Publications Server of the Weierstrass Institute for Applied Analysis and Stochastics
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Ratio limits and simulation algorithms for the Palm version of stationary iterated tessellations
Full text linkDistributional properties and a simulation algorithm for the Palm version of stationary iterated tessellations are considered. In particular we study the limit behavior of functionals related to Cox-Voronoi cells (such as typical shortest path lengths) if either the intensity γ0 of the initial tessellation or the intensity γ1 of the component tessellation converges to 0. We develop an explicit description of the Palm version of Poisson-Delaunay tessellations (PDT) which provides a new direct simulation algorithm for the typical Cox-Voronoi cell based on PDT. It allows us to simulate the Palm version of stationary iterated tessellations where either the initial or component tessellation is a PDT and can furthermore be used in order to show numerically that the qualitative and quantitative behavior of certain functionals related to Cox-Voronoi cells strongly depends on the type of the underlying iterated tessellation
Mirror Descent and Convex Optimization Problems with Non-smooth Inequality Constraints
No full textWe consider the problem of minimization of a convex function on a simple set with convex non-smooth inequality constraint and describe first-order methods to solve such problems in different situations: smooth or non-smooth objective function; convex or strongly convex objective and constraint; deterministic or randomized information about the objective and constraint. Described methods are based on Mirror Descent algorithm and switching subgradient scheme. One of our focus is to propose, for the listed different settings, a Mirror Descent with adaptive stepsizes and adaptive stopping rule. We also construct Mirror Descent for problems with objective function, which is not Lipschitz, e.g., is a quadratic function. Besides that, we address the question of recovering the dual solution in the considered problem
Theoretical investigation of optical polarisation in alloy disordered (Al,Ga)N quantum well systems
No full textAluminium Gallium Nitride ((Al,Ga)N) is an ideal material for light emitting devices in the UV spectral range. However, these devices still suffer from low external quantum efficiencies, particularly in the deep-UV range. A contributor to the low external quantum efficiency is low light extraction efficiency (LEE), which is tightly linked to the valence band structure of (Al,Ga)N quantum wells. Theoretical studies that account for alloy disorder-induced valence band mixing effects in these structures are sparse. Here, we utilise an atomistic multiband tight-binding model to gain insight into the degree of optical polarisation in (Al,Ga)N quantum well systems. Special attention is paid to the impact of Al content, well width and carrier density in the wells
Quantum circuit simulation with a local time-dependent variational principle
Full text linkClassical simulations of quantum circuits are vital for assessing potential quantum advantage and benchmarking devices, yet they require sophisticated methods to avoid the exponential growth of resources. Tensor network approaches, in particular matrix product states (MPS) combined with the time-evolving block decimation (TEBD) algorithm, currently dominate large-scale circuit simulations. These methods scale efficiently when entanglement is limited but suffer rapid bond dimension growth with increasing entanglement and handle long-range gates via costly SWAP insertions. Motivated by the success of the time-dependent variational principle (TDVP) in many-body physics, we reinterpret quantum circuits as series of discrete time evolutions, using gate generators to construct an MPS-based circuit simulation via a local TDVP formulation. This addresses TEBD's key limitations by (1) naturally accommodating long-range gates and (2) optimally representing states on the MPS manifold. By diffusing entanglement more globally, the method suppresses local bond growth and reduces memory and runtime costs. We benchmark the approach on five 49-qubit circuits: three Hamiltonian circuits (1D open and periodic Heisenberg, 2D 7 x 7 Ising) and two algorithmic ones (quantum approximate optimization, hardware-efficient ansatz). Across all cases, our method yields substantial resource reductions over standard tools, establishing a new state-of-the-art for circuit simulation and enabling advances across quantum computing, condensed matter, and beyond
Derivation of a thermo-visco-elastic plate model at small strains
Full text linkWe investigate a three-dimensional thermo-visco-elastic model with Kelvin--Voigt rheology under small strains confined to a thin domain. The model comprises a quasistatic linear momentum equation, with viscous stresses adhering to a Kelvin--Voigt viscosity law, coupled with a nonlinear heat equation governing temperature. The heat equation incorporates source terms arising from viscous dissipation and adiabatic heat sources due to thermal expansion. The model ensures thermodynamic consistency, maintaining energy conservation, positive temperature, and entropy production. We analyze the asymptotic behavior of solutions as the domain thickness approaches zero, deriving an effective two-dimensional model. This derivation involves rescaling the domain to a fixed thickness and establishing uniform a priori estimates relative to the plate's thickness. In the limit, the temperature becomes vertically constant, and displacement are of Kirchhoff--Love type, enabling meaningful interpretation of the limiting objects within the plate's two-dimensional cross-section. The mechanical equations consist of two parabolic equations, one for the membrane part and one for the bending part. Notably, the viscosity law in the limiting model departs from the Kelvin--Voigt form, reflecting nontrivial kinematic constraints on the rescaled out-of-plane strains. The bending of the plate does not depend on the temperature in the limit
