Publications Server of the Weierstrass Institute for Applied Analysis and Stochastics
Not a member yet
7600 research outputs found
Sort by
Reconstruction of Flow Domain Boundaries from Velocity Data via Multi-Step Optimization of Distributed Resistance
No full textWe reconstruct the unknown shape of a flow domain using partially available internal velocity measurements. This inverse problem is motivated by applications in cardiovascular imaging where motion-sensitive protocols, such as phase-contrast MRI, can be used to recover three-dimensional velocity fields inside blood vessels. In this context, the information about the domain shape serves to quantify the severity of pathological conditions, such as vessel obstructions. We consider a flow modeled by a linear Brinkman problem with a fictitious resistance accounting for the presence of additional boundaries. To reconstruct these boundaries, we employ a multi-step gradient-based variational method to compute a resistance that minimizes the difference between the computed flow velocity and the available data. Afterward, we apply different post-processing steps to reconstruct the shape of the internal boundaries. To limit the overall computational cost, we use a stabilized equal-order finite element method. We prove the stability and the well-posedness of the considered optimization problem. We validate our method on three-dimensional examples based on synthetic velocity data and using realistic geometries obtained from cardiovascular imaging
Impact of Viscosity Modeling on the Simulation of Aortic Blood Flow
No full textModeling issues for the simulation of blood flow in an aortic coarctation are studied in this paper. From the physical point of view, several viscosity models for non-Newtonian fluids as well as a Newtonian fluid model will be considered. From the numerical point of view, two different turbulence models are utilized in the simulations. The impact of both, the physical and the numerical modeling, on clinically relevant biomarkers is investigated and compared
Unifying Framework for Accelerated Randomized Methods in Convex Optimization
No full textIn this paper, we consider smooth convex optimization problems with simple constraints and inexactness in the oracle information such as value, partial or directional derivatives of the objective function. We introduce a unifying framework, which allows to construct different types of accelerated randomized methods for such problems and to prove convergence rate theorems for them. We focus on accelerated random block-coordinate descent, accelerated random directional search, accelerated random derivative-free method and, using our framework, provide their versions for problems with inexact oracle information. Our contribution also includes accelerated random block-coordinate descent with inexact oracle and entropy proximal setup as well as derivative-free version of this method. Moreover, we present an extension of our framework for strongly convex optimization problems. We also discuss an extension for the case of inexact model of the objective function
Explicit fractional Laplacians and Riesz potentials of classical functions
Get PDFWe prove and collect numerous explicit and computable results for the fractional Laplacian with as well as its whole space inverse, the Riesz potential, with . Choices of include weighted classical orthogonal polynomials such as the Legendre, Chebyshev, Jacobi, Laguerre and Hermite polynomials, or first and second kind Bessel functions with or without sinusoid weights. Some higher dimensional fractional Laplacians and Riesz potentials of generalized Zernike polynomials on the unit ball and its complement as well as whole space generalized Laguerre polynomials are also discussed. The aim of this paper is to aid in the continued development of numerical methods for problems involving the fractional Laplacian or the Riesz potential in bounded and unbounded domains -- both directly by providing useful basis or frame functions for spectral method approaches and indirectly by providing accessible ways to construct computable toy problems on which to test new numerical methods
Weighted sparsity and sparse tensor networks for least squares approximation
Get PDFThe approximation of high-dimensional functions is a ubiquitous problem in many scientific fields that is only feasible practically if advantageous structural properties can be exploited. One prominent structure is sparsity relatively to some basis. For the analysis of these best n-term approximations a relevant tool is the Stechkin's lemma. In its standard form, however, this lemma does not allow to explain convergence rates for a wide range of relevant function classes. This work presents a new weighted version of Stechkin's lemma that improves the best n-term rates for weighted ℓp-spaces and associated function classes such as Sobolev or Besov spaces. For the class of holomorphic functions, which for example occur as solutions of common high-dimensional parameter dependent PDEs, we recover exponential rates that are not directly obtainable with Stechkin's lemma. This sparsity can be used to devise weighted sparse least squares approximation algorithms as known from compressed sensing. However, in high-dimensional settings, classical algorithms for sparse approximation suffer the curse of dimensionality. We demonstrate that sparse approximations can be encoded efficiently using tensor networks with sparse component tensors. This representation gives rise to a new alternating algorithm for best n-term approximation with a complexity scaling polynomially in n and the dimension. We also demonstrate that weighted ℓpsummability not only induces sparsity of the tensor but also low ranks. This is not exploited by the previous format. We thus propose a new low-rank tensor train format with a single weighted sparse core tensor and an ad-hoc algorithm for approximation in this format. To analyse the sample complexity for this new model class we derive a novel result of independent interest that allows to transfer the restricted isometry property from one set to another sufficiently close set. We then prove that the new model class is close enough to the set of weighted sparse vectors such that the restricted isometry property transfers. Numerical examples illustrate the theoretical results for a benchmark problem from uncertainty quantification. Although they lead up to the analysis of our final model class, our contributions on weighted Stechkin and the restricted isometry property are of independent interest and can be read independently
Randomized Gradient-Free Methods in Convex Optimization
No full textThis review presents modern gradient-free methods to solve convex optimization problems. By gradient-free methods, we mean those that use only (noisy) realizations of the objective value. We are motivated by various applications where gradient information is prohibitively expensive or even unavailable. We mainly focus on three criteria: oracle complexity, iteration complexity, and the maximum permissible noise level
Law of the SLE Tip
No full textWe analyze the law of the SLE tip at a fixed time in capacity parametrization. We describe it as the stationary law of a suitable diffusion process, and show that it has a density which is a unique solution of a certain PDE. Moreover, we identify the phases in which the even negative moments of the imaginary value are finite. For the negative second and negative fourth moments we provide closed-form expressions
Fiber Reinforced Hydrated Networks Recapitulate the Poroelastic Mechanics of Articular Cartilage.
