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- ItemAnisotropic solid-liquid interface kinetics in silicon: An atomistically informed phase-field model(Bristol : IOP Publ., 2017) Bergmann, S.; Albe, K.; Flege, E.; Barragan-Yani, D.A.; Wagner, B.We present an atomistically informed parametrization of a phase-field model for describing the anisotropic mobility of liquid–solid interfaces in silicon. The model is derived from a consistent set of atomistic data and thus allows to directly link molecular dynamics and phase field simulations. Expressions for the free energy density, the interfacial energy and the temperature and orientation dependent interface mobility are systematically fitted to data from molecular dynamics simulations based on the Stillinger–Weber interatomic potential. The temperature-dependent interface velocity follows a Vogel–Fulcher type behavior and allows to properly account for the dynamics in the undercooled melt.
- ItemBoundary conditions for electrochemical interfaces(Bristol : IOP Publishing, 2017) Landstorfer, ManuelConsistent boundary conditions for electrochemical interfaces, which cover double layer charging, pseudo-capacitive effects and transfer reactions, are of high demand in electrochemistry and adjacent disciplines. Mathematical modeling and optimization of electrochemical systems is a strongly emerging approach to reduce cost and increase efficiency of super-capacitors, batteries, fuel cells, and electro-catalysis. However, many mathematical models which are used to describe such systems lack a real predictive value. Origin of this shortcoming is the usage of oversimplified boundary conditions. In this work we derive the boundary conditions for some general electrode-electrolyte interface based on non-equilibrium thermodynamics for volumes and surfaces. The resulting equations are widely applicable and cover also tangential transport. The general framework is then applied to a specific material model which allows the deduction of a current-voltage relation and thus a comparison to experimental data. Some simplified 1D examples show the range of applicability of the new approach.
- ItemScattering matrices and Dirichlet-to-Neumann maps(Amsterdam [u.a.] : Elsevier, 2017) Behrndt, Jussi; Malamud, Mark M.; Neidhardt, HagenA general representation formula for the scattering matrix of a scattering system consisting of two self-adjoint operators in terms of an abstract operator valued Titchmarsh–Weyl m-function is proved. This result is applied to scattering problems for different self-adjoint realizations of Schrödinger operators on unbounded domains, Schrödinger operators with singular potentials supported on hypersurfaces, and orthogonal couplings of Schrödinger operators. In these applications the scattering matrix is expressed in an explicit form with the help of Dirichlet-to-Neumann maps.
- ItemOptimal Entropy-Transport problems and a new Hellinger–Kantorovich distance between positive measures(Berlin ; Heidelberg : Springer, 2017) Liero, Matthias; Mielke, Alexander; Savaré, GiuseppeWe develop a full theory for the new class of Optimal Entropy-Transport problems between nonnegative and finite Radon measures in general topological spaces. These problems arise quite naturally by relaxing the marginal constraints typical of Optimal Transport problems: given a pair of finite measures (with possibly different total mass), one looks for minimizers of the sum of a linear transport functional and two convex entropy functionals, which quantify in some way the deviation of the marginals of the transport plan from the assigned measures. As a powerful application of this theory, we study the particular case of Logarithmic Entropy-Transport problems and introduce the new Hellinger–Kantorovich distance between measures in metric spaces. The striking connection between these two seemingly far topics allows for a deep analysis of the geometric properties of the new geodesic distance, which lies somehow between the well-known Hellinger–Kakutani and Kantorovich–Wasserstein distances.
- ItemOption pricing in the moderate deviations regime(Oxford [u.a.] : Wiley-Blackwell, 2017) Friz, Peter; Gerhold, Stefan; Pinter, ArpadWe consider call option prices close to expiry in diffusion models, in an asymptotic regime (“moderately out of the money”) that interpolates between the well-studied cases of at-the-money and out-of-the-money regimes. First and higher order small-time moderate deviation estimates of call prices and implied volatilities are obtained. The expansions involve only simple expressions of the model parameters, and we show how to calculate them for generic local and stochastic volatility models. Some numerical computations for the Heston model illustrate the accuracy of our results.
- ItemOn the regularity of SLE trace(Cambridge : Cambridge Univ. Press, 2017) Friz, Peter K.; Tran, HuyWe revisit regularity of SLE trace, for all κ≠8, and establish Besov regularity under the usual half-space capacity parametrization. With an embedding theorem of Garsia–Rodemich–Rumsey type, we obtain finite moments (and hence almost surely) optimal variation regularity with index min(1+κ/8,2), improving on previous works of Werness, and also (optimal) Hölder regularity à la Johansson Viklund and Lawler.
