Weierstrass Institute for Applied Analysis and Stochastics

TetGen is a C++ program for generating good quality tetrahedral meshes aimed to support numerical methods and scientific computing. The problem of quality tetrahedral mesh generation is challenged by many theoretical and practical issues. TetGen uses Delaunay-based algorithms which have theoretical guarantee of correctness. It can robustly handle arbitrary complex 3D geometries and is fast in practice. The source code of TetGen is freely available. This article presents the essential algorithms and techniques used to develop TetGen. The intended audience are researchers or developers in mesh generation or other related areas. It describes the key software components of TetGen, including an efficient tetrahedral mesh data structure, a set of enhanced local mesh operations (combination of flips and edge removal), and filtered exact geometric predicates. The essential algorithms include incremental Delaunay algorithms for inserting vertices, constrained Delaunay algorithms for inserting constraints (edges and triangles), a new edge recovery algorithm for recovering constraints, and a new constrained Delaunay refinement algorithm for adaptive quality tetrahedral mesh generation. Experimental examples as well as comparisons with other softwares are presented.

The aim of this paper is to deal with the non-local fractional counterpart of the Laplace equation involving critical non-linearities studied in the famous paper of Brezis and Nirenberg (1983). Namely, our model is the equation \[ { ( − Δ ) s u − λ u = | u | 2 ∗ − 2 u a m p ; in Ω , u = 0 a m p ; in R n ∖ Ω , \left \{ \begin {array}{ll} (-\Delta )^s u-\lambda u=|u|^{2^*-2}u & {\mbox { in }} \Omega ,\\ u=0 & {\mbox { in }} \mathbb {R}^n\setminus \Omega \,, \end {array} \right . \] where ( − Δ ) s (-\Delta )^s is the fractional Laplace operator, s ∈ ( 0 , 1 ) s\in (0,1) , Ω \Omega is an open bounded set of

Since the beginning of the 1990s, hysteresis operators have been employed on a larger scale for the linearisation of hysteretic transducers. One reason for this is the increasing number of mechatronic applications that use solid-state actuators based on magnetostrictive or piezo-electric material or shape memory alloys. All of these actuator types show strong hysteretic effects. In addition to hysteresis, piezo-electric actuators show strong creep effects. Thus, the objective of the paper is to enlarge the operator-based methodology of hysteresis operators by elements that allow the description of systems with hysteresis and creep. To reach this objective, following the procedure used for hysteretic systems, creep operators are introduced to form, together with the hysteresis operators, a system operator for the simultaneous consideration of both phenomena. With regard to applications in control and measurement technology, the existence, uniqueness, Lipschitz continuity and thus input–output stability of its inverse operator are theoretically supported by functional analytical methods. Subsequently, the efficiency of this new concept is demonstrated, in practice, by a real-time inverse feedforward controller for piezo-electric actuators. Using this control concept, the tracking errors caused by hysteretic and creep effects are reduced by approximately one order of magnitude.

From an analysis of the time series of realized variance using recent high-frequency data, Gatheral et al. [Volatility is rough, 2014] previously showed that the logarithm of realized variance behaves essentially as a fractional Brownian motion with Hurst exponent H of order 0.1, at any reasonable timescale. The resulting Rough Fractional Stochastic Volatility (RFSV) model is remarkably consistent with financial time series data. We now show how the RFSV model can be used to price claims on both the underlying and integrated variance. We analyse in detail a simple case of this model, the rBergomi model. In particular, we find that the rBergomi model fits the SPX volatility markedly better than conventional Markovian stochastic volatility models, and with fewer parameters. Finally, we show that actual SPX variance swap curves seem to be consistent with model forecasts, with particular dramatic examples from the weekend of the collapse of Lehman Brothers and the Flash Crash.

Systems of nonlocally coupled oscillators can exhibit complex spatiotemporal patterns, called chimera states, which consist of coexisting domains of spatially coherent (synchronized) and incoherent dynamics. We report on a novel form of these states, found in a widely used model of a limit-cycle oscillator if one goes beyond the limit of weak coupling typical for phase oscillators. Then patches of synchronized dynamics appear within the incoherent domain giving rise to a multi-chimera state. We find that, depending on the coupling strength and range, different multichimera states arise in a transition from classical chimera states. The additional spatial modulation is due to strong coupling interaction and thus cannot be observed in simple phase-oscillator models.

We develop a new test of a parametric model of a conditional mean function against a nonparametric alternative. The test adapts to the unknown smoothness of the alternative model and is uniformly consistent against alternatives whose distance from the parametric model converges to zero at the fastest possible rate. This rate is slower than n[superscript -1/2]. Some existing tests have nontrivial power against restricted classes of alternatives whose distance from the parametric model decreases at the rate n[superscript -1/2]. There are, however, sequences of alternatives against which these tests are inconsistent and ours is consistent. As a consequence, there are alternative models for which the finite-sample power of our test greatly exceeds that of existing tests. This conclusion is illustrated by the results of some Monte Carlo experiments.

