Aicke Hinrichs, Lev Markhasin, Jens Oettershagen and Tino Ullrich Optimal quasi-Monte Carlo rules on higher order digital nets for the numerical integration of multivariate periodic functions 2015 http://arxiv.org/pdf/1501.01800v1
Juhi Jang, Juan J. L. Velázquez and Hyung Ju Hwang On the structure of the singular set for the kinetic Fokker-Planck equations in domains with boundaries 2015 http://arxiv.org/abs/1509.03366
Barbara Niethammer, Juan J. L. Velázquez and Michael Helmers Mathematical analysis of a coarsening model with local interactions 2015 http://arxiv.org/abs/1509.04917
2014
Mathias Beiglböck, Alexander M.G. Cox and Martin Huesmann Optimal Transport and Skorokhod Embedding ArXiv eprints 2014 http://arxiv.org/abs/1307.3656
Abstract: The Skorokhod embedding problem is to represent a given probability as the distribution of Brownian motion at a chosen stopping time. Over the last 50 years this has become one of the important classical problems in probability theory and a number of authors have constructed solutions with particular optimality properties. These constructions employ a variety of techniques ranging from excursion theory to potential and PDE theory and have been used in many different branches of pure and applied probability.
We develop a new approach to Skorokhod embedding based on ideas and concepts from optimal mass transport. In analogy to the celebrated article of Gangbo and McCann on the geometry of optimal transport, we establish a geometric characterization of Skorokhod embeddings with desired optimality properties. This leads to a systematic method to construct optimal embeddings. It allows us, for the first time, to derive all known optimal Skorokhod embeddings as special cases of one unified construction and leads to a variety of new embeddings. While previous constructions typically used particular properties of Brownian motion, our approach applies to all sufficiently regular Markov processes.
Anton Bovier and Lisa B. Hartung The extremal process of two-speed branching Brownian motion Electron. J. Probab., 19(Art. 18): 1-28 2014 http://dx.doi.org/10.1214/EJP.v19-2982
Abstract: We construct and describe the extremal process for variable speed branching Brownian motion, studied recently by Fang and Zeitouni \citeFZ_BM, for the case of piecewise constant speeds; in fact for simplicity we concentrate on the case when the speed is \(\sigma_1\) for \(s\leq bt\) and \(\sigma_2\) when \(bt\leq s\leq t\). In the case \(\sigma_1>\sigma_2\), the process is the concatenation of two BBM extremal processes, as expected. In the case \(\sigma_1<\sigma_2\), a new family of cluster point processes arises, that are similar, but distinctively different from the BBM process. Our proofs follow the strategy of Arguin, Bovier, and Kistler.
Anton Bovier and Lisa B. Hartung Extended Convergence of the Extremal Process of Branching Brownian Motion ArXiv e-prints 2014 http://arxiv.org/abs/1412.5975
Abstract: We extend the results of Arguin et al and A\"\i{}d\'ekon et al on the convergence of the extremal process of branching Brownian motion by adding an extra dimension that encodes the "location" of the particle in the underlying Galton-Watson tree. We show that the limit is a cluster point process on R+×R where each cluster is the atom of a Poisson point process on R+×R with a random intensity measure Z(dz)×Ce−2√x, where the random measure is explicitly constructed from the derivative martingale. This work is motivated by an analogous conjecture for the Gaussian free field by Biskup and Louidor.
Fabio Cavalletti and Martin Huesmann Self-intersection of optimal geodesics Bulletin of the London Mathematical Society, 46(3): 653-656 2014 http://dx.doi.org/10.1112/blms/bdu026
Abstract: Let (X,d,m)(X,d,m) be a geodesic metric measure space. Consider a geodesic μtμt in the L2L2-Wasserstein space. Then as ss goes to tt, the support of μsμs and the support of μtμt have to overlap, provided an upper bound on the densities holds. We give a more precise formulation of this self-intersection property. Given a geodesic of (X,d,m)(X,d,m) and t∈[0,1]t∈[0,1], we consider the set of times for which this geodesic belongs to the support of μtμt. We prove that tt is a point of Lebesgue density 1 for this set, in the integral sense. Our result applies to spaces satisfying CD(K,∞)CD(K,∞). The non-branching property is not needed.
Abstract: Let (X,d,m) be a proper, non-branching, metric measure space. We show existence and uniqueness of optimal transport maps for cost written as non-decreasing and strictly convex functions of the distance, provided (X,d,m) satisfies a new weak property concerning the behavior of m under the shrinking of sets to points, see Assumption 1. This in particular covers spaces satisfying the measure contraction property.
We also prove a stability property for Assumption 1: If (X,d,m) satisfies Assumption 1 and View the MathML source, for some continuous function g>0, then also View the MathML source verifies Assumption 1. Since these changes in the reference measures do not preserve any Ricci type curvature bounds, this shows that our condition is strictly weaker than measure contraction property.
Michael Griebel, Jan Hamaekers and Frederik Heber A bond order dissection ANOVA approach for efficient electronic structure calculations In Extraction of Quantifiable Information from Complex Systems, Volume 102 of Lecture Notes in Computational Science and Engineering
Chapter 11, page 211-235.
