Publications Server of the Weierstrass Institute for Applied Analysis and Stochastics
Not a member yet
7600 research outputs found
Sort by
An Eulerian formulation for dissipative materials using Lie derivatives and GENERIC
Full text linkWe recall the systematic formulation of Eulerian mechanics in terms of Lie derivatives along the vector field of the material points. Using the abstract properties of Lie derivatives we show that the transport via Lie derivatives generates in a natural way a Poisson structure on the chosen phase space. The evolution equations for thermo-viscoelastic-viscoplastic materials in the Eulerian setting is formulated in the abstract framework of GENERIC (General Equations for Non-Equilibrium Reversible Irreversible Coupling). The equations may not be new, but the systematic splitting between reversible Hamiltonian and dissipative effects allows us to see the equations in a new light that is especially useful for future generalizing of the system, e.g.for adding new effects like reactive species
Hellinger--Kantorovich gradient flows: Global exponential decay of entropy functionals
Full text linkWe investigate a family of gradient flows of positive and probability measures, focusing on the Hellinger--Kantorovich (HK) geometry, which unifies transport mechanism of Otto--Wasserstein, and the birth-death mechanism of Hellinger (or Fisher--Rao). A central contribution is a complete characterization of global exponential decay behaviors of entropy functionals under Otto--Wasserstein and Hellinger-type gradient flows. In particular, for the more challenging analysis of HK gradient flows on positive measures---where the typical log-Sobolev arguments fail---we develop a specialized shape-mass decomposition that enables new analysis results. Our approach also leverages the Polyak--Łojasiewicz-type functional inequalities and a careful extension of classical dissipation estimates. These findings provide a unified and complete theoretical framework for gradient flows and underpin applications in computational algorithms for statistical inference, optimization, and machine learning
Research data to Ch. 4 "Machine-learning based multigrid methods" of the dissertation "Turbulence modeling and blood flow: impacts and alternatives" (Freie Universität Berlin, 2025)
No full textThis is the dataset related to Ch. 4 "Machine-learning based multigrid methods" of the dissertation "Turbulence modeling and blood flow: impacts and alternatives" (Freie Universität Berlin, 2025). Following data are available here: - data.tar.gz: Simulation inputs and results. - scripts.tar.gz: Data handling and machine learning scripts. - dnn_k.tar.gz: DNN-K learning results. - dnn_k5.tar.gz: DNN-K-5 learning results. - dnn_h_*.tar.gz: DNN-H learning results, separated by node-level hidden layer width - gnn.tar.gz: GNN-DOF learning results. - gnn_k.tar.gz: GNN-K learning results
Uniqueness and regularity of weak solutions of a drift-diffusion system for perovskite solar cells
No full textWe establish a novel uniqueness result for an instationary drift-diffusion model for perovskite solar cells. This model for vacancy-assisted charge transport uses Fermi--Dirac statistics for electrons and holes and Blakemore statistics for the mobile ionic vacancies in the perovskite. Existence of weak solutions and their boundedness was proven in a previous work. For the uniqueness proof, we establish improved integrability of the gradients of the charge-carrier densities. Based on estimates obtained in the previous paper, we consider suitably regularized continuity equations with partly frozen arguments and apply the regularity results for scalar quasilinear elliptic equations by Meinlschmidt & Rehberg, Evolution Equations and Control Theory, 2016, 5(1):147-184
Dictionary learning based regularization in quantitative MRI: A nested alternating optimization framework
No full textIn this article we propose a novel regularization method for a class of nonlinear inverse problems that is inspired by an application in quantitative magnetic resonance imaging (MRI). It is a special instance of a general dynamical image reconstruction problem with an underlying time discrete physical model. Our regularization strategy is based on dictionary learning, a method that has been proven to be effective in classical MRI. To address the resulting non-convex and non-smooth optimization problem, we alternate between updating the physical parameters of interest via a Levenberg-Marquardt approach and performing several iterations of a dictionary learning algorithm. This process falls under the category of nested alternating optimization schemes. We develop a general such algorithmic framework, integrated with the Levenberg-Marquardt method, of which the convergence theory is not directly available from the literature. Global sub-linear and local strong linear convergence in infinite dimensions under certain regularity conditions for the sub-differentials are investigated based on the Kurdyka?Lojasiewicz inequality. Eventually, numerical experiments demonstrate the practical potential and unresolved challenges of the method
Bifurcations and intermittency in coupled dissipative kicked rotors.