Reversible Saddle-Node Separatrix-Loop Bifurcation
No full textWe describe the unfolding of a special variant of the codimension-two Saddle-Node Separatrix-Loop (SNSL) bifurcation that occurs in systems with time-reversibility. While the classical SNSL bifurcation can be characterized as the collision of a saddle-node equilibrium with a limit cycle, the reversible variant (R-SNSL) is characterised by as the collision of a saddle-node equilibrium with a boundary separating a dissipative and a conservative region in phase space. Moreover, we present several reversible versions of the SNIC (Saddle-Node on Invariant Circle) bifurcation and discuss the role of an additional reversible saddle equilibrium in all these scenarios. As an example, we provide a detailed bifurcation scenario for a reversible system of two coupled phase rotators (a system on a 2D torus) involving a R-SNSL bifurcation
Convergence of a finite-volume scheme and dissipative measure valued-strong stability for a hyperbolic-parabolic cross-diffusion system
No full textThis article is concerned with the development of a theoretical framework of global measure-valued solutions for a class of hyperbolic–parabolic cross-diffusion systems, and its application to the convergence analysis of a fully discrete finite-volume scheme. After introducing an appropriate notion of dissipative measure-valued solutions to the PDE system, a numerical scheme is proposed which is shown to generate, in the continuum limit, a dissipative measure-valued solution. The “parabolic density part” of the limiting measure-valued solution is atomic and converges to its constant state for long times. Furthermore, it is proved that whenever the PDE system possesses a strong solution, the convergence of the approximation scheme holds in the strong sense. The results are based on Young measure theory and a weak–strong stability estimate combining Shannon and Rao entropies. The convergence of the numerical scheme is achieved by means of discrete entropy dissipation inequalities and an artificial diffusion, which vanishes in the continuum limit
Modeling hydrogen embrittlement for pricing degradation in gas pipelines
Full text linkThis paper addresses aspects of the critical challenge of hydrogen embrittlement in the context of Germany's transition to a sustainable, hydrogen-inclusive energy system. As hydrogen infrastructure expands, estimating and pricing embrittlement become paramount due to safety, operational, and economic concerns. We present a twofold contribution: We discuss hydrogen embrittlement modeling using both continuum models and simplified approximations. Based on these models, we propose optimization-based pricing schemes for market makers, considering simplified cyclic loading and more complex digital twin models. Our approaches leverage widely-used subcritical crack growth models in steel pipelines, with parameters derived from experiments. The study highlights the challenges and potential solutions for incorporating hydrogen embrittlement into gas transportation planning and pricing, ultimately aiming to enhance the safety and economic viability of Germany's future energy infrastructure
Quantum dynamics of coupled excitons and phonons in chain-like systems: Tensor train approaches and higher-order propagators
No full textWe investigate tensor-train approaches to the solution of the time-dependent Schrödinger equation for chain-like quantum systems with on-site and nearest-neighbor interactions only. Using efficient low-rank tensor train representations, we aim at reducing memory consumption and computational costs. As an example, coupled excitons and phonons modeled in terms of Fröhlich-Holstein type Hamiltonians are studied here. By comparing our tensor-train-based results with semi-analytical results, we demonstrate the key role of the ranks of the quantum state vectors. Typically, an excellent quality of solutions is found only when the maximum number of ranks exceeds a certain value. One class of propagation schemes builds on splitting the Hamiltonian into two groups of interleaved nearest-neighbor interactions commutating within each of the groups. In particular, the fourth-order Yoshida-Neri and the eighth-order Kahan-Li symplectic composition yield results close to machine precision. Similar results are found for fourth and eighth order global Krylov scheme. However, the computational effort currently restricts the use of these four propagators to rather short chains, which also applies to propagators based on the time-dependent variational principle, typically used for matrix product states. Yet, another class of propagators involves explicit, time-symmetrized Euler integrators. Especially, the fourth-order variant is recommended for quantum simulations of longer chains, even though the high precision of the splitting schemes cannot be reached. Moreover, the scaling of the computational effort with the dimensions of the local Hilbert spaces is much more favorable for the differencing than for splitting or variational schemes