Publisher: Springer International
2014 http://dx.doi.org/10.1007/978-3-319-08159-5
Michael Griebel and Alexander Hullmann Dimensionality Reduction of High-Dimensional Data with a NonLinear Principal Component Aligned Generative Topographic Mapping SIAM J. Sci. Comput., 36(3): A1027-A1047 2014 http://dx.doi.org/10.1137/130931382
Michael Griebel and Helmut Harbrecht On the convergence of the combination technique In Sparse grids and Applications, Volume 97 of Lecture Notes in Computational Science and Engineering
page 55-74.
2014 http://dx.doi.org/10.1007/978-3-319-04537-5_3
Michael Griebel and Alexander Hullmann A Sparse Grid Based Generative Topographic Mapping for the Dimensionality Reduction of High-Dimensional Data In Modeling, Simulation and Optimization of Complex Processes - HPSC 2012
page 51-62.
2014 http://dx.doi.org/10.1007/978-3-319-09063-4_5
Michael Griebel and Jan Hamaekers Fast Discrete Fourier Transform on Generalized Sparse Grids In Sparse grids and Applications, Lecture Notes in Computational Science and Engineering Vol. 97, Springer, Volume 97 of Lecture Notes in Computational Science and Engineering
page 75-108.
2014 http://dx.doi.org/10.1007/978-3-319-04537-5_4
Thomas Hangelbroek, Francis J. Narcowich, Christian Rieger and Joseph D. Ward An inverse theorem on bounded domains for meshless methods using localized bases 2014 http://arxiv.org/pdf/1406.1435v1
Aicke Hinrichs and Jens Oettershagen Optimal point sets for quasi-Monte Carlo integration of bivariate periodic functions with bounded mixed derivatives 2014 http://arxiv.org/pdf/1409.5894v1
Martin Huesmann Optimal transport between random measures Annales de l’Institut Henri Poincaré (B) 2014 http://arxiv.org/abs/1206.3672
Abstract: We study couplings q∙ of two equivariant random measures λ∙ and μ∙ on a Riemannian manifold (M,d,m). Given a cost function we ask for minimizers of the mean transportation cost per volume. In case the minimal/optimal cost is finite and λω≪m we prove that there is a unique equivariant coupling minimizing the mean transportation cost per volume. Moreover, the optimal coupling is induced by a transportation map, q∙=(id,T)∗λ∙. We show that the optimal transportation map can be approximated by solutions to classical optimal transportation problems on bounded regions. In case of Lp−cost the optimal transportation cost per volume defines a metric on the space of equivariant random measure with unit intensity.
2013
Benjamin Berkels, Tom Fletcher, Behrend Heeren, Martin Rumpf and Benedikt Wirth Discrete geodesic regression in shape space In Anders Heyden, Fredrik Kahl, Carl Olsson, Magnus Oskarsson, Xue-Cheng Tai, editor, Energy Minimization Methods in Computer Vision and Pattern Recognition, Volume 8081 of Lecture Notes in Computer Science
page 108-122.
Publisher: Springer International
2013 http://dx.doi.org/10.1007/978-3-642-40395-8_9
Abstract: A new approach for the effective computation of geodesic re- gression curves in shape spaces is presented. Here, one asks for a geodesic curve on the shape manifold that minimizes a sum of dissimilarity mea- sures between given two- or three-dimensional input shapes and corre- sponding shapes along the regression curve. The proposed method is based on a variational time discretization of geodesics. Curves in shape space are represented as deformations of suitable reference shapes, which renders the computation of a discrete geodesic as a PDE constrained optimization for a family of deformations. The PDE constraint is de- duced from the discretization of the covariant derivative of the velocity in the tangential direction along a geodesic. Finite elements are used for the spatial discretization, and a hierarchical minimization strategy together with a Lagrangian multiplier type gradient descent scheme is implemented. The method is applied to the analysis of root growth in botany and the morphological changes of brain structures due to aging.
Michael Griebel and Helmut Harbrecht A note on the construction of L-fold sparse tensor product spaces Constructive Approximation, 38(2): 235-251 2013 http://dx.doi.org/10.1007/s00365-012-9178-7
Abstract: In the present paper, we consider the construction of general sparse tensor product spaces in arbitrary space dimensions when the single subdomains are of different dimensionality and the associated ansatz spaces possess different approximation properties. Our theory extends the results from Griebel and Harbrecht (Math. Comput., 2013) for the construction of two-fold sparse tensor product space to arbitrary L-fold sparse tensor product spaces.
Martin Huesmann and Karl-Theodor Sturm Optimal transport from Lebesgue to Poisson The Annals of Probability, 41(4): 2426-2478 2013 http://dx.doi.org/10.1214/12-AOP814
Abstract: This paper is devoted to the study of couplings of the Lebesgue measure and the Poisson point process. We prove existence and uniqueness of an optimal coupling whenever the asymptotic mean transportation cost is finite. Moreover, we give precise conditions for the latter which demonstrate a sharp threshold at d=2d=2. The cost will be defined in terms of an arbitrary increasing function of the distance.
The coupling will be realized by means of a transport map (“allocation map”) which assigns to each Poisson point a set (“cell”) of Lebesgue measure 1. In the case of quadratic costs, all these cells will be convex polytopes.