- ItemDeath and rebirth of neural activity in sparse inhibitory networks([London] : IOP, 2017) Angulo-Garcia, David; Luccioli, Stefano; Olmi, Simona; Torcini, AlessandroInhibition is a key aspect of neural dynamics playing a fundamental role for the emergence of neural rhythms and the implementation of various information coding strategies. Inhibitory populations are present in several brain structures, and the comprehension of their dynamics is strategical for the understanding of neural processing. In this paper, we clarify the mechanisms underlying a general phenomenon present in pulse-coupled heterogeneous inhibitory networks: inhibition can induce not only suppression of neural activity, as expected, but can also promote neural re-activation. In particular, for globally coupled systems, the number of firing neurons monotonically reduces upon increasing the strength of inhibition (neuronal death). However, the random pruning of connections is able to reverse the action of inhibition, i.e. in a random sparse network a sufficiently strong synaptic strength can surprisingly promote, rather than depress, the activity of neurons (neuronal rebirth). Thus, the number of firing neurons reaches a minimum value at some intermediate synaptic strength. We show that this minimum signals a transition from a regime dominated by neurons with a higher firing activity to a phase where all neurons are effectively sub-threshold and their irregular firing is driven by current fluctuations. We explain the origin of the transition by deriving a mean field formulation of the problem able to provide the fraction of active neurons as well as the first two moments of their firing statistics. The introduction of a synaptic time scale does not modify the main aspects of the reported phenomenon. However, for sufficiently slow synapses the transition becomes dramatic, and the system passes from a perfectly regular evolution to irregular bursting dynamics. In this latter regime the model provides predictions consistent with experimental findings for a specific class of neurons, namely the medium spiny neurons in the striatum.
- ItemType II singular perturbation approximation for linear systems with Lévy noise(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Redmann, MartinWhen solving linear stochastic partial differential equations numerically, usually a high order spatial discretisation is needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretised systems and hence reduce computational complexity. A particular MOR technique to obtain a reduced order model (ROM) is singular perturbation approximation (SPA), a method which has been extensively studied for deterministic systems. As so-called type I SPA it has already been extended to stochastic equations. We provide an alternative generalisation of the deterministic setting to linear systems with Lévy noise which is called type II SPA. It turns out that the ROM from applying type II SPA has better properties than the one of using type I SPA. In this paper, we provide new energy interpretations for stochastic reachability Gramians, show the preservation of mean square stability in the ROM by type II SPA and prove two different error bounds for type II SPA when applied to Lévy driven systems.
- ItemTowards time-limited H2-optimal model order reduction(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Goyal, Pawan; Redmann, MartinIn order to solve partial differential equations numerically and accurately, a high order spatial discretization is usually needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretized systems and hence reduce computational complexity. A particular class of MOR techniques are H2-optimal methods such as the iterative rational Krylov subspace algorithm (IRKA) and related schemes. However, these methods are used to obtain good approximations on a infinite time-horizon. Thus, in this work, our main goal is to discuss MOR schemes for time-limited linear systems. For this, we propose an alternative time-limited H2-norm and show its connection with the time-limited Gramians. We then provide first-order optimality conditions for an optimal reduced order model (ROM) with respect to the time-limited H2-norm. Based on these optimality conditions, we propose an iterative scheme which upon convergences aims at satisfying these conditions. Then, we analyze how far away the obtained ROM is from satisfying the optimality conditions.We test the efficiency of the proposed iterative scheme using various numerical examples and illustrate that the newly proposed iterative method can lead to a better reduced-order compared to unrestricted IRKA in the time interval of interest.
- ItemMathematical models as research data via flexiformal theory graphs(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Kohlhase, Michael; Koprucki, Thomas; Müller, Dennis; Tabelow, KarstenMathematical modeling and simulation (MMS) has now been established as an essential part of the scientific work in many disciplines. It is common to categorize the involved numerical data and to some extent the corresponding scientific software as research data. But both have their origin in mathematical models, therefore any holistic approach to research data in MMS should cover all three aspects: data, software, and models. While the problems of classifying, archiving and making accessible are largely solved for data and first frameworks and systems are emerging for software, the question of how to deal with mathematical models is completely open. In this paper we propose a solution to cover all aspects of mathematical models: the underlying mathematical knowledge, the equations, boundary conditions, numeric approximations, and documents in a flexiformal framework, which has enough structure to support the various uses of models in scientific and technology workflows. Concretely we propose to use the OMDoc/MMT framework to formalize mathematical models and show the adequacy of this approach by modeling a simple, but non-trivial model: van Roosbroecks drift-diffusion model for one-dimensional devices. This formalization and future extensions allows us to support the modeler by e.g. flexibly composing models, visualizing Model Pathway Diagrams, and annotating model equations in documents as induced from the formalized documents by flattening. This directly solves some of the problems in treating MMS as research data and opens the way towards more MKM services for models.