Publications Server of the Weierstrass Institute for Applied Analysis and Stochastics
Not a member yet
7600 research outputs found
Sort by
On the formation of microstructure and the occurrence of vortices in a singularly perturbed energy related to helimagnetism: A scaling law result
Full text linkIn this work, singularly perturbed energies arising from discrete spin models are studied. The energies under consideration consist of a non-convex bulk term and a higher-order regularizing term and are subject to incompatible boundary conditions. In contrast to existing results in the literature, in this work, admissible fields are not necessarily gradient fields, instead their curl is linked to topological singularities, so-called vortices, in the discrete spin model. The main result of this work is a scaling law for the minimal energy with respect to three parameters: one measuring the incompatibility of the boundary conditions, the second measuring the strength of the regularizing term, and the third being related to the interatomic distance in the discrete model. The shown result implies in particular that in certain parameter regimes, minimizers necessarily develop vortices. A key tool in the analysis is a careful modification of the celebrated ball-construction technique that, due to a lack of rigidity, considers simultaneously both the bulk energy and the regularizing term
Genetic variability under the seedbank coalescent
Full text linkWe analyse patterns of genetic variability of populations in the presence of a large seedbank with the help of a new coalescent structure called the seedbank coalescent. This ancestral process appears naturally as scaling limit of the genealogy of large populations that sustain seedbanks, if the seedbank size and individual dormancy times are of the same order as the active population. Mutations appear as Poisson processes on the active lineages, and potentially at reduced rate also on the dormant lineages. The presence of ‘dormant’ lineages leads to qualitatively altered times to the most recent common ancestor and non-classical patterns of genetic diversity. To illustrate this we provide a Wright-Fisher model with seedbank component and mutation, motivated from recent models of microbial dormancy, whose genealogy can be described by the seedbank coalescent. Based on our coalescent model, we derive recursions for the expectation and variance of the time to most recent common ancestor, number of segregating sites, pairwise differences, and singletons. Estimates (obtained by simulations) of the distributions of commonly employed distance statistics, in the presence and absence of a seedbank, are compared. The effect of a seedbank on the expected site-frequency spectrum is also investigated using simulations. Our results indicate that the presence of a large seedbank considerably alters the distribution of some distance statistics, as well as the site-frequency spectrum. Thus, one should be able to detect from genetic data the presence of a large seedbank in natural populations
Denoising for Improved Parametric MRI of the Kidney: Protocol for Nonlocal Means Filtering
No full textIn order to tackle the challenges caused by the variability in estimated MRI parameters (e.g., T2* and T2) due to low SNR a number of strategies can be followed. One approach is postprocessing of the acquired data with a filter. The basic idea is that MR images possess a local spatial structure that is characterized by equal, or at least similar, noise-free signal values in vicinities of a location. Then, local averaging of the signal reduces the noise component of the signal. In contrast, nonlocal means filtering defines the weights for averaging not only within the local vicinity, bur it compares the image intensities between all voxels to define “nonlocal” weights. Furthermore, it generally compares not only single-voxel intensities but small spatial patches of the data to better account for extended similar patterns. Here we describe how to use an open source NLM filter tool to denoise 2D MR image series of the kidney used for parametric mapping of the relaxation times T2* and T2. This chapter is based upon work from the COST Action PARENCHIMA, a community-driven network funded by the European Cooperation in Science and Technology (COST) program of the European Union, which aims to improve the reproducibility and standardization of renal MRI biomarkers
Reference map approach to Eulerian thermomechanics using GENERIC
No full textAn Eulerian GENERIC model for thermo-viscoelastic materials with diffusive components is derived based on a transformation framework that maps a Lagrangian formulation to corresponding Eulerian coordinates. The key quantity describing the deformation in Eulerian coordinates is the inverse of the deformation, i.e., the reference map. The Eulerian model is formally constructed, and by reducing the GENERIC system to a damped Hamiltonian system, the isothermal limit is derived. A structure-preserving weak formulation is developed. As an example, the coupling of finite strain viscoelasticity and diffusion in a multiphase system governed by Lagrangian indicator functions is demonstrated
Numerical analysis of the SIMP model for the topology optimization problem of minimizing compliance in linear elasticity
