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Connectivity in Mobile Device-to-Device Networks in Urban Environments
No full textIn this article we setup a dynamic device-to-device communication system where devices, given as a Poisson point process, move in an environment, given by a street system of random planar-tessellation type, via a random-waypoint model. Every device independently picks a target location on the street system using a general waypoint kernel, and travels to the target along the shortest path on the streets with an individual velocity. Then, any pair of devices becomes connected whenever they are on the same street in sufficiently close proximity, for a sufficiently long time. In our main results we isolate regimes for the almost-sure absence of percolation if, for example, the device intensity is too small, or the connectivity time is too large. On the other hand, we exhibit parameter regimes of sufficiently large intensities of devices, under favorable choices of the other parameters, such that percolation is possible with positive probability. Most interestingly, we also show an in-and-out of percolation as the velocity increases. The rigorous analysis of the system mainly rests on comparison arguments with simplified models via spatial coarse graining and thinning approaches. Here we also make contact to geostatistical percolation models with infinite-range dependencies
On a Two-Scale Phasefield Model for Topology Optimization
No full textIn this article, we consider a gradient flow stemming from a problem in two-scale topology optimization. We use the phase-field method, where a Ginzburg–Landau term with obstacle potential is added to the cost functional, which contains the usual compliance but also an additional contribution including a local volume constraint in a penalty term. The minimization of such an energy by its gradient-flow is analyzed in this paper. We use an regularization and discretization of the associated state-variable to show the existence of weak solutions to the considered system
Estimation of Frequency Noise Characteristics and Data-Driven Modeling of Narrow-Linewidth Semiconductor Lasers
No full textThe design of narrow-linewidth lasers requires stochastic laser models providing a realistic description of the noise in the device. We present a statistical inference approach to extract the frequency noise characteristics and model parameters of narrow-linewidth lasers from delayed self-heterodyne beat note experiments. By exploiting prior knowledge about the statistical distribution of the measurement data, accurate estimates of the parameters of the free running laser can be achieved even in the presence of considerable detector noise. The approach is demonstrated for simulated time series data using a stochastic laser rate equation model including 1/f–type noise
Computational Optimal Transport
No full textThe optimal transport (OT) problem is a classical optimization problem having the form of linear programming. Machine learning applications put forward new computational challenges in their solution. In particular, the OT problem defines a distance between real-world objects such as images, videos, texts, etc., modeled as probability distributions. In this case, the large dimension of the corresponding optimization problem does not allow applying classical methods such as network simplex or interior-point methods. This challenge was overcome by introducing entropic regularization and using the efficient Sinkhorn algorithm to solve the regularized problem. A flexible alternative is the accelerated primal–dual gradient method, which can use any strongly convex regularization. These algorithms and other related problems such as approximating the Wasserstein barycenter together with efficient algorithms for its solution, including decentralized distributed algorithms, are discussed
Volume Exclusion Effects in Perovskite Charge Transport Modeling
No full textDue to their flexible material properties, perovskite materials are a promising candidate for many semiconductor devices such as lasers, memristors, LEDs and solar cells. For example, perovskite-based solar cells have recently become one of the fastest growing photovoltaic technologies. Unfortunately, perovskite devices are far from commercialization due to challenges such as fast degradation. Mathematical models can be used as tools to explain the behavior of such devices, for example drift-diffusion equations portray the ionic and electric motion in perovskites. In this work, we take volume exclusion effects on ion migration within a perovskite crystal lattice into account. This results in the formulation of two different ionic current densities for such a drift-diffusion model – treating either the mobility or the diffusivity as density-dependent while the other quantity remains constant. The influence of incorporating each current density description into a model for a typical perovskite solar cell configuration is investigated numerically, through simulations performed using two different open source tools
On collocation points for physics-informed neural networks applied to convection-dominated convection-diffusion problems
