@article{1617,
abstract = {We study the discrepancy of jittered sampling sets: such a set P⊂ [0,1]d is generated for fixed m∈ℕ by partitioning [0,1]d into md axis aligned cubes of equal measure and placing a random point inside each of the N=md cubes. We prove that, for N sufficiently large, 1/10 d/N1/2+1/2d ≤EDN∗(P)≤ √d(log N) 1/2/N1/2+1/2d, where the upper bound with an unspecified constant Cd was proven earlier by Beck. Our proof makes crucial use of the sharp Dvoretzky-Kiefer-Wolfowitz inequality and a suitably taylored Bernstein inequality; we have reasons to believe that the upper bound has the sharp scaling in N. Additional heuristics suggest that jittered sampling should be able to improve known bounds on the inverse of the star-discrepancy in the regime N≳dd. We also prove a partition principle showing that every partition of [0,1]d combined with a jittered sampling construction gives rise to a set whose expected squared L2-discrepancy is smaller than that of purely random points.},
author = {Pausinger, Florian and Steinerberger, Stefan},
journal = {Journal of Complexity},
pages = {199 -- 216},
publisher = {Academic Press},
title = {{On the discrepancy of jittered sampling}},
doi = {10.1016/j.jco.2015.11.003},
volume = {33},
year = {2016},
}
@article{1662,
abstract = {We introduce a modification of the classic notion of intrinsic volume using persistence moments of height functions. Evaluating the modified first intrinsic volume on digital approximations of a compact body with smoothly embedded boundary in Rn, we prove convergence to the first intrinsic volume of the body as the resolution of the approximation improves. We have weaker results for the other modified intrinsic volumes, proving they converge to the corresponding intrinsic volumes of the n-dimensional unit ball.},
author = {Edelsbrunner, Herbert and Pausinger, Florian},
journal = {Advances in Mathematics},
pages = {674 -- 703},
publisher = {Academic Press},
title = {{Approximation and convergence of the intrinsic volume}},
doi = {10.1016/j.aim.2015.10.004},
volume = {287},
year = {2016},
}
@article{1149,
abstract = {We study the usefulness of two most prominent publicly available rigorous ODE integrators: one provided by the CAPD group (capd.ii.uj.edu.pl), the other based on the COSY Infinity project (cosyinfinity.org). Both integrators are capable of handling entire sets of initial conditions and provide tight rigorous outer enclosures of the images under a time-T map. We conduct extensive benchmark computations using the well-known Lorenz system, and compare the computation time against the final accuracy achieved. We also discuss the effect of a few technical parameters, such as the order of the numerical integration method, the value of T, and the phase space resolution. We conclude that COSY may provide more precise results due to its ability of avoiding the variable dependency problem. However, the overall cost of computations conducted using CAPD is typically lower, especially when intervals of parameters are involved. Moreover, access to COSY is limited (registration required) and the rigorous ODE integrators are not publicly available, while CAPD is an open source free software project. Therefore, we recommend the latter integrator for this kind of computations. Nevertheless, proper choice of the various integration parameters turns out to be of even greater importance than the choice of the integrator itself. © 2016 IMACS. Published by Elsevier B.V. All rights reserved.},
author = {Miyaji, Tomoyuki and Pilarczyk, Pawel and Gameiro, Marcio and Kokubu, Hiroshi and Mischaikow, Konstantin},
journal = {Applied Numerical Mathematics},
pages = {34 -- 47},
publisher = {Elsevier},
title = {{A study of rigorous ODE integrators for multi scale set oriented computations}},
doi = {10.1016/j.apnum.2016.04.005},
volume = {107},
year = {2016},
}
@article{1216,