No full textWe investigate the emergence of complex dynamics in a system of coupled dissipative kicked rotors and show that critical transitions can be understood via bifurcations of simple states. We study multistability and bifurcations in the single-rotor model, demonstrating how these give rise to a variety of coexisting spatial patterns in a coupled system. A combined order parameter is introduced to characterize different spatial patterns and to reveal the coexistence of chaotic and regular attractors. Finally, we illustrate an intermittent phenomenon near the onset of chaos
Stable time rondeau crystals in dissipative many-body systems
Full text linkDriven systems offer the potential to realize a wide range of non-equilibrium phenomena that are inaccessible in static systems, such as the discrete time crystals. Time rondeau crystals with a partial temporal order have been proposed as a distinctive prethermal phase of matter in systems driven by structured random protocols. Yet, heating is inevitable in closed systems and time rondeau crystals eventually melt. We introduce dissipation to counteract heating and demonstrate stable time rondeau crystals, which persist indefinitely, in a many-body interacting system. A key ingredient is synchronization in the non-interacting limit, which allows for stable time rondeau order without generating excessive heating. The presence of many-body interaction competes with synchronization and a de-synchronization phase transition occurs at a finite interaction strength. This transition is well captured via a linear stability analysis of the underlying stochastic processes
A Posteriori Error Control for Stochastic Galerkin FEM with High-Dimensional Random Parametric PDEs
No full textPDEs with random data are investigated and simulated in the field of Uncertainty Quantification (UQ), where uncertainties or (planned) variations of coefficients, forces, domains and boundary conditions in differential equations formally depend on random events with respect to a pre-determined probability distribution. The discretization of these PDEs typically leads to high-dimensional (deterministic) systems, where in addition to the physical space also the (often much larger) parameter space has to be considered. A proven technique for this task is the Stochastic Galerkin Finite Element Method (SGFEM), for which a review of the state of the art is provided. Moreover, important concepts and results are summarized. A special focus lies on the a posteriori error estimation and the derivation of an adaptive algorithm that controls all discretization parameters. In addition to an explicit residual based error estimator, also an equilibration estimator with guaranteed bounds is discussed. Under certain mild assumptions it can be shown that the successive refinement produced by such an adaptive algorithm leads to a sequence of approximations with guaranteed convergence to the true solution. Numerical examples illustrate the practical behavior for some common benchmark problems. Additionally, an adaptive algorithm for a problem with a non-affine coefficient is shown. By transforming the original PDE a convection-diffusion problem is obtained, which can be treated similarly to the standard affine case
Developing a hybrid single band carrier transport model for (Al,Ga)N heterostructures
No full textAluminium gallium nitride (Al,Ga)N alloys and heterostructures are used in the development of UV light emitting devices, and can emit at energies extending into the UV-C spectral range. In the UV-C wavelength window and thus at high AlN content, devices exhibit poor quantum efciencies. In order to aid the development of these devices, simulation techniques which capture the essential physics of these materials and heterostructures should be used. Due to a change in band ordering in a quantum well at compositions close to Al0.75Ga0.25N, special attention should be given to the treatment of valence band states in device simulation. In this work we develop a hybrid single band efective mass model which is informed by degree of optical polarization data obtained from atomistic multi-band calculations. Overall, the hybrid single band efective mass model is benchmarked against tight-binding electronic structure calculations. To do so a confining energy landscape is extracted from the tight-binding model and used as input for the single band efective mass calculations. Moreover, the extracted tight-binding energy landscape is transferred to a drift-difusion model, allowing therefore for a multi-scale study of transport properties of a single (Al,Ga)N quantum well embedded in a p-i-n junction. Our results show that wider wells lead to a lower turn-on voltage due to a reduction of the band gap, but the internal quantum efciency of these wells is lower than in narrower wells. Alloy disorder leads to carrier localization and an uneven distribution of recombination within the quantum well plane, which gives rise to percolation currents. A comparison of results with 'pure' band simulations shows that when TE emission dominated, the heavy hole mass is a good approximation. In contrast, where band mixing was strong between heavy hole and split-of bands the mass from the split of band was very efective