Publications Server of the Weierstrass Institute for Applied Analysis and Stochastics
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Continuum Percolation in a Nonstabilizing Environment
No full textWe prove phase transitions for continuum percolation in a Boolean model based on a Cox point process with nonstabilizing directing measure. The directing measure, which can be seen as a stationary random environment for the classical Poisson–Boolean model, is given by a planar rectangular Poisson line process. This Manhattan grid type construction features long-range dependencies in the environment, leading to absence of a sharp phase transition for the associated Cox–Boolean model. The phase transitions are established under individually as well as jointly varying parameters. Our proofs rest on discretization arguments and a comparison to percolation on randomly stretched lattices established in [Hof05]
Local Volatility under Rough Volatility
No full textSeveral asymptotic results for the implied volatility generated by a rough volatility model have been obtained in recent years (notably in the small-maturity regime), providing a better understanding of the shapes of the volatility surface induced by rough volatility models, supporting their calibration power to SP500 option data. Rough volatility models also generate a local volatility surface, via the so-called Markovian projection of the stochastic volatility. We complement the existing results on implied volatility by studying the asymptotic behavior of the local volatility surface generated by a class of rough stochastic volatility models, encompassing the rough Bergomi model. Notably, we observe that the celebrated “1/2 skew rule” linking the short-term at-the-money skew of the implied volatility to the short-term at-the-money skew of the local volatility, a consequence of the celebrated “harmonic mean formula” of [Berestycki et al. (2002). Quantitative Finance, 2, 61–69], is replaced by a new rule: the ratio of the at-the-money implied and local volatility skews tends to the constant (as opposed to the constant 1/2), where H is the regularity index of the underlying instantaneous volatility process
Local Surrogate Responses in the Schwarz Alternating Method for Elastic Problems on Random Voided Domains
No full textImperfections and inaccuracies in real technical products often influence the mechanical behavior and the overall structural reliability. The prediction of real stress states and possibly resulting failure mechanisms is essential and a real challenge, e.g. in the design process. In this contribution, imperfections in elastic materials such as air voids in adhesive bonds between fiber-reinforced composites are investigated. They are modeled as arbitrarily shaped and positioned. The focus is on local displacement values as well as on associated stress concentrations caused by the imperfections. For this purpose, the resulting complex random one-scale finite element model is numerically solved by a new developed surrogate model using an overlapping domain decomposition scheme based on Schwarz alternating method. Here, the actual response of local subproblems associated with isolated material imperfections is determined by a single appropriate surrogate model, that allows for an accelerated propagation of randomness. The efficiency of the method is demonstrated for imperfections with elliptical and ellipsoidal shape in 2D and 3D and extended to arbitrarily shaped voids. For the latter one, a local surrogate model based on artificial neural networks (ANN) is constructed. Finally, a comparison to experimental results validates the numerical predictions for a real engineering problem
On the structure of genealogical trees associated with explosive Crump--Mode--Jagers branching processes
Get PDFWe study the structure of genealogical trees associated with explosive Crump--Mode-Jagers branching processes (stopped at the explosion time), proving criteria for the associated tree to contain a node of infinite degree (a emphstar) or an infinite path. Next, we provide uniqueness criteria under which with probability 1 there exists exactly one of a unique star or a unique infinite path. Under the latter uniqueness criteria we also provide an example where, with strictly positive probability less than 1, there exists a unique star in the model. We thus illustrate that this probability is not restricted to being 0 or 1. Moreover, we provide structure theorems when there is a star, where we prove that certain trees appear as sub-trees in the tree infinitely often. We apply our results to a general discrete evolving tree model, named emphexplosive recursive trees with fitness. As a particular case, we study a family of emphsuper-linear preferential attachment models with fitness. For these models, we derive phase transitions in the model parameters in three different examples, leading to either exactly one star with probability 1 or one infinite path with probability 1 with every node having finite degree. Furthermore, we highlight examples where sub-trees T of empharbitrary size can appear infinitely often; behaviour that is markedly distinct from super-linear preferential attachment models studied in the literature so far
Exponential equivalence for misanthrope processes in contact with weak reservoirs and applications to totally asymmetric exclusion processes
Get PDFWe provide a short proof for the exponential equivalence between misanthrope processes in contact with weak reservoirs and those with impermeable boundaries. As a consequence, we can derive both the hydrodynamic limit and the large deviations of the totally asymmetric exclusion process (TASEP) in contact with weak reservoirs. This extends a recent result which proved the hydrodynamic behaviour of a vanishing viscocity approximation of the TASEP in contact with weak reservoirs. Furthermore, applications to a class of asymmetric exclusion processes with long jumps is discussed
