TY - THES A1 - Wulkow, Hanna T1 - Regularization of Elliptic Partial Differential Equations using Neural Networks N2 - This thesis presents a method for interpolating data using a neural network. The data is sparse and perturbed and is used as training data for a small neural network. For severely perturbed data, the network does not manage to find a smooth interpolation. But as the data resembles the solution to the one-dimensional and time-independent heat equation, the weak form of this PDE and subsequently its functional can be written down. If the functional is minimized, a solution to the weak form of the heat equation is found. The functional is now added to the traditional loss function of a neural network, the mean squared error between the network prediction and the given data, in order to smooth out fluctuations and interpolate between distanced grid points. This way, the network minimizes both the mean squared error and the functional, resulting in a smoother curve that can be used to predict u(x) for any grid point x. KW - elliptic partial differential equations KW - neural networks Y1 - 2020 ER - TY - THES A1 - Baumann, Felix T1 - Impact of Contact Surfaces on the Death Estimation N2 - The determination of time of death is one of the central tasks in forensic medicine. A standard method of time of death estimation elies on matching temperature measurements of the corpse with a post-mortem cooling model. In addition to widely used empirical post-mortem models, modelling based on a precise mathematical simulation of the cooling process have been gaining popularity. The simulation based cooling models and the resulting time of death estimates dependon a large variety of parameters. These include hermal properties for different body tissue types, environmental conditions such as temperature and air flow, and the presence of clothing and coverings. In this thesis we focus on a specific arameter - the contact between corpse and underground - and investigate its influence on the time of death estimation. Resulting we aim to answer the question whether it is necessary to consider contact mechanics in the underlying mathematical cooling model. KW - time of death estimation KW - contact mechanics KW - heat equation KW - inverse problems Y1 - 2020 ER - TY - THES A1 - Semler, Phillip T1 - Mixed finite elements in cloth simulation N2 - This master thesis investigates the use and behaviour of a mixed finite element formulation for the simulation of garments. The garment is modelled as an isotropic shell and is related to its mid-surface by energetic degeneration. Based on this, an energy functional is constructed, which contains the deformation and the mid-surface vector as degree of freedom. It is then shown why this problem does not correspond to a saddle point problem, but to a non-convex energy minimization. The implementation of the energy minimization takes place with the ZIB-internal FE framework Kaskade7.4, whereby a geometric linear and different geometric non-linear problems are examined, whereby for a selected, non-linear example a comparison is made with an existing implementation on basis of Morley elements. The further evaluations include the analysis of the quantitative and qualitative results, the used solution method, the behaviour of the system energy as well as the used CPU time. KW - cloth simulation KW - shell elements KW - mixed finite elements Y1 - 2020 ER - TY - THES A1 - Betz, Fridtjof T1 - Investigation and application of contour integral methods for resonance expansion in nanophotonics Y1 - 2020 ER - TY - THES A1 - Mathew, Maria T1 - Numerical treatment of radiative heat transfer on cooling process N2 - This thesis examines how taking into account surface to surface radiation impacts the cooling process in general. We formulate the non local bound- ary condition after introducing the general setting for the cooling model. In section 3, the mathematical description of the radiative heat transfer is dis- cussed. We cover the implementation of the radiative matrix in section 4, which is followed by a brief explanation of the radiative matrix’s structure and several techniques to dealing with the accompanying challenges. We investigate the importance of radiative heat transport by applying the given approach to a two-dimensional geometry and computing the ensuing cooling curves. We compare the findings of our computation to those ac- quired from experiment conducted and find that they are extremely similar. There is a considerable difference (of about 35%) in the time of cooling of the surface where there is a possibility of influence of radiation from the second surface to that of the surface with no influence at all. Although it is possible to infer that heat convection plays a role in the total result, this has yet to be proved. However, one can clearly see the significance of the surface to surface radiative heat transfer on these parts confirming the research question posed at the begining. The effect of the surface to surface radiative heat transfer has an influence on the resulting cooling time and should be considered in the model. Y1 - 2021 ER - TY - THES A1 - Rabben, Robert Julian T1 - Ein holistischer Ansatz zur Analyse molekularer Konformationen auf der Basis von ISOKANN Y1 - 2022 ER - TY - THES A1 - Villani, Paolo T1 - Regression Techniques for Surrogate Modelling in Bayesian Inverse Problems N2 - For many real-world applications, a system of interest can be represented via a mathematical model which depends on a set of parameters. In order to identify the parameters, a set of observations is available and an Inverse Problem is formulated. Identifying the parameters from the observations is often a challenging task, especially when the model is expensive to evaluate. This is the case for Partial Differential Equations models, where numerical simulations which are both inexact and computationally expensive are required to obtain the model output. To ease the computational costs, surrogate models can be used to approximate the forward model. In this work, we present two different regression techniques, Gaussian Process Regression and Lipschitz Regression. After reformulating the Inverse Problem to account for the surrogate model, we develope an adaptive training strategy to train the surrogate model. The proposed training strategy aims at optimizing not only the training points’ positions but also their evaluation accuracies. Moreover, interleaved sampling of the posterior distribution of the unknown parameters is performed while the surrogate model is trained, providing a solution for the Inverse Problem. The quality of the surrogating techniques as well as the effectiveness of the adaptive training strategy are tested through different numerical experiments. Y1 - 2025 ER -