WIAS Preprints

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1D symmetry for semilinear pdes from the limit interface of the solution
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Farina, Alberto; Valdinoci, Enrico
We study bounded, entire, monotone solutions of the Allen-Cahn equation. We prove that under suitable assumptions on the limit interface and on the energy growth, the solution is 1D. In particular, differently from the previous literature, the solution is not assumed to have minimal properties. We think that this approach could be fruitful in concrete situations, where one can observe the phase separation at a large scale and whishes to deduce the values of the state parameter in the vicinity of the interface. As a simple example of the results obtained with this point of view, we mention that monotone solutions with energy bounds, whose limit interface does not contain a vertical line through the origin, are 1D, at least up to dimension 4.
3D boundary recovery by constrained Delaunay tetrahedralization
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Si, Hang; Gärtner, Klaus
Three-dimensional boundary recovery is a fundamental problem in mesh generation. In this paper, we propose a practical algorithm for solving this problem. Our algorithm is based on the construction of a it constrained Delaunay tetrahedralization (CDT) for a set of constraints (segments and facets). The algorithm adds additional points (so-called Steiner points) on segments only. The Steiner points are chosen in such a way that the resulting subsegments are Delaunay and their lengths are not unnecessarily short. It is theoretically guaranteed that the facets can be recovered without using Steiner points. The complexity of this algorithm is analyzed. The proposed algorithm has been implemented. Its performance is reported through various application examples
3D electrothermal simulations of organic LEDs showing negative differential resistance
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Liero, Matthias; Fuhrmann, Jürgen; Glitzky, Annegret; Koprucki, Thomas; Fischer, Axel; Reineke, Sebastian
Organic semiconductor devices show a pronounced interplay between temperature-activated conductivity and self-heating which in particular causes inhomogeneities in the brightness of large-area OLEDs at high power. We consider a 3D thermistor model based on partial differential equations for the electrothermal behavior of organic devices and introduce an extension to multiple layers with nonlinear conductivity laws, which also take the diode-like behavior in recombination zones into account. We present a numerical simulation study for a red OLED using a finite-volume approximation of this model. The appearance of S-shaped current-voltage characteristics with regions of negative differential resistance in a measured device can be quantitatively reproduced. Furthermore, this simulation study reveals a propagation of spatial zones of negative differential resistance in the electron and hole transport layers toward the contact.
3D numerical simulations of THz generation by two-color laser filaments
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Bergé, Luc; Skupin, Stefan; Köhler, Christian; Babushkin, Ihar; Herrmann, Joachim
Terahertz (THz) radiation produced by the filamentation of two-color pulses over long distances in argon is numerically investigated using a comprehensive model in full spacetime resolved geometry. We show that the dominant physical mechanism for THz generation in the filamentation regime at clamping intensity is based on quasi-dc plasma currents. The calculated THz spectra for different pump pulse energies and pulse durations are in agreement with previously reported experimental observations. For the same pulse parameters, near-infrared pump pulses at 2 m are shown to generate a more than one order of magnitude larger THz yield than pumps centered at 800 nm.
The 3D transient semiconductor equations with gradient-dependent and interfacial recombination
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Disser, Karoline; Rehberg, Joachim
We establish the well-posedness of the transient van Roosbroeck system in three space dimensions under realistic assumptions on the data: non-smooth domains, discontinuous coefficient functions and mixed boundary conditions. Moreover, within this analysis, recombination terms may be concentrated on surfaces and interfaces and may not only depend on chargecarrier densities, but also on the electric field and currents. In particular, this includes Avalanche recombination. The proofs are based on recent abstract results on maximal parabolic and optimal elliptic regularity of divergence-form operators.