Frequency Noise Characterization of Narrow-Linewidth Lasers: A Bayesian Approach
No full textWe describe a Bayesian estimation approach to infer on the frequency noise characteristics of narrow-linewidth semiconductor lasers from delayed self-heterodyne beat note measurements. Our technique is grounded in a statistical model of the measurement process that accounts for both the impact of the interferometer and the detector noise. The approach yields accurate results, even in scenarios where the intrinsic linewidth plateau is obscured by detector noise. The analysis is carried out using a Markov-chain Monte Carlo method in the frequency domain and exploits prior knowledge about the statistical distribution of the data. The method is validated using simulated time series data from a stochastic laser rate equation model incorporating 1/f -type non-Markovian noise
A hybrid physics-informed neural network based multiscale solver as a partial differential equation constrained optimization problem
Get PDFIn this work, we study physics-informed neural networks (PINNs) constrained by partial differential equations (PDEs) and their application in approximating multiscale PDEs. From a continuous perspective, our formulation corresponds to a non-standard PDE-constrained optimization problem with a PINN-type objective. From a discrete standpoint, the formulation represents a hybrid numerical solver that utilizes both neural networks and finite elements. We propose a function space framework for the problem and develop an algorithm for its numerical solution, combining an adjoint-based technique from optimal control with automatic differentiation. The multiscale solver is applied to a heat transfer problem with oscillating coefficients, where the neural network approximates a fine-scale problem, and a coarse-scale problem constrains the learning process. We show that incorporating coarse-scale information into the neural network training process through our modelling framework acts as a preconditioner for the low-frequency component of the fine-scale PDE, resulting in improved convergence properties and accuracy of the PINN method. The relevance of the hybrid solver to numerical homogenization is discussed
IWCN 223
No full textI would like to welcome all participants to the International Workshop on Computational Nanotechnology (IWCN 2023) that will take place in Barcelona (Spain). This is the 22nd edition of a series of conferences that started more than 30 years ago. The workshop serves as a unique platform for researchers, scholars, and practitioners to exchange ideas, present cutting-edge advancements, and foster collaborations that push the boundaries of this interdisciplinary field. The workshop's agenda has been thoughtfully crafted to encompass a broad spectrum of topics, ranging from first principles modeling and simulation methods to the development of advanced algorithms and software tools for nanoscale phenomena. Your help is needed for a fruitful conference. International conferences are the modern forums for exchanging forefront scientific knowledge. At the frontiers of research our knowledge is still unstable, somehow immature and, certainly, not free from controversies. I encourage all participants to promote discussions and ask and answer questions in a friendly atmosphere. We are all on the same team, playing the fascinating game of unraveling the mysteries of the nanoscale world. I would also like to encourage all participants (especially those who are in Barcelona for the first time) to discover our city. Barcelona is famous worldwide for its modernist architecture. Indeed, the conference site (casa Convalescència) is one example of the modernist style. This is one of the reasons for choosing this place to hold the conference. Finally, I would like to thank our sponsors SILVACO and MINISTERIO DE CIENCIA E INNOVACION. Special thanks to the local, national, program and advisory committees for their constant help and support. I really wish all of you a pleasant and scientifically fruitful stay in Barcelona during this week
Research-Data Management Planning in the German Mathematical Community
No full textIn this paper we discuss the notion of research data for the field of mathematics and report on the status quo of research-data management and planning. A number of decentralized approaches are presented and compared to needs and challenges faced in three use cases from different mathematical subdisciplines. We highlight the importance of tailoring research-data management plans to mathematicians research processes and discuss their usage all along the data life cycle
Galilean Bulk-Surface Electrothermodynamics and Applications to Electrochemistry
Get PDFIn this work, the balance equations of non-equilibrium thermodynamics are coupled to Galilean limit systems of the Maxwell equations, i.e., either to (i) the quasi-electrostatic limit or (ii) the quasi-magnetostatic limit. We explicitly consider a volume Ω, which is divided into Ω+ and Ω− by a possibly moving singular surface S, where a charged reacting mixture of a viscous medium can be present on each geometrical entity (Ω+,S,Ω−). By the restriction to the Galilean limits of the Maxwell equations, we achieve that only subsystems of equations for matter and electromagnetic fields are coupled that share identical transformation properties with respect to observer transformations. Moreover, the application of an entropy principle becomes more straightforward and finally helps estimate the limitations of the more general approach based the full set of Maxwell equations. Constitutive relations are provided based on an entropy principle, and particular care is taken in the analysis of the stress tensor and the momentum balance in the general case of non-constant scalar susceptibility. Finally, we summarise the application of the derived model framework to an electrochemical system with surface reactions