 2015Roland Donninger
Strichartz estimates in similarity coordinates and stable blowup for the critical wave equation
2015
http://arxiv.org/abs/1509.02041

 
 Roland Donninger and Birgit Schörkhuber
Stable blowup for wave equations in odd space dimensions
2015
http://arxiv.org/abs/1504.00808

 
 Dinh Dũng and Michael Griebel
Hyperbolic cross approximation in infinite dimensions
Journal of Complexity 2015
http://arxiv.org/pdf/1501.01119v1
Abstract: We give tight upper and lower bounds of the cardinality of
the index sets of certain hyperbolic crosses which reflect mixed
Sobolev–Korobovtype smoothness and mixed Sobolevanalytictype
smoothness in the infinitedimensional case where specific
summability properties of the smoothness indices are fulfilled.
These estimates are then applied to the linear approximation of
functions from the associated spaces in terms of the εdimension
of their unit balls. Here, the approximation is based on linear
information. Such function spaces appear for example for the
solution of parametric and stochastic PDEs. The obtained upper
and lower bounds of the approximation error as well as of the
associated εcomplexities are completely independent of any parametric
or stochastic dimension. Moreover, the rates are independent
of the parameters which define the smoothness properties
of the infinitevariate parametric or stochastic part of the solution.
These parameters are only contained in the order constants.
This way, linear approximation theory becomes possible in the
infinitedimensional case and corresponding infinitedimensional
problems get tractable. 
 
 Alexander Effland, Martin Rumpf, Stefan Simon, Kirsten Stahn and Benedikt Wirth
Bézier curves in the space of images
In Proceedings Scale Space and Variational Methods in Computer Vision, Volume 9087 of Lecture Notes in Computer Science
page 372384.
Publisher: Springer International
2015
http://dx.doi.org/10.1007/9783319184616_30
Abstract: Bézier curves are a widespread tool for the design of curves in Euclidian space. This paper generalizes the notion of Bézier curves to the infinitedimensional space of images. To this end the space of images is equipped with a Riemannian metric which measures the cost of image transport and intensity variation in the sense of the metamorphosis model by Miller and Younes. Bézier curves are then computed via the Riemannian version of de Casteljau's algorithm, which is based on a hierarchical scheme of convex combination along geodesic curves. Geodesics are approximated using a variational discretization of the Riemannian path energy. This leads to a generalized de Casteljau method to compute suitable discrete Bézier curves in image space. Selected test cases demonstrate qualitative properties of the approach. Furthermore, a Bézier approach for the modulation of face interpolation and shape animation via image sketches is presented. 
 
 Alberto Enciso, Daniel PeraltaSalas and Stefan Steinerberger
Prescribing the nodal set of the first eigenfunction in each conformal class
2015
http://arxiv.org/abs/1503.05105

 
 Matthias Erbar, Jan Maas and Prasad Tetali
Ricci curvature bounds for BernoulliLaplace and random transposition models
Ann. Fac. Sci. Toulouse Math., ArXiv eprints, 24(4): 781800 2015
http://arxiv.org/abs/1409.8605
Abstract: We calculate a Ricci curvature lower bound for some classical examples of random walks, namely, a chain on a slice of the $n$dimensional discrete cube (the socalled BernoulliLaplace model) and the random transposition shuffle of the symmetric group of permutations on $n$ letters. 
 
 Matthias Erbar, Kazumasa Kuwada and KarlTheodor Sturm
On the equivalence of the entropic curvaturedimension condition and Bochner's inequality on metric measure spaces
Invent. Math., 201(3): 9931071 2015
http://dx.doi.org/10.1007/s0022201405637

 
 Matthias Erbar and Martin Huesmann
Curvature bounds for configuration spaces
Calculus of Variations and Partial Differential Equations, 54(1): 397430 2015
http://dx.doi.org//10.1007/s0052601407901

 
 Max Fathi and Jan Maas
Entropic Ricci curvature bounds for discrete interacting systems
Ann. Appl. Prob. 2015
http://arxiv.org/abs/1501.00562

 
 Patrik L. Ferrari and Peter Nejjar
Anomalous shock fluctuations in TASEP and last passage percolation models
Probab. Theory Related Fields, 161(1): 61109 2015
http://dx.doi.org/10.1007/s0044001305446
Abstract: We consider the totally asymmetric simple exclusion process with initial conditions and/or jump rates such that shocks are generated. If the initial condition is deterministic, then the shock at time \(t\) will have a width of order \(t^{1/3}\). We determine the law of particle positions in the large time limit around the shock in a few models. In particular, we cover the case where at both sides of the shock the process of the particle positions is asymptotically described by the Airy\(_1\) process. The limiting distribution is a product of two distribution functions, which is a consequence of the fact that at the shock two characteristics merge and of the slow decorrelation along the characteristics. We show that the result generalizes to generic last passage percolation models. 
 
 Patrik L. Ferrari, Herbert Spohn and Thomas Weiss
Scaling Limit for Brownian Motions with Onesided Collisions
Ann. Appl. Probab., 25(3): 13491382 2015
http://dx.doi.org/10.1214/14AAP1025
Abstract: We consider Brownian motions with onesided collisions, meaning that each particle is reflected at its right neighbour. For a finite number of particles a Schütztype formula is derived for the transition probability. We investigate an infinite system with periodic initial configuration, i.e., particles are located at the integer lattice at time zero. The joint distribution of the positions of a finite subset of particles is expressed as a Fredholm determinant with a kernel defining a signed determinantal point process. In the appropriate large time scaling limit, the fluctuations in the particle positions are described by the Airy\(_1\) process. 
 