abstract = {A framework fo r extracting features in 2D transient flows, based on the acceleration field to ensure Galilean invariance is proposed in this paper. The minima of the acceleration magnitude (a superset of acceleration zeros) are extracted and discriminated into vortices and saddle points, based on the spectral properties of the velocity Jacobian. The extraction of topological features is performed with purely combinatorial algorithms from discrete computational topology. The feature points are prioritized with persistence, as a physically meaningful importance measure. These feature points are tracked in time with a robust algorithm for tracking features. Thus, a space-time hierarchy of the minima is built and vortex merging events are detected. We apply the acceleration feature extraction strategy to three two-dimensional shear flows: (1) an incompressible periodic cylinder wake, (2) an incompressible planar mixing layer and (3) a weakly compressible planar jet. The vortex-like acceleration feature points are shown to be well aligned with acceleration zeros, maxima of the vorticity magnitude, minima of the pressure field and minima of λ2.},
author = {Kasten, Jens and Reininghaus, Jan and Hotz, Ingrid and Hege, Hans and Noack, Bernd and Daviller, Guillaume and Morzyński, Marek},
journal = {Archives of Mechanics},
number = {1},
pages = {55 -- 80},
publisher = {Polish Academy of Sciences Publishing House},
title = {{Acceleration feature points of unsteady shear flows}},
volume = {68},
year = {2016},
}
@article{1222,
abstract = {We consider packings of congruent circles on a square flat torus, i.e., periodic (w.r.t. a square lattice) planar circle packings, with the maximal circle radius. This problem is interesting due to a practical reason—the problem of “super resolution of images.” We have found optimal arrangements for N=6, 7 and 8 circles. Surprisingly, for the case N=7 there are three different optimal arrangements. Our proof is based on a computer enumeration of toroidal irreducible contact graphs.},
author = {Musin, Oleg and Nikitenko, Anton},
journal = {Discrete & Computational Geometry},
number = {1},
pages = {1 -- 20},
publisher = {Springer},
title = {{Optimal packings of congruent circles on a square flat torus}},
doi = {10.1007/s00454-015-9742-6},
volume = {55},
year = {2016},
}
@inproceedings{1237,
abstract = {Bitmap images of arbitrary dimension may be formally perceived as unions of m-dimensional boxes aligned with respect to a rectangular grid in ℝm. Cohomology and homology groups are well known topological invariants of such sets. Cohomological operations, such as the cup product, provide higher-order algebraic topological invariants, especially important for digital images of dimension higher than 3. If such an operation is determined at the level of simplicial chains [see e.g. González-Díaz, Real, Homology, Homotopy Appl, 2003, 83-93], then it is effectively computable. However, decomposing a cubical complex into a simplicial one deleteriously affects the efficiency of such an approach. In order to avoid this overhead, a direct cubical approach was applied in [Pilarczyk, Real, Adv. Comput. Math., 2015, 253-275] for the cup product in cohomology, and implemented in the ChainCon software package [http://www.pawelpilarczyk.com/chaincon/]. We establish a formula for the Steenrod square operations [see Steenrod, Annals of Mathematics. Second Series, 1947, 290-320] directly at the level of cubical chains, and we prove the correctness of this formula. An implementation of this formula is programmed in C++ within the ChainCon software framework. We provide a few examples and discuss the effectiveness of this approach. One specific application follows from the fact that Steenrod squares yield tests for the topological extension problem: Can a given map A → Sd to a sphere Sd be extended to a given super-complex X of A? In particular, the ROB-SAT problem, which is to decide for a given function f: X → ℝm and a value r > 0 whether every g: X → ℝm with ∥g - f ∥∞ ≤ r has a root, reduces to the extension problem.},
author = {Krcál, Marek and Pilarczyk, Pawel},
location = {Marseille, France},
pages = {140 -- 151},
publisher = {Springer},
title = {{Computation of cubical Steenrod squares}},
doi = {10.1007/978-3-319-39441-1_13},
volume = {9667},
year = {2016},
}
@article{1252,
abstract = {We study the homomorphism induced in homology by a closed correspondence between topological spaces, using projections from the graph of the correspondence to its domain and codomain. We provide assumptions under which the homomorphism induced by an outer approximation of a continuous map coincides with the homomorphism induced in homology by the map. In contrast to more classical results we do not require that the projection to the domain have acyclic preimages. Moreover, we show that it is possible to retrieve correct homological information from a correspondence even if some data is missing or perturbed. Finally, we describe an application to combinatorial maps that are either outer approximations of continuous maps or reconstructions of such maps from a finite set of data points.},
