Publications Server of the Weierstrass Institute for Applied Analysis and Stochastics
Not a member yet
7600 research outputs found
Sort by
Impact of Random Alloy Fluctuations on the Carrier Distribution in Multi-Color (In,Ga)N/GaN Quantum Well Systems
No full textThe efficiency of (In,Ga)N-based light-emitting diodes (LEDs) is limited by the failure of holes to evenly distribute across the (In,Ga)N/GaN multiquantum well stack that forms the active region. To tackle this problem, it is important to understand carrier transport in these alloys. In this work, we study the impact that random alloy fluctuations have on the distribution of electrons and holes in such devices. To do so, an atomistic tight-binding model is employed to account for alloy fluctuations on a microscopic level and the resulting tight-binding energy landscape forms input to a quantum corrected drift-diffusion model. Here, quantum corrections are introduced via localization-landscape theory. Similar to experimental studies in the literature, we have focused on a multiquantum well system in which two of the three wells have the same In content, while the third well differs in In content. By changing the order of wells in this “multicolor” quantum well structure and looking at the relative radiative-recombination rates of the different emitted wavelengths, we (i) gain insight into the distribution of carriers in such a system and (ii) can compare our findings to trends observed in experiment. We focus on three factors and evaluate the impact that each have on carrier distribution: an electron blocking layer, quantum corrections, and random alloy fluctuations. We find that the electron blocking layer is of secondary importance. However, in order to recover experimentally observed features—namely, that the ��-side quantum well dominates the light emission—both quantum corrections and random alloy fluctuations should be considered. The widely assumed homogeneous virtual-crystal approximation fails to capture the characteristic light emission distribution across a multiquantum well stack
Stokes Problem with Slip Boundary Conditions using Stabilized Finite Elements Combined with Nitsche
No full textWe discuss how slip conditions for the Stokes equation can be handled using Nitsche method, for a stabilized finite element discretization. Emphasis is made on the interplay between stabilization and Nitsche terms. Well-posedness of the discrete problem and optimal convergence rates, in natural norm for the velocity and the pressure, are established, and illustrated with various numerical experiments. The proposed method fits naturally in the context of a finite element implementation while being accurate, and allows an increased flexibility in the choice of the finite element pairs
Multilevel Monte Carlo with Numerical Smoothing for Robust and Efficient Computation of Probabilities and Densities
No full textThe multilevel Monte Carlo (MLMC) method is highly efficient for estimating expectations of a functional of a solution to a stochastic differential equation (SDE). However, MLMC estimators may be unstable and have a poor (noncanonical) complexity in the case of low regularity of the functional. To overcome this issue, we extend our previously introduced idea of numerical smoothing in (Quantitative Finance, 23(2), 209-227, 2023), in the context of deterministic quadrature methods to the MLMC setting. The numerical smoothing technique is based on root-finding methods combined with one-dimensional numerical integration with respect to a single well-chosen variable. This study is motivated by the computation of probabilities of events, pricing options with a discontinuous payoff, and density estimation problems for dynamics where the discretization of the underlying stochastic processes is necessary. The analysis and numerical experiments reveal that the numerical smoothing significantly improves the strong convergence, and consequently, the complexity and robustness (by making the kurtosis at deep levels bounded) of the MLMC method. In particular, we show that numerical smoothing enables recovering the MLMC complexities obtained for Lipschitz functionals due to the optimal variance decay rate when using the Euler--Maruyama scheme. For the Milstein scheme, numerical smoothing recovers the canonical MLMC complexity even for the nonsmooth integrand mentioned above. Finally, our approach efficiently estimates univariate and multivariate density functions
Cluster sizes in subcritical soft Boolean models
Get PDFWe consider the soft Boolean model, a model that interpolates between the Boolean model and long-range percolation, where vertices are given via a stationary Poisson point process. Each vertex carries an independent Pareto-distributed radius and each pair of vertices is assigned another independent Pareto weight with a potentially different tail exponent. Two vertices are now connected if they are within distance of the larger radius multiplied by the edge weight. We determine the tail behaviour of the Euclidean diameter and the number of points of a typical maximally connected component in a subcritical percolation phase. For this, we present a sharp criterion in terms of the tail exponents of the edge-weight and radius distributions that distinguish a regime where the tail behaviour is controlled only by the edge exponent from a regime in which both exponents are relevant. Our proofs rely on fine path-counting arguments identifying the precise order of decay of the probability that far-away vertices are connected
