Margenberg, Nils

In this work, we present an anisotropic multi-goal error control based on the Dual Weighted Residual (DWR) method for time-dependent convection-diffusion-reaction (CDR) equations. This multi-goal oriented approach allows for an accurate and efficient error control with regard to several quantities of interest simultaneously. Using anisotropic interpolation and restriction operators, we obtain elementwise error indicators in space and time, where the spatial indicators are additionally separated with respect to the single directions. The directional error indicators quantify anisotropy of the solution with respect to the goals, and produce adaptive, anisotropic meshes that efficiently capture layers. To prevent spurious oscillations the streamline upwind Petrov-Galerkin (SUPG) method is applied to stabilize the underlying system in the case of high Péclet numbers. Numerical examples show efficiency and robustness of the proposed approach for several goal quantities using established benchmarks for convection-dominated transport.

We present an anisotropic goal-oriented error estimator based on the Dual Weighted Residual (DWR) method for time-dependent convection-diffusion-reaction (CDR) equations. Using anisotropic interpolation operators the estimator is elementwise separated with respect to the single directions in space and time leading to adaptive, anisotropic mesh refinement in a natural way. To prevent spurious oscillations the streamline upwind Petrov-Galerkin (SUPG) method is applied to stabilize the underlying system in the case of high Péclet numbers. Efficiency and robustness of the underlying algorithm are demonstrated for different goal functionals. The directional error indicators quantify anisotropy of the solution with respect to the goal, and produce meshes that efficiently capture sharp layers. Numerical examples show the superiority of the proposed approach over isotropic adaptive and global mesh refinement using established benchmarks for convection-dominated transport.

This paper presents a mathematical foundation for physical models in nonlinear optics through the lens of evolutionary equations. It focuses on two key concepts: well-posedness and exponential stability of Maxwell equations, with models that include materials with complex dielectric properties, dispersion, and discontinuities. We use a Hilbert space framework to address these complex physical models in nonlinear optics. While our focus is on the first-order formulation in space and time, higher solution regularity recovers and equates to the second-order formulation. We incorporate perfectly matched layers (PMLs), which model absorbing boundary conditions, to facilitate the development of numerical methods. We demonstrate that the combined system remains well-posed and exponentially stable. Our approach applies to a broad class of partial differential equations (PDEs) and accommodates materials with nonlocal behavior in space and time. The contribution of this work is a unified framework for analyzing wave interactions in advanced optical materials.

We consider Biot’s equations of poroelasticity where the development of viscous boundary layers in the pores is allowed for by using a dynamic permeability convolution operator in the time domain. This system with memory effects is also referred to as the dynamic Biot–Allard model. We use a series representation of the dynamic permeability in the frequency domain to rewrite the equations in the time domain in a coupled system without convolution integrals, which is also suitable for designing efficient numerical approximation schemes. The main result here is the well-posedness of the system, rewritten in evolutionary form, which is proved by an abstract theory for evolutionary problems.

The subject of this thesis is the efficient simulation of nonlinear optical phenomena, especially frequency mixing processes. Unlike linear optics, where the response of the material to light remains constant regardless of the light's intensity, in nonlinear optics, this response varies as a function of the light's intensity. Nonlinear optical phenomena have paved the way for significant scientific innovations, e.g. the development of novel optical sources, and have become central to a wide range of technological applications. This work is located in the emerging field of scientific machine learning and contributes to the efficient simulation of nonlinear optical phenomena by combining the rigor of finite element methods with the empirical strength of deep neural networks. Specifically, we focus on frequency mixing processes in the context of THz generation in periodically poled nonlinear crystals through Quasi-Phase-Matching. We develop a mathematical model and numerical simulation techniques in the time domain to simulate complex nonlinear optical processes without the usual slowly varying envelope approximation. The accuracy is illustrated through numerical simulations and comparisons to experimental data. The simulations elucidate the THz generation in periodically poled Lithium Niobate (PPLN), including optical harmonic generation. This thesis introduces a novel approach for solving optimal Dirichlet boundary control problems of nonlinear optics using deep learning. For computing high-resolution approximations of the solution to the nonlinear wave model, we propose higher order space-time finite element methods combined with collocation techniques, ensuring C^l-regularity in time of the global discrete solution. The simulation data trains solution operators that effectively leverage the higher regularity of the training data. The operators, represented by Fourier Neural Operators, are used as the forward solver in the optimal Dirichlet boundary control problem. This approach demonstrates an innovative integration of traditional numerical methods with data-driven techniques. The proposed algorithm is implemented and tested on high-performance computing platforms, with a focus on efficiency and scalability. We showcase its effectiveness on the problem of generating Terahertz radiation in PPLN, optimizing the parametrization of the optical input pulse to maximize the yield of 0.3THz-frequency radiation. By exploiting the periodic layering of the crystal, the networks are trained to learn propagation through one period of the layers. The recursive application of the network onto itself yields an approximation to the full problem. The results indicate a significant speedup compared to classical methods and, when compared to experimental data, demonstrate the potential to revolutionize optimization in nonlinear optics. The proposed approach of data-driven acceleration contributes to a paradigm shift towards more efficient and innovative problem-solving methods in the realm of nonlinear optics and beyond. In addition, we review a promising approach to showing the well-posedness of the nonlinear Maxwell equation and show how it can be applied in the settings we consider throughout this thesis. To this end, the nonlinear wave equations are rewritten as a first-order system in space and time. The well-posedness is ensured by a combination of the abstract theory for evolutionary problems by R. Picard and fixed point arguments.