Full text linkWe study the finite element approximation of the solid isotropic material with penalization method (SIMP) for the topology optimization problem of minimizing the compliance of a linearly elastic structure. To ensure the existence of a local minimizer to the infinite-dimensional problem, we consider two popular regularization methods: -type penalty methods and density filtering. Previous results prove weak(-*) convergence in the space of the material distribution to a local minimizer of the infinite-dimensional problem. Notably, convergence was not guaranteed to \emph{all} the isolated local minimizers. In this work, we show that, for every isolated local or global minimizer, there exists a sequence of finite element local minimizers that strongly converges to the minimizer in the appropriate space. As a by-product, this ensures that there exists a sequence of unfiltered discretized material distributions that does not exhibit checkerboarding
A sparse hierarchical hp-finite element method on disks and annuli
No full textWe develop a sparse hierarchical hp-finite element method (hp-FEM) for the Helmholtz equation with variable coefficients posed on a two-dimensional disk or annulus. The mesh is an inner disk cell (omitted if on an annulus domain) and concentric annuli cells. The discretization preserves the Fourier mode decoupling of rotationally invariant operators, such as the Laplacian, which manifests as block diagonal mass and stiffness matrices. Moreover, the matrices have a sparsity pattern independent of the order of the discretization and admit an optimal complexity factorization. The sparse hp-FEM can handle radial discontinuities in the right-hand side and in rotationally invariant Helmholtz coefficients. Rotationally anisotropic coefficients that are approximated by low-degree polynomials in Cartesian coordinates also result in sparse linear systems. e consider examples such as a high-frequency Helmholtz equation with radial discontinuities and rotationally anisotropic coefficients, singular source terms, țhe time-dependent Schrödinger equation, and an extension to a three-dimensional cylinder domain, with a quasi-optimal solve, via the Alternating Direction Implicit (ADI) algorithm
On Brinkman flows with curvature-induced phase separation in binary mixtures
Full text linkThe mathematical analysis of diffuse-interface models for multiphase flows has attracted significant attention due to their ability to capture complex interfacial dynamics, including curvature effects, within a unified, energetically consistent framework. In this work, we study a novel Brinkman--Cahn--Hilliard system, coupling a sixth-order phase-field evolution with a Brinkman-type momentum equation featuring variable shear viscosity. The Cahn--Hilliard equation includes a nonconservative source term accounting for mass exchange, and the velocity equation contains a non divergence-free forcing term. We establish the existence of weak solutions in a divergence-free variational framework, and, in the case of constant mobility and shear viscosity, prove uniqueness and continuous dependence on the forcing. Additionally, we analyze the Darcy limit, providing existence results for the corresponding reduced system
Hierarchical clustering in mean-field coupled Stuart--Landau oscillators
Full text linkClustered solutions in oscillator networks provide an important insight into how a system might diversify from a synchronous solution into spatiotemporal complex solutions. They can therefore form a link between fully synchronized and incoherent states. Despite their fundamental role in coupled oscillator dynamics, our understanding of how these clusters form and differentiate is still quite limited. Here, we study an ensemble of globally coupled Stuart--Landau oscillators and focus on the question of how 3-cluster solutions emerge from 2-cluster solutions and how the different 3-cluster solutions are organized in parameter space. We show that the arrangement of the clusters is dictated by a co-dimension 2 point, which we coin Type-II cluster singularity. Furthermore, our study points to a hierarchical structure of higher cluster solutions
Theoretical study of the impact of carrier density screening on Urbach tail energies and optical polarization in (Al,Ga)N quantum well systems
No full textAluminium Gallium Nitride ((Al,Ga)N) presents an ideal platform for designing ultra-violet (UV) light emitters across the entire UV spectral range. However, in the deep-UV spectral range (<280 nm) these emitters exhibit very low quantum efficiencies, which in part is linked to the light polarization characteristics of (Al,Ga)N quantum wells (QWs). In this study we provide insight into the degree of optical polarization of (Al,Ga)N QW systems operating across the UV-C spectral range by means of an atomistic, multi-band electronic structure model. Our model not only captures the difference in valence band ordering in AlN and GaN, it accounts also for alloy disorder induced band mixing effects originating from random alloy fluctuations in (Al,Ga)N QWs. The latter aspect is not captured in widely employed continuum based models. The impact of alloy disorder on the electronic structure is studied in terms of Urbach tail energies, which reflect the broadening of the valence band density of states due to carrier localization effects. We find that especially in wider wells, Urbach tail energies are reduced with increasing carrier densities in the well, highlighting that alloy disorder induced carrier localization effects in (Al,Ga)N QWs are also tightly linked to electrostatic built-in fields. Our calculations show that for QWs designed to emit at the longer wavelength end of the UV-C spectrum, carrier density and well width are of secondary importance for their light emission properties, meaning that one observes mainly transverse electrical polarization. However, for (Al,Ga)N QWs with high Al contents, we find that both well width and carrier density will impact the degree of optical polarization. Our calculations suggest that wider wells will increase the degree of optical polarization and may therefore be a viable option to improve the light extraction efficiency in deep UV light emitters