A stabilized total pressure-formulation of the Biot's poroelasticity equations in frequency domain: Numerical analysis and applications
Get PDFThis work focuses on the numerical solution of the dynamics of a poroelastic material in the frequency domain. We provide a detailed stability analysis based on the application of the Fredholm alternative in the continuous case, considering a total pressure formulation of the Biot's equations. In the discrete setting, we propose a stabilized equal order finite element method complemented by an additional pressure stabilization to enhance the robustness of the numerical scheme with respect to the fluid permeability. Utilizing the Fredholm alternative, we extend the well-posedness results to the discrete setting, obtaining theoretical optimal convergence for the case of linear finite elements. We present different numerical experiments to validate the proposed method. First, we consider model problems with known analytic solutions in two and three dimensions. As next, we show that the method is robust for a wide range of permeabilities, including the case of discontinuous coefficients. Lastly, we show the application for the simulation of brain elastography on a realistic brain geometry obtained from medical imaging
Traveling wave mode analysis of a coherence collapse regime semiconductor laser with optical feedback
No full textA highly developed traveling wave model for a semiconductor laser system supports sophisticated mode analysis of the coherence collapse regime in semiconductor lasers with delayed optical feedback. The concept of instantaneous optical modes is used. Time-frequency representations of chaotic trajectories are constructed and interpreted. This involves synthesizing the calculated optical modes with their corresponding steady states, analysis of the mode driving and coupling sources, and field expansion into modal components. The results support detailed physical interpretation of the optical and radiofrequency spectra throughout the coherence collapse regime. This includes simulation results for the transition out of coherence collapse at high optical feedback. Also key frequencies observed in the radiofrequency spectrum of the chaotic output are linked to the modes that mix to generate them
On the equilibrium solutions of electro-energy-reaction-diffusion systems
Get PDFElectro-energy-reaction-diffusion systems are thermodynamically consistent continuum models for reaction-diffusion processes that account for temperature and electrostatic effects in a way that total charge and energy are conserved. The question of the long-time asymptotic behavior in electro-energy-reaction-diffusion systems and the characterization of their equilibrium solutions leads to a maximization problem of the entropy on the manifold of states with fixed values for the linear charge and the nonlinear convex energy functional. As the main result, we establish the existence, uniqueness, and regularity of solutions to this constrained optimization problem. We give two conceptually different proofs, which are related to different perspectives on the constrained maximization problem. The first one is based on the Lagrangian approach, whereas the second one employs the direct method of the calculus of variations
Simulation of the Mode Dynamics in Broad-Ridge Laser Diodes
No full textIn this publication a method to simulate the mode dynamics in broad-ridge laser diode is presented. These devices exhibit rich lateral mode dynamics in addition to longitudinal mode dynamics observed in narrow-ridge laser diodes. The mode dynamics are strongly influenced by higher order effects, which are described by effective interaction terms and can derived from the band structure and the carrier scattering in the quantum well. The spatial dependency of pump current densities plays a crucial role in lateral mode dynamics, and thus, a Drift-Diffusion model is employed to calculate the current densities with an additional capturing term
ideal.II: a Galerkin space-time extension to the finite element library deal.II
Get PDFThe C++ library deal.II provides classes and functions to solve stationary problems with finite elements on one- to threedimensional domains. It also supports the typical way to solve time-dependent problems using time-stepping schemes, either with an implementation by hand or through the use of external libraries like SUNDIALS. A different approach is the usage of finite elements in time as well, which results in space-time finite element schemes. The library ideal.II (short for instationary deal.II) aims to extend deal.II to simplify implementations of the second approach
Matlab code to solve state and control constrained linear quadratic Mean-Field Games
No full textThis repository provides a Matlab code to approximate a state and control constrained linear quadratic Mean-Field Game. This code was developed with Matlab version R2023a in the context of the Ph.D. thesis of Mike Theiß. The theoretical background can be found in the thesis with the name "First Order Mean-Field Games and Mean-Field Optimal Control with State and Control Constraints", which is available at the bibliography of the Humboldt University