A system of N diffusions on R$SUP:D$ESUP:in which the interaction is expressed in terms of the empirical measure is considered. The limiting behavior as N →∞ is described by a McKean_Vlasov equation. The purpose of this paper is to show that the large deviations from the McKean-Vlasov limit can be described by a generalization of the theory of Freidlin and Wentzell and to obtain a characterization of the action functional. In order to obtain this action functional we first obtain results on projective limits of large deviation systems, large deviations on dual vector spaces and a Sanov type theorem for vectors of empirical measures

Abstract Using Lyapunov functionals the global behaviour of the solutions of a reaction‐diffusion system modelling chemotaxis is studied for bounded piecewise smooth domains in the plane. Geometric criteria can be given so that this dynamical system tends to a (not necessarily trivial) stationary state.

We develop a potential theoretic approach to the problem of metastability for reversible diffusion processes with generators of the form -\epsilon \Delta +\nabla F(\cdot)\nabla on \mathbb R^d or subsets of \mathbb R^d , where F is a smooth function with finitely many local minima. In analogy to previous work on discrete Markov chains, we show that metastable exit times from the attractive domains of the minima of F can be related, up to multiplicative errors that tend to one as \epsilon\downarrow 0 , to the capacities of suitably constructed sets. We show that this capacities can be computed, again up to multiplicative errors that tend to one, in terms of local characteristics of F at the starting minimum and the relevant saddle points . As a result, we are able to give the first rigorous proof of the classical Eyring–Kramers formula in dimension larger than 1 . The estimates on capacities make use of their variational representation and monotonicity properties of Dirichlet forms. The methods developed here are extensions of our earlier work on discrete Markov chains to continuous diffusion processes.

In this paper we deal with two non-local operators that are both well known and widely studied in the literature in connection with elliptic problems of fractional type. More precisely, for a fixed s ∈ (0,1) we consider the integral definition of the fractional Laplacian given by where c(n, s) is a positive normalizing constant, and another fractional operator obtained via a spectral definition, that is, where ei, λi are the eigenfunctions and the eigenvalues of the Laplace operator −Δ in Ω with homogeneous Dirichlet boundary data, while ai represents the projection of u on the direction ei. The aim of this paper is to compare these two operators, with particular reference to their spectrum, in order to emphasize their differences.

We derive and study a model describing passive mode locking---a set of differential equations with time delay. Unlike classical mode locking models based on the Haus master equation, this model does not assume small gain and loss per cavity round trip. Therefore, it is valid in a parameter range typical of semiconductor lasers. The limit of a slow saturable absorber is analyzed analytically. Bifurcations responsible for the appearance and breakup of the mode locking regime are studied numerically.

Neuroscience and clinical researchers are increasingly interested in quantitative magnetic resonance imaging (qMRI) due to its sensitivity to micro-structural properties of brain tissue such as axon, myelin, iron and water concentration. We introduce the hMRI-toolbox, an open-source, easy-to-use tool available on GitHub, for qMRI data handling and processing, presented together with a tutorial and example dataset. This toolbox allows the estimation of high-quality multi-parameter qMRI maps (longitudinal and effective transverse relaxation rates R 1 and R 2 ⋆ , proton density P D and magnetisation transfer M T saturation) that can be used for quantitative parameter analysis and accurate delineation of subcortical brain structures. The qMRI maps generated by the toolbox are key input parameters for biophysical models designed to estimate tissue microstructure properties such as the MR g-ratio and to derive standard and novel MRI biomarkers. Thus, the current version of the toolbox is a first step towards in vivo histology using MRI (hMRI) and is being extended further in this direction. Embedded in the Statistical Parametric Mapping (SPM) framework, it benefits from the extensive range of established SPM tools for high-accuracy spatial registration and statistical inferences and can be readily combined with existing SPM toolboxes for estimating diffusion MRI parameter maps. From a user's perspective, the hMRI-toolbox is an efficient, robust and simple framework for investigating qMRI data in neuroscience and clinical research.

The authors study the best possible accuracy of recovering the solution from linear ill-posed problems in variable Hilbert scales. A priori smoothness of the solution is expressed in terms of general source conditions, given through index functions. The emphasis is on geometric concepts. The notion of regularization is appropriately generalized, and the interplay between qualification of regularization and index function becomes visible. A general adaptation strategy is presented and its optimality properties are studied.

Functional neuroimaging methods hold promise for the identification of cognitive function and communication capacity in some severely brain-injured patients who may not retain sufficient motor function to demonstrate their abilities. We studied seven severely brain-injured patients and a control group of 14 subjects using a novel hierarchical functional magnetic resonance imaging assessment utilizing mental imagery responses. Whereas the control group showed consistent and accurate (for communication) blood-oxygen-level-dependent responses without exception, the brain-injured subjects showed a wide variation in the correlation of blood-oxygen-level-dependent responses and overt behavioural responses. Specifically, the brain-injured subjects dissociated bedside and functional magnetic resonance imaging-based command following and communication capabilities. These observations reveal significant challenges in developing validated functional magnetic resonance imaging-based methods for clinical use and raise interesting questions about underlying brain function assayed using these methods in brain-injured subjects.