 Patrik L. Ferrari and Balint Vető
TracyWidom asymptotics for qTASEP
Ann. Inst. H. Poincaré Probab. Statist., 51(4): 14651485 2015
http://dx.doi.org/10.1214/14AIHP614
Abstract: We consider the qTASEP, that is a qdeformation of the totally asymmetric simple exclusion process (TASEP) on \(\mathbb{Z}\) for \(q \in [0,1)\) where the jump rates depend on the gap to the next particle. For step initial condition, we prove that the current fluctuation of \(q\)TASEP at time \(\tau\) are of order \(\tau^{1/3}\) and asymptotically distributed as the GUE TracyWidom distribution. 
 
 Patrik L. Ferrari, Herbert Spohn and Thomas Weiss
Brownian motions with onesided collisions: the stationary case
Electronic Journal of Probability, 20(Art. 69): 141 2015
http://dx.doi.org/10.1214/EJP.v204177
Abstract: We consider an infinite system of Brownian motions which interact through a given Brownian motion being reflected from its left neighbor. Earlier we studied this system for deterministic periodic initial configurations. In this contribution we consider initial configurations distributed according to a Poisson point process with constant intensity, which makes the process spacetime stationary. We prove convergence to the Airy process for stationary the case. As a byproduct we obtain a novel representation of the finitedimensional distributions of this process. Our method differs from the one used for the TASEP and the KPZ equation by removing the initial step only after the large time limit. This leads to a new universal crossover process. 
 
 Patrik L. Ferrari and Peter Nejjar
Shock fluctuations in flat TASEP under critical scaling
J. Stat. Phys., 160(4): 9851004 2015
http://arxiv.org/abs/1408.4850
Abstract: We consider TASEP with two types of particles starting at every second site. Particles to the left of the origin have jump rate $1$, while particles to the right have jump rate $\alpha$. When $\alpha<1$ there is a formation of a shock where the density jumps to $(1\alpha)/2$. For $\alpha<1$ fixed, the statistics of the associated height functions around the shock is asymptotically (as time $t\to\infty$) a maximum of two independent random variables as shown in [arXiv:1306.3336]. In this paper we consider the critical scaling when $1\alpha=a t^{1/3}$, where $t\gg 1$ is the observation time. In that case the decoupling does not occur anymore. We determine the limiting distributions of the shock and numerically study its convergence as a function of $a$. We see that the convergence to product $F_{\rm GOE}^2$ occurs quite rapidly as $a$ increases. The critical scaling is analogue to the one used in the last passage percolation to obtain the BBP transition processes. 
 
 Benedict Geihe and Martin Rumpf
A posteriori error estimates for sequential laminates in shape optimization
In DCDSS Special issue on HomogenizationBased Numerical Methods 2015
http://arxiv.org/abs/1501.07461
Abstract: A posteriori error estimates are derived in the context of twodimensional structural elastic shape optimization under the compliance objective. It is known that the optimal shape features are microstructures that can be constructed using sequential lamination. The descriptive parameters explicitly depend on the stress. To derive error estimates the dual weighted residual approach for control problems in PDE constrained optimization is employed, involving the elastic solution and the microstructure parameters. Rigorous estimation of interpolation errors ensures robustness of the estimates while local approximations are used to obtain fully practical error indicators. Numerical results show sharply resolved interfaces between regions of full and intermediate material density. 
 
 Nicola Gigli, Tapio Rajala and KarlTheodor Sturm
Optimal maps and exponentiation on finite dimensional spaces with Ricci curvature bounded from below
J. Geom. Anal. 2015
http://arxiv.org/abs/1305.4849
Abstract: We prove existence and uniqueness of optimal maps on RCD∗(K,N) spaces under the assumption that the starting measure is absolutely continuous. We also discuss how this result naturally leads to the notion of exponentiation and to the localtoglobal property of RCD∗(K,N) bounds. 
 
 Michael Griebel, Christian Rieger and Barbara Zwicknagl
Multiscale approximation and reproducing kernel Hilbert space methods
SIAM Journal on Numerical Analysis, 53(2): 852873 2015
http://dx.doi.org/10.1137/130932144

 
 Michael Griebel, Alexander Hullmann and Oeter Oswald
Optimal scaling parameters for sparse grid discretizations
Numerical Linear Algebra with Applications, 22(1): 76100 2015
http://dx.doi.org/10.1002/nla.1939
Abstract: We apply iterative subspace correction methods to elliptic PDE problems discretized by generalized sparse grid systems. The involved subspace solvers are based on the combination of all anisotropic full grid spaces that are contained in the sparse grid space. Their relative scaling is at our disposal and has significant influence on the performance of the iterative solver. In this paper, we follow three approaches to obtain closetooptimal or even optimal scaling parameters of the subspace solvers and thus of the overall subspace correction method. We employ a Linear Program that we derive from the theory of additive subspace splittings, an algebraic transformation that produces partially negative scaling parameters which result in improved asymptotic convergence properties, and finally we use the OptiCom method as a variable nonlinear preconditioner. 
 
 Lisa B. Hartung and Anton Klimovsky
The glassy phase of the complex branching Brownian motion energy model
Electron. Commun. Probab., 20(Art. 78): 115 2015
http://dx.doi.org/10.1214/ECP.v204360

 
 Stefanie Heyden, Bo Li, Kerstin Weinberg, Sergio Conti and Michael Ortiz
A micromechanical damage and fracture model for polymers based on fractional straingradient elasticity
J. Mech. Phys. Solids, 74: 175195 2015
http://dx.doi.org/10.1016/j.jmps.2014.08.005

 