexities in the proof of energy stability and mass conservation for the DLRA scheme. In a second step, the gained insights are transferred to the linear Boltzmann-Bhatnagar-Gross-Krook (BGK) equation. Being different in structure, a distinct notion of numerical stability is required and new ideas for basis augmentations and an appropriate truncation strategy are introduced into the mathematically rigorous proof of stability. Various numerical experiments confirm the efficiency and the accuracy of the proposed DLRA schemes and validate the theoretical results. In the second part of this thesis, the method of DLRA is applied to parameter identification inverse problems. For the reconstruction of the scattering coefficient in the RTE, a PDE constrained optimization problem together with a gradient-based iterative update scheme is formulated. The optimization procedure requires the solution of the forward and the adjoint kinetic equations in each step of the algorithm, rendering numerical computations especially in higher dimensions extremely expensive. For the reduction of computational demands a DLRA approach is applied to the fully discrete forward and adjoint equations. Its efficiency is further enhanced by using an adaptive choice of the optimization step size and of the DLRA truncation tolerance. Numerical test examples underline the applicability of DLRA to inverse problems and confirm the efficiency of the proposed method.
Lena Baumann (Thu,) studied this question.