Publications Server of the Weierstrass Institute for Applied Analysis and Stochastics
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On self-similar pattern in coupled parabolic systems as non-equilibrium steady states
No full textWe consider reaction-diffusion systems and other related dissipative systems on unbounded domains which would have a Liapunov function (and gradient structure) when posed on a finite domain. In this situation, the system may reach local equilibrium on a rather fast time scale but the infinite amount of mass or energy leads to persistent mass or energy flow for all times. In suitably rescaled variables the system converges to a steady state that corresponds to asymptotically self-similar behavior in the original system
Convergence to self-similar profiles in reaction-diffusion systems
No full textWe study a reaction-diffusion system on the real line, where the reactions of the species are given by one reversible reaction pair satisfying the mass-action law. Under prescribed (different) positive limits at and we investigate the long-time behavior of solutions. Rescaling space and time according to the parabolic scaling with and , we show that solutions converge exponentially for to a self-similar profile. In the original variables, these profiles correspond to asymptotically self-similar behavior describing the phenomenon of diffusive mixing of the different states at infinity. Our method provides global exponential convergence for all initial states with finite entropy relative to the self-similar profile. For the case we can allow for profiles with arbitrary limiting states at , while for we need to assume that the two states at infinity are sufficiently close such that the profile is flat enough
Minimal and maximal solution maps of elliptic QVIs: Penalisation, Lipschitz stability, differentiability and optimal control
Get PDFQuasi-variational inequalities (QVIs) of obstacle type in many cases have multiple solutions that can be ordered. We study a multitude of properties of the operator mapping the source term to the minimal or maximal solution of such QVIs. We prove that the solution maps are locally Lipschitz continuous and directionally differentiable and show existence of optimal controls for problems that incorporate these maps as the control-to-state operator. We also consider a Moreau?Yosida-type penalisation for the QVI wherein we show that it is possible to approximate the minimal and maximal solutions by sequences of minimal and maximal solutions (respectively) of certain PDEs, which have a simpler structure and offer a convenient characterisation in particular for computation. For solution mappings of these penalised problems, we prove a number of properties including Lipschitz and differential stability. Making use of the penalised equations, we derive (in the limit) C-stationarity conditions for the control problem, in addition to the Bouligand stationarity we get from the differentiability result
Transport of heat and mass for reactive gas mixtures in porous media: Modeling and application
Get PDFWe present a modeling framework for multi-component, reactive gas mixtures and heat transport in porous media based on the Maxwell--Stefan and Darcy equations for multi-component diffusion and forced, viscous flow through porous media. Analysis of the model equations reveals thermodynamic consistency and uniqueness of steady states, while their mathematical structure facilitates discretization via the Finite-Volume approach resulting in an open- source based implementation of the modeling framework in Julia. The model allows to impose boundary conditions that accurately reflect the conditions prevailing in a photo-thermal chemical reactor that is subsequently introduced as a case study for the modeling framework. Comparison of numerical with experimental results reveals good agreement. Improvement options for the physical reactor are derived from simulation results demonstrating the practical utility of the modeling framework. Additionally, the framework is used for the simulation of thermodiffusion in a ternary gas mixture and has been verified with published numerical results with very good agreement
Probabilistic Maximization of Time-Dependent Capacities in a Gas Network
No full textThe determination of free technical capacities belongs to the core tasks of a gas network owner. Since gas loads are uncertain by nature, it makes sense to understand this as a probabilistic problem provided that stochastic modeling of available historical data is possible. Future clients, however, do not have a history or they do not behave in a random way, as is the case, for instance, in gas reservoir management. Therefore, capacity maximization becomes an optimization problem with uncertainty-related constraints which are partially of probabilistic and partially of robust (worst case) type. While previous attempts to solve this problem were devoted to models with static (time-independent) gas flow, we aim at considering here transient gas flow subordinate to the isothermal Euler equations. The basic challenge addressed in the manuscript is two-fold: first, a proper way of formulating probabilistic constraints in terms of the differential equations has to be provided. This will be realized on the basis of the so-called spherical-radial decomposition of Gaussian random vectors. Second, a suitable characterization of the worst-case load behaviour of future customers has to be found. It will be shown, that this is possible for quasi-static flow and can be transferred to the transient case. The complexity of the problem forces us to constrain ourselves in this first analysis to simple pipes or to a V-like structure of the network. Numerical solutions are presented and show that the differences between quasi-static and transient solutions are small, at least in these elementary examples