Abelian theorems for stochastic volatility models with application to the estimation of jump activity of volatility
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Belomestny, Denis; Panov, Vladimir
In this paper, we prove a kind of Abelian theorem for a class of stochastic volatility models $(X, V)$, where both the state process $X$ and the volatility process $V$ may have jumps. Our results relate the asymptotic behavior of the characteristic function of $X_Delta$ for some $Delta > 0$ in a stationary regime to the Blumenthal-Getoor indexes of the Lévy processes driving the jumps in $X$ and $V$ . The results obtained are used to construct consistent estimators for the above Blumenthal-Getoor indexes based on low-frequency observations of the state process $X$. We derive the convergence rates for the corresponding estimator and prove that these rates can not be improved in general.
Absence of percolation in graphs based on stationary point processes with degrees bounded by two
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Jahnel, Benedikt; Tóbiás, András
We consider undirected graphs that arise as deterministic functions of stationary point processes such that each point has degree bounded by two. For a large class of point processes and edge-drawing rules, we show that the arising graph has no infinite connected component, almost surely. In particular, this extends our previous result for SINR graphs based on stabilizing Cox point processes and verifies the conjecture of Balister and Bollobás that the bidirectional $k$-nearest neighbor graph of a two-dimensional homogeneous Poisson point process does not percolate for k=2.
Absolute stability and absolute hyperbolicity in systems with discrete time-delays
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Yanchuk, Serhiy; Wolfrum, Matthias; Pereira, Tiago; Turaev, Dmitry
An equilibrium of a delay differential equation (DDE) is absolutely stable, if it is locally asymptotically stable for all delays. We present criteria for absolute stability of DDEs with discrete timedelays. In the case of a single delay, the absolute stability is shown to be equivalent to asymptotic stability for sufficiently large delays. Similarly, for multiple delays, the absolute stability is equivalent to asymptotic stability for hierarchically large delays. Additionally, we give necessary and sufficient conditions for a linear DDE to be hyperbolic for all delays. The latter conditions are crucial for determining whether a system can have stabilizing or destabilizing bifurcations by varying time delays.
Absorbing boundary condition for hyperbolic systems
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Erhardt, Matthias
This paper deals with absorbing boundary conditions for hyperbolic systems in one and two space dimensions. We prove the strict well-posedness of the resulting initial boundary value problem in 1D. Afterwards we establish the GKS-stability of the corresponding Lax-Wendroff-type finite difference scheme. Hereby, we have to extend the classical proofs, since the (discretized) absorbing boundary conditions do not fit the standard form of boundary conditions for hyperbolic systems.
Accelerated rogue solitons triggered by background radiation
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Demircan, Ayhan; Amiranashvili, Shalva; Brée, Carsten; Morgner, Uwe; Steinmeyer, Günter
[no abstract available]
Accurate localization of brain activity in presurgical fMRI by structure adaptive smoothing
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Tabelow, Karsten; Polzehl, Jörg; Uluğ, Aziz M.; Dyke, Jonathan P.; Watts, Richard; Heier, Linda A.; Voss, Henning U.
An important problem of the analysis of fMRI experiments is to achieve some noise reduction of the data without blurring the shape of the activation areas. As a novel solution to this problem, the Propagation-Separation approach (PS), a structure adaptive smoothing method, has been proposed recently. PS adapts to different shapes of activation areas by generating a spatial structure corresponding to similarities and differences between time series in adjacent locations. In this paper we demonstrate how this method results in more accurate localization of brain activity. First, it is shown in numerical simulations that PS is superior over Gaussian smoothing with respect to the accurate description of the shape of activation clusters and and results in less false detections. Second, in a study of 37 presurgical planning cases we found that PS and Gaussian smoothing often yield different results, and we present examples showing aspects of the superiority of PS as applied to presurgical planning.