No full textThe role of poroelasticity on the functional performance of articular cartilage has been established in the scientific literature since the 1960s. Despite the extensive knowledge on this topic there remain few attempts to design for poroelasticity and to our knowledge no demonstration of an engineered poroelastic material that approaches the physiological performance. In this paper, we report on the development of an engineered material that begins to approach physiological poroelasticity. We quantify poroelasticity using the fluid load fraction, apply mixture theory to model the material system, and determine cytocompatibility using primary human mesenchymal stem cells. The design approach is based on a fiber reinforced hydrated network and uses routine fabrication methods (electrohydrodynamic deposition) and materials (poly[ɛ-caprolactone] and gelatin) to develop the engineered poroelastic material. This composite material achieved a mean peak fluid load fraction of 68%, displayed consistency with mixture theory, and demonstrated cytocompatibility. This work creates a foundation for designing poroelastic cartilage implants and developing scaffold systems to study chondrocyte mechanobiology and tissue engineering. STATEMENT OF SIGNIFICANCE: Poroelasticity drives the functional mechanics of articular cartilage (load bearing and lubrication). In this work we develop the design rationale and approach to produce a poroelastic material, known as a fiber reinforced hydrated network (FiHy™), that begins to approach the native performance of articular cartilage. This is the first engineered material system capable of exceeding isotropic linear poroelastic theory. The framework developed here enables fundamental studies of poroelasticity and the development of translational materials for cartilage repair
The Role of Vacancy Dynamics in Two‐Dimensional Memristive Devices
No full textTwo-dimensional layered transition metal dichalcogenides (TMDCs) are promising memristive materials for neuromorphic computing systems. Despite extensive experimental work, the underlying switching mechanisms are still not understood, impeding progress in material and device functionality. This study reveals the dominant role of defect dynamics in the switching process of 2D TMDC materials. The switching process is governed by the formation and annihilation dynamics of a local vacancy depletion zone. It explains the distinct features of the device characteristics observed experimentally, including fundamentally different device behavior previously thought to originate from multiple mechanisms. Key influence factors are identified and discussed with a fully coupled and dynamic charge transport model for electrons, holes, and ionic point defects, including image-charge-induced Schottky barrier lowering (SBL). Thermal effects and local Joule heating are considered by coupling the transient heat transfer equation to the electronic properties. The model is validated with hysteresis and pulse measurements for various lateral 2D MoS2-based devices, strongly corroborating the relevance of vacancy dynamics in TMDC devices and offering a new perspective on the switching mechanisms. The insights gained from this study can be used to extend the functional behavior of 2D TMDC memristive devices in future neuromorphic computing applications
Theoretical Study of the Impact of Alloy Disorder on Carrier Transport and Recombination Processes in Deep UV (Al,Ga)N Light Emitters
No full textAluminum gallium nitride [(Al,Ga)N] has gained significant attention in recent years due to its potential for highly efficient light emitters operating in the deep ultra-violet (UV) range (<280 nm). However, given that current devices exhibit extremely low efficiencies, understanding the fundamental properties of (Al,Ga)N-based systems is of key importance. Here, using a multi-scale simulation framework, we study the impact of alloy disorder on carrier transport, radiative and non-radiative recombination processes in a c-plane Al0.7Ga0.3N/Al0.8Ga0.2N quantum well embedded in a p–n junction. Our calculations reveal that alloy fluctuations can open “percolative” pathways that promote transport for the electrons and holes into the quantum well region. Such an effect is neglected in conventional and widely used transport simulations. Moreover, we find that the resulting increased carrier density and alloy induced carrier localization effects significantly increase non-radiative Auger–Meitner recombination in comparison to the radiative process. Thus, to suppress such non-radiative process and potentially related material degradation, a careful design (wider well, multi-quantum wells) of the active region is required to improve the efficiency of deep UV light emitters