Cardiac dynamics of a human ventricular tissue model with focus on early afterdepolarizations
Get PDFThe paper is aimed to investigate computationally complex cardiac dynamics of the famous human ventricular model of ten Tusscher and Panfilov from 2006. The corresponding physical system is modeled by a set of nonlinear differential equations containing various of system pa- rameters. In case a specific physical parameter crosses a certain threshold, the system is forced to change dynamics, which might result in dangerous cardiac dynamics and can be precursors to cardiac death. For the performance of an efficient numerical analysis the original model is remod- eled and simplified in such a way that the modified models perfectly matches the trajectory of the original model. Moreover, it is demonstrated that the simplified models have the same dynamics. Furthermore, using the lowest dimensional model it is systematically shown by means of bifur- cation analysis that combinations of reduced slow and rapid potassium channels and enhanced sodium channel may lead to early afterdepolarizations. Finally, synchronization and the effect of EADs on larger scale (macro scale) is investigated numerically by studying the corresponding monodomain model. To this end we study the pattern formation of an one dimensional network of epi-, mid-myo- and endocardial cells and a two dimensional epicardial monodomain equation
Metaheuristic algorithms for enhancing multicepstral representation in voice spoofing detection: An experimental approach: Advances in Swarm Intelligence
No full textThe problem of voice spoofing detection is critical for identity authentication within biometric systems. Among the existing countermeasures, those based on soft computing have received attention from researchers in the last few years. However, it is known that spoofing representation is only effective when many features are used, which limits its applicability due to the curse of dimensionality. Accordingly, we focus on strategies to reduce the dimensionality of multicepstral features while maintaining reasonable accuracy in distinguishing between real and spoofed voices. Given the complexity of voice data, identifying and prioritizing the features with the highest information content is of utmost relevance. The study utilized four metaheuristic algorithms-GA, DA, PSO, and GWO for dimension reduction. The findings indicate that all algorithms, particularly GWO, exceed baseline performance levels. This demonstrates their efficacy in detecting voice spoofing. Moreover, it was found that certain combinations of cepstral coefficients when applied with principal component analysis projection, notably enhanced the model’s performance of voice spoofing detection
Diffusion Dynamics for an Infinite System of Two-Type Spheres and the Associated Depletion Effect
No full textWe consider a random diffusion dynamics for an infinite system of hard spheres of two different sizes evolving in ℝd, its reversible probability measure, and its projection on the subset of the large spheres. The main feature is the occurrence of an attractive short-range dynamical interaction --- known in the physics literature as a depletion interaction -- between the large spheres, which is induced by the hidden presence of the small ones. By considering the asymptotic limit for such a system when the density of the particles is high, we also obtain a constructive dynamical approach to the famous discrete geometry problem of maximisation of the contact number of n identical spheres in ℝd. As support material, we propose numerical simulations in the form of movies
Optimal control under uncertainty with joint chance state constraints: Almost-everywhere bounds, variance reduction, and application to (bi-)linear elliptic PDEs
Get PDFWe study optimal control of PDEs under uncertainty with the state variable subject to joint chance constraints. The controls are deterministic, but the states are probabilistic due to random variables in the governing equation. Joint chance constraints ensure that the random state variable meets pointwise bounds with high probability. For linear governing PDEs and elliptically distributed random parameters, we prove existence and uniqueness results for almost-everywhere state bounds. Using the spherical-radial decomposition (SRD) of the uncertain variable, we prove that when the probability is very large or small, the resulting Monte Carlo estimator for the chance constraint probability exhibits substantially reduced variance compared to the standard Monte Carlo estimator. We further illustrate how the SRD can be leveraged to efficiently compute derivatives of the probability function, and discuss different expansions of the uncertain variable in the governing equation. Numerical examples for linear and bilinear PDEs compare the performance of Monte Carlo and quasi-Monte Carlo sampling methods, examining probability estimation convergence as the number of samples increases. We also study how the accuracy of the probabilities depends on the truncation of the random variable expansion, and numerically illustrate the variance reduction of the SRD
Generalized bootstrap in the Bures-Wasserstein space
Get PDFThis study focuses on finite-sample inference on the non-linear Bures-Wasserstein manifold and introduces a generalized bootstrap procedure for estimating Bures-Wasserstein barycenters. We provide non-asymptotic statistical guarantees for the resulting bootstrap confidence sets. The proposed approach incorporates classical resampling methods, including the multiplier bootstrap highlighted as a specific example. Additionally, the paper compares bootstrap-based confidence sets with asymptotic confidence sets obtained in the work of Kroshnin et al. [2021], evaluating their statistical performance and computational complexities. The methodology is validated through experiments on synthetic datasets and real-world applications