An active poroelastic model for mechanochemical patterns in protoplasmic droplets of Physarum polycephalum
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Radszuweit, Markus; Engel, Harald; Bär, Markus
Motivated by recent experimental studies, we derive and analyze a two-dimensional model for the contraction patterns observed in protoplasmic droplets of Physarum polycephalum. The model couples a description of an active poroelastic two-phase medium with equations describing the spatiotemporal dynamics of the intracellular free calcium concentration. The poroelastic medium is assumed to consist of an active viscoelastic solid representing the cytoskeleton and a viscous fluid describing the cytosol. The equations for the poroelastic medium are obtained from continuum force balance and include the relevant mechanical fields and an incompressibility condition for the two-phase medium. The reaction-diffusion equations for the calcium dynamics in the protoplasm of Physarum are extended by advective transport due to the flow of the cytosol generated by mechanical stress. Moreover, we assume that the active tension in the solid cytoskeleton is regulated by the calcium concentration in the fluid phase at the same location, which introduces a mechanochemical coupling. A linear stability analysis of the homogeneous state without deformation and cytosolic flows exhibits an oscillatory Turing instability for a large enough mechanochemical coupling strength. Numerical simulations of the model equations reproduce a large variety of wave patterns, including traveling and standing waves, turbulent patterns, rotating spirals and antiphase oscillations in line with experimental observations of contraction patterns in the protoplasmic droplets.
Adaptive behaviour in a predator-prey model leads to multiple equilibrium states
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Pimenov, Alexander; Korobeinikov, Andrei; Rachinskii, Dmitrii
There is evidence that multiple stable equilibrium states are possible in real-life ecological systems. In order to verify a hypothesis that such a multitude of equilibrium states can be caused by adapting of animal behaviour to changes of environmental conditions, we consider a simple predator-prey model where prey changes a mode of behaviour in response to the pressure of predation. This model exhibits two stable coexisting equilibrium states with basins of attraction separated by a separatrix of a saddle point.
Adaptive goodness-of-fit tests based on signed ranks
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Rohde, Angelika
Within the nonparametric regression model with unknown regression function $l$ and independent, symmetric errors, a new multiscale signed rank statistic is introduced and a conditional multiple test of the simple hypothesis $l = 0$ against a nonparametric alternative is proposed. This test is distribution-free and exact for finite samples even in the heteroscedastic case. It adapts in a certain sense to the unknown smoothness of the regression function under the alternative, and it is uniformly consistent against alternatives whose sup-norm tends to zero at the fastest possible rate. The test is shown to be asymptotically optimal in two senses: It is rate-optimal adaptive against Hölder classes. Furthermore, its relative asymptotic efficiency with respect to an asymptotically minimax optimal test under sup-norm loss is close to one in case of homoscedastic Gaussian errors within a broad range of Hölder classes simultaneously.
Adaptive gradient descent for convex and non-convex stochastic optimization
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Ogaltsov, Aleksandr; Dvinskikh, Darina; Dvurechensky, Pavel; Gasnikov, Alexander; Spokoiny, Vladimir
In this paper we propose several adaptive gradient methods for stochastic optimization. Our methods are based on Armijo-type line search and they simultaneously adapt to the unknown Lipschitz constant of the gradient and variance of the stochastic approximation for the gradient. We consider an accelerated gradient descent for convex problems and gradient descent for non-convex problems. In the experiments we demonstrate superiority of our methods to existing adaptive methods, e.g. AdaGrad and Adam.
Adaptive manifold clustering
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Besold, Franz; Spokoiny, Vladimir
Clustering methods seek to partition data such that elements are more similar to elements in the same cluster than to elements in different clusters. The main challenge in this task is the lack of a unified definition of a cluster, especially for high dimensional data. Different methods and approaches have been proposed to address this problem. This paper continues the study originated by [6] where a novel approach to adaptive nonparametric clustering called Adaptive Weights Clustering (AWC) was offered. The method allows analyzing high-dimensional data with an unknown number of unbalanced clusters of arbitrary shape under very weak modeling as-sumptions. The procedure demonstrates a state-of-the-art performance and is very efficient even for large data dimension D. However, the theoretical study in [6] is very limited and did not re-ally address the question of efficiency. This paper makes a significant step in understanding the remarkable performance of the AWC procedure, particularly in high dimension. The approach is based on combining the ideas of adaptive clustering and manifold learning. The manifold hypoth-esis means that high dimensional data can be well approximated by a d-dimensional manifold for small d helping to overcome the curse of dimensionality problem and to get sharp bounds on the cluster separation which only depend on the intrinsic dimension d. We also address the problem of parameter tuning. Our general theoretical results are illustrated by some numerical experiments.