Roadmap on optical rogue waves and extreme events, Akhmediev, Nail, Kibler, Bertrand, Baronio, Fabio, Belić, Milivoj, Zhong, Wei-Ping, Zhang, Yiqi, Chang, Wonkeun, Soto-Crespo, Jose M, Vouzas, Peter, Grelu, Philippe, Lecaplain, Caroline, Hammani, K, Rica, S, Picozzi, A, Tlidi, Mustapha, Panajotov, Krassimir, Mussot, Arnaud, Bendahmane, Abdelkrim, Szriftgiser, Pascal, Genty, Goery, Dudley, John, Kudlinski, Alexandre, Demircan, Ayhan, Morgner, Uwe, Amiraranashvili, Shalva, Bree, Carsten, Steinmeyer, Günter, Masoller, C, Broderick, Neil G R, Runge, Antoine F J, Erkintalo, Miro, Residori, S, Bortolozzo, U, Arecchi, F T, Wabnitz, Stefan, Tiofack, C G, Coulibaly, S, Taki, M

Spatiotemporal chaos and turbulence are universal concepts for the explanation of irregular behavior in various physical systems. Recently, a remarkable new phenomenon, called "chimera states," has been described, where in a spatially homogeneous system, regions of irregular incoherent motion coexist with regular synchronized motion, forming a self-organized pattern in a population of nonlocally coupled oscillators. Whereas most previous studies of chimera states focused their attention on the case of large numbers of oscillators employing the thermodynamic limit of infinitely many oscillators, here we investigate the properties of chimera states in populations of finite size using concepts from deterministic chaos. Our calculations of the Lyapunov spectrum show that the incoherent motion, which is described in the thermodynamic limit as a stationary behavior, in finite size systems appears as weak spatially extensive chaos. Moreover, for sufficiently small populations the chimera states reveal their transient nature: after a certain time span we observe a sudden collapse of the chimera pattern and a transition to the completely coherent state. Our results indicate that chimera states can be considered as chaotic transients, showing the same properties as type-II supertransients in coupled map lattices.

Abstract Optimal control problems with delays in state and control variables are studied. Constraints are imposed as mixed control–state inequality constraints. Necessary optimality conditions in the form of Pontryagin's minimum principle are established. The proof proceeds by augmenting the delayed control problem to a nondelayed problem with mixed terminal boundary conditions to which Pontryagin's minimum principle is applicable. Discretization methods are discussed by which the delayed optimal control problem is transformed into a large‐scale nonlinear programming problem. It is shown that the Lagrange multipliers associated with the programming problem provide a consistent discretization of the advanced adjoint equation for the delayed control problem. An analytical example and numerical examples from chemical engineering and economics illustrate the results. Copyright © 2008 John Wiley & Sons, Ltd.

Signal recovery in Gaussian white noise with variance tending to zero has served for some time as a representative model for nonparametric curve estimation, having all the essential traits in a pure form. The equivalence has mostly been stated informally, but an approximation in the sense of Le Cam's deficiency distance $\Delta$ would make it precise. The models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. In nonparametrics, a first result of this kind has recently been established for Gaussian regression. We consider the analogous problem for the experiment given by n i.i.d. observations having density f on the unit interval. Our basic result concerns the parameter space of densities which are in a Hölder ball with exponent $\alpha > 1/2$ and which are uniformly bounded away from zero. We show that an i. i. d. sample of size n with density f is globally asymptotically equivalent to a white noise experiment with drift $f^{1/2}$ and variance $1/4 n^{-1}$. This represents a nonparametric analog of Le Cam's heteroscedastic Gaussian approximation in the finite dimensional case. The proof utilizes empirical process techniques related to the Hungarian construction. White noise models on f and log f are also considered, allowing for various "automatic" asymptotic risk bounds in the i.i.d. model from white noise.

In this review article we discuss different techniques to solve numerically the time-dependent Schr\"odinger equation on unbounded domains. We present in detail the most recent approaches and describe briefly alternative ideas pointing out the relations bet\-ween these works. We conclude with several numerical examples from different application areas to compare the presented techniques. We mainly focus on the one-dimensional problem but also touch upon the situation in two space dimensions and the cubic nonlinear case.

Suppose that one observes a process Y on the unit interval, where dY(t) =n1/2 f(t)dt +dW (t) with an unknown function parameter f, given scale parameter n <=1 ("sample size") and standard Brownian motion W. We propose two classes of tests of qualitative nonparametric hypotheses about f such as monotonicity or concavity. These tests are asymptotically optimal and adaptive in a certain sense. They are constructed via a new class of multiscale statistics and an extension of Lévy's modulus of continuity of Brownian motion.