A geometric multigrid solver for the incompressible Navier-Stokes equations using discretely divergence-free finite elements in 3D

A geometric multigrid solution technique for the incompressible Navier-Stokes equations in three dimensions is presented, utilizing the concept of discretely divergence-free finite elements without requiring the explicit construction of a basis on each mesh level. For this purpose, functions are constructed in an a priori manner spanning the subspace of discretely divergence-free functions for the Rannacher-Turek finite element pair under consideration. Compared to mixed formulations, this approach yields smaller system matrices with no saddle point structure. This prevents the use of complex Schur complement solution techniques and more general preconditioners can be employed. While constructing a basis for discretely divergence-free finite elements may pose significant challenges and its use prevents a structured assembly routine, a basis is utilized only on the coarsest mesh level of the multigrid algorithm. On finer grids, this information is extrapolated to prescribe boundary conditions efficiently. Here, special attention is required for geometries introducing bifurcations in the flow. In such cases, so called ‘global’ functions with an extended support are defined, which can be used to prescribe the net flux through different branches. Various numerical examples for meshes with different shapes and boundary conditions illustrate the strengths, limitations, and future challenges of this solution concept

Investigation of lncRNA GRASLND in the context of melanoma differentiation and IFNy response

Melanoma, originating from malignant melanocytes, is the leading cause of skin cancer-related deaths arises from its high metastatic potential, heterogeneity, and phenotypic plasticity. This plasticity drives tumor progression, metastasis, and therapy resistance. Immune checkpoint blockade (ICB) immunotherapy, the primary treatment for metastatic melanoma, enhances cytotoxic T lymphocyte (CTL) activity but faces challenges due to resistance. Resistance is linked to impaired HLA class I antigen presentation and loss of melanocytic differentiation antigens via phenotype switching, allowing immune evasion. Long non-coding RNAs (lncRNAs) regulate key melanoma processes, including immune escape, proliferation, metastasis, and drug resistance, but their role in melanoma plasticity remains largely unknown. This study investigated lncRNA GRASLND in melanoma, building on its known role in mesenchymal stem cells (MSCs) as an inhibitor of IFNγ signaling, a key pathway regulating HLA class I-APM and CTL immunogenicity via interaction with PKR. GRASLND was overexpressed in melanoma tumors, associated with a differentiated cell state, and linked to poor prognosis. Its knockdown induced a shift from a proliferative melanocytic state to a dedifferentiated, invasive state. GRASLND was also enriched in immune “cold” tumors and inversely correlated with immune response activation. Notably, IFNγ reduced GRASLND expression, leading to increased ISG and HLA-I expression, confirming its suppressive effect on IFNγ signaling in melanoma. The GRASLND-PKR interaction was validated, suggesting an immune evasion mechanism. Targeting this interaction may help counteract immune escape in melanoma

Compilation-based explainability of tree ensembles

Machine learning, particularly deep learning, has achieved remarkable success in areas such as image recognition, natural language processing, and speech recognition. However, for structured or tabular data, tree ensemble approaches, such as random forests and gradient boosted trees, often outperform deep learning-based approaches. Although they perform strongly, tree ensembles are in general considered to be black boxes, because the complexity of combining multiple decision trees makes it difficult to trace the reasoning behind individual predictions. Recent advancements in Explainable Artificial Intelligence have presented heuristic post-hoc explanation methods like LIME and SHAP. Nevertheless, these methods often rely on a model's input-output behavior and therefore only approximate how it internally arrives at its predictions. As a result, there is an urgent need for efficient and precise approaches to explaining ensembles, especially in domains where safety or fairness is critical. To tackle the interpretability gap, this thesis explores compilation-based techniques that transform an entire tree ensemble into a single, semantically equivalent structure such as a directed acyclic graph. This single graph representation reveals the ensemble's underlying logic, making it amenable to formal analysis. Once compiled, the model's internal logic becomes more transparent, allowing efficient generation of formal explanations, as well as support for verification tasks such as pre- and postcondition checks and model equivalence checking. One significant advantage is that, after the one-time cost of building the unified representation, subsequent explanations can be generated efficiently. This makes the proposed solutions particularly well-suited for real-time or interactive settings where many explanations are requested in sequence. The main focus of this thesis is efficiency and scalability. Existing compilation-based approaches can be very expensive on large ensembles. To address this, the thesis introduces novel compilation algorithms and optimizations, significantly reducing transformation time and memory usage while maintaining exact equivalence with the original tree ensemble. The experimental results show over an order of magnitude speedup in model compilation and multiple orders of magnitude in explanation generation compared to state-of-the-art solver-based approaches. Furthermore, the thesis introduces a user-friendly, web-based tool (Forest GUMP) that allows non-experts to train, visualize, verify, and explain tree ensembles interactively. Overall, these contributions advance the field of explainable AI by delivering efficient, formally grounded, and practical solutions for tree ensemble interpretability and explainability

A posteriori error analysis for optimization with PDE constraints

We consider finite element solutions to optimization problems, where the state depends on the possibly constrained control through a linear partial differential equation. Basing upon a reduced and rescaled optimality system, we derive a posteriori bounds capturing the approximation of the state, the adjoint state, the control and the observation. The upper and lower bounds show a gap, which grows with decreasing cost or Tikhonov regularization parameter. This growth is mitigated compared to previous results and can be countered by refinement if control and observation involve compact operators. Numerical results illustrate these properties for model problems with distributed and boundary control

Development of subjective well-being in adolescents before and during the COVID-19 pandemic

Previous studies have already revealed detrimental effects of the COVID-19 pandemic on school students’ subjective well-being (SWB). However, there is a lack of studies examining the development of various facets of SWB such as life satisfaction, mood as well as domain satisfactions regarding peers, family, or school before and during the pandemic among adolescents longitudinally. Furthermore, the present study aims to shed further light on various moderators such as gender, age, migration background and socioeconomic status. Data from N = 207 students (Grade 5 to 9) from two German schools were assessed on four measurement time points, three before and one after the onset of the pandemic. Piecewise latent growth curve models with three time slopes were conducted to investigate the development of SWB and its moderators. They showed significant declines in general mood and domain-specific satisfaction with family, peers and school before the COVID-19 pandemic. However, during the COVID-19 pandemic, only satisfaction with family decreased significantly. Among the moderators, especially the socioeconomic status indicated interindividual differences in the variation of different SWB facets