An adaptive multi level Monte-Carlo method with stochastic bounds for quantities of interest in groundwater flow with uncertain data
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Eigel, Martin; Merdon, Christian; Neumann, Johannes
The focus of this work is the introduction of some computable a posteriori error control to the popular multilevel Monte Carlo sampling for PDE with stochastic data. We are especially interested in applications in the geosciences such as groundwater flow with rather rough stochastic fields for the conductive permeability. With a spatial discretisation based on finite elements, a goal functional is defined which encodes the quantity of interest. The devised goal-oriented error estimator enables to determine guaranteed a posteriori error bounds for this quantity. In particular, it allows for the adaptive refinement of the mesh hierarchy used in the multilevel Monte Carlo simulation. In addition to controlling the deterministic error, we also suggest how to treat the stochastic error in probability. Numerical experiments illustrate the performance of the presented adaptive algorithm for a posteriori error control in multilevel Monte Carlo methods. These include a localised goal with problem-adapted meshes and a slit domain example. The latter demonstrates the refinement of regions with low solution regularity based on an inexpensive explicit error estimator in the multilevel algorithm.
Adaptive non-intrusive reconstruction of solutions to high-dimensional parametric PDEs
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Eigel, Martin; Farchmin, Nando; Heidenreich, Sebastian; Trunschke, Philipp
Numerical methods for random parametric PDEs can greatly benefit from adaptive refinement schemes, in particular when functional approximations are computed as in stochastic Galerkin and stochastic collocations methods. This work is concerned with a non-intrusive generalization of the adaptive Galerkin FEM with residual based error estimation. It combines the non-intrusive character of a randomized least-squares method with the a posteriori error analysis of stochastic Galerkin methods. The proposed approach uses the Variational Monte Carlo method to obtain a quasi-optimal low-rank approximation of the Galerkin projection in a highly efficient hierarchical tensor format. We derive an adaptive refinement algorithm which is steered by a reliable error estimator. Opposite to stochastic Galerkin methods, the approach is easily applicable to a wide range of problems, enabling a fully automated adjustment of all discretization parameters. Benchmark examples with affine and (unbounded) lognormal coefficient fields illustrate the performance of the non-intrusive adaptive algorithm, showing best-in-class performance
Adaptive regularization for image reconstruction from subsampled data
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Hintermüller, Michael; Langer, Andreas; Rautenberg, Carlos N.; Wu, Tao
Choices of regularization parameters are central to variational methods for image restoration. In this paper, a spatially adaptive (or distributed) regularization scheme is developed based on localized residuals, which properly balances the regularization weight between regions containing image details and homogeneous regions. Surrogate iterative methods are employed to handle given subsampled data in transformed domains, such as Fourier or wavelet data. In this respect, this work extends the spatially variant regularization technique previously established in [15], which depends on the fact that the given data are degraded images only. Numerical experiments for the reconstruction from partial Fourier data and for wavelet inpainting prove the efficiency of the newly proposed approach.
Adaptive smoothing as inference strategy: More specificity for unequally sized or neighboring regions
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Welvaert, Marijke; Tabelow, Karsten; Seurinck, Ruth; Rosseel, Yves
Although spatial smoothing of fMRI data can serve multiple purposes, increasing the sensitivity of activation detection is probably its greatest benefit. However, this increased detection power comes with a loss of specificity when non-adaptive smoothing (i.e. the standard in most software packages) is used. Simulation studies and analysis of experimental data was performed using the R packages neu-Rosim and fmri. In these studies, we systematically investigated the effect of spatial smoothing on the power and number of false positives in two particular cases that are often encountered in fMRI research: (1) Single condition activation detection for regions that differ in size, and (2) multiple condition activation detection for neighbouring regions. Our results demonstrate that adaptive smoothing is superior in both cases because less false positives are introduced by the spatial smoothing process compared to standard Gaussian smoothing or FDR inference of unsmoothed data.

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