Get PDFIn recent years physics-informed neural networks (PINNs) for approximating the solution to (initial-)boundary value problems gained a lot of interest. PINNs are trained to minimize several residuals of the problem in collocation points. In this work we tackle convection-dominated convection-diffusion problems, whose solutions usually possess layers, which are small regions where the solution has a steep gradient. Inspired by classical Shishkin meshes, we compare hard- constrained PINNs trained with layer-adapted collocation points with ones trained with equispaced and uniformly randomly chosen points. We observe that layer-adapted points work the best for a problem with an interior layer and the worst for a problem with boundary layers. For both problems at most acceptable solutions can be obtained with PINNs
An energy-based finite-strain model for 3D heterostructured materials and its validation by curvature analysis
Get PDFThis paper presents a comprehensive study of the intrinsic strain response of 3D het- erostructures arising from lattice mismatch. Combining materials with different lattice constants induces strain, leading to the bending of these heterostructures. We propose a model for nonlinear elastic heterostructures such as bimetallic beams or nanowires that takes into account local prestrain within each distinct material region. The resulting system of partial differential equations (PDEs) in Lagrangian coordinates incorporates a nonlinear strain and a linear stress-strain relationship governed by Hooke?s law. To validate our model, we apply it to bimetallic beams and hexagonal hetero-nanowires and perform numerical simulations using finite element methods (FEM). Our simulations ex- amine how these structures undergo bending under varying material compositions and cross-sectional geometries. In order to assess the fidelity of the model and the accuracy of simulations, we compare the calculated curvature with analytically derived formula- tions. We derive these analytical expressions through an energy-based approach as well as a kinetic framework, adeptly accounting for the lattice constant mismatch present at each compound material of the heterostructures. The outcomes of our study yield valuable insights into the behavior of strained bent heterostructures. This is particularly significant as the strain has the potential to influence the electronic band structure, piezoelectricity, and the dynamics of charge carriers
On loss functionals for physics-informed neural networks for convection-dominated convection-diffusion problems
Get PDFIn the convection-dominated regime, solutions of convection-diffusion problems usually possesses layers, which are regions where the solution has a steep gradient. It is well known that many classical numerical discretization techniques face difficulties when approximating the solution to these problems. In recent years, physics-informed neural networks (PINNs) for approximating the solution to (initial-)boundary value problems received a lot of interest. In this work, we study various loss functionals for PINNs that are novel in the context of PINNs and are especially designed for convection-dominated convection-diffusion problems. They are numerically compared to the vanilla and a -variational loss functional from the literature based on two benchmark problems whose solutions possess different types of layers. We observe that the best novel loss functionals reduce the error by for the first and for the second problem compared to the methods from the literature
Optimal control of a phase field system of Caginalp type with fractional operators
No full textIn their recent work ``Well-posedness, regularity and asymptotic analyses for a fractional phase field system'' (Asymptot. Anal. 114 (2019), 93--128), two of the present authors have studied phase field systems of Caginalp type, which model nonconserved, nonisothermal phase transitions and in which the occurring diffusional operators are given by fractional versions in the spectral sense of unbounded, monotone, selfadjoint, linear operators having compact resolvents. In this paper, we complement this analysis by investigating distributed optimal control problems for such systems. It is shown that the associated control-to-state operator is Fréchet differentiable between suitable Banach spaces, and meaningful first-order necessary optimality conditions are derived in terms of a variational inequality and the associated adjoint state variables
Weakly self-avoiding walk in a Pareto-distributed random potential
Get PDFWe investigate a model of continuous-time simple random walk paths in ℤ d undergoing two competing interactions: an attractive one towards the large values of a random potential, and a self-repellent one in the spirit of the well-known weakly self-avoiding random walk. We take the potential to be i.i.d. Pareto-distributed with parameter α > d, and we tune the strength of the interactions in such a way that they both contribute on the same scale as t → ∞. Our main results are (1) the identification of the logarithmic asymptotics of the partition function of the model in terms of a random variational formula, and, (2) the identification of the path behaviour that gives the overwhelming contribution to the partition function for α > 2d: the random-walk path follows an optimal trajectory that visits each of a finite number of random lattice sites for a positive random fraction of time. We prove a law of large numbers for this behaviour, i.e., that all other path behaviours give strictly less contribution to the partition function.The joint distribution of the variational problem and of the optimal path can be expressed in terms of a limiting Poisson point process arising by a rescaling of the random potential. The latter convergence is in distribution?and is in the spirit of a standard extreme-value setting for a rescaling of an i.i.d. potential in large boxes, like in KLMS09