author = {Harker, Shaun and Kokubu, Hiroshi and Mischaikow, Konstantin and Pilarczyk, Pawel},
journal = {Proceedings of the American Mathematical Society},
number = {4},
pages = {1787 -- 1801},
publisher = {American Mathematical Society},
title = {{Inducing a map on homology from a correspondence}},
doi = {10.1090/proc/12812},
volume = {144},
year = {2016},
}
@article{1254,
abstract = {We use rigorous numerical techniques to compute a lower bound for the exponent of expansivity outside a neighborhood of the critical point for thousands of intervals of parameter values in the quadratic family. We first compute a radius of the critical neighborhood outside which the map is uniformly expanding. This radius is taken as small as possible, yet large enough for our numerical procedure to succeed in proving that the expansivity exponent outside this neighborhood is positive. Then, for each of the intervals, we compute a lower bound for this expansivity exponent, valid for all the parameters in that interval. We illustrate and study the distribution of the radii and the expansivity exponents. The results of our computations are mathematically rigorous. The source code of the software and the results of the computations are made publicly available at http://www.pawelpilarczyk.com/quadratic/.},
author = {Golmakani, Ali and Luzzatto, Stefano and Pilarczyk, Pawel},
journal = {Experimental Mathematics},
number = {2},
pages = {116 -- 124},
publisher = {Taylor and Francis},
title = {{Uniform expansivity outside a critical neighborhood in the quadratic family}},
doi = {10.1080/10586458.2015.1048011},
volume = {25},
year = {2016},
}
@article{1710,
abstract = {We consider the hollow on the half-plane {(x, y) : y ≤ 0} ⊂ ℝ2 defined by a function u : (-1, 1) → ℝ, u(x) < 0, and a vertical flow of point particles incident on the hollow. It is assumed that u satisfies the so-called single impact condition (SIC): each incident particle is elastically reflected by graph(u) and goes away without hitting the graph of u anymore. We solve the problem: find the function u minimizing the force of resistance created by the flow. We show that the graph of the minimizer is formed by two arcs of parabolas symmetric to each other with respect to the y-axis. Assuming that the resistance of u ≡ 0 equals 1, we show that the minimal resistance equals π/2 - 2arctan(1/2) ≈ 0.6435. This result completes the previously obtained result [SIAM J. Math. Anal., 46 (2014), pp. 2730-2742] stating in particular that the minimal resistance of a hollow in higher dimensions equals 0.5. We additionally consider a similar problem of minimal resistance, where the hollow in the half-space {(x1,...,xd,y) : y ≤ 0} ⊂ ℝd+1 is defined by a radial function U satisfying the SIC, U(x) = u(|x|), with x = (x1,...,xd), u(ξ) < 0 for 0 ≤ ξ < 1, and u(ξ) = 0 for ξ ≥ 1, and the flow is parallel to the y-axis. The minimal resistance is greater than 0.5 (and coincides with 0.6435 when d = 1) and converges to 0.5 as d → ∞.},
author = {Akopyan, Arseniy and Plakhov, Alexander},
journal = {Society for Industrial and Applied Mathematics},
number = {4},
pages = {2754 -- 2769},
publisher = {SIAM},
title = {{Minimal resistance of curves under the single impact assumption}},
doi = {10.1137/140993843},
volume = {47},
year = {2015},
}
@article{1792,
abstract = {Motivated by recent ideas of Harman (Unif. Distrib. Theory, 2010) we develop a new concept of variation of multivariate functions on a compact Hausdorff space with respect to a collection D of subsets. We prove a general version of the Koksma-Hlawka theorem that holds for this notion of variation and discrepancy with respect to D. As special cases, we obtain Koksma-Hlawka inequalities for classical notions, such as extreme or isotropic discrepancy. For extreme discrepancy, our result coincides with the usual Koksma-Hlawka theorem. We show that the space of functions of bounded D-variation contains important discontinuous functions and is closed under natural algebraic operations. Finally, we illustrate the results on concrete integration problems from integral geometry and stereology.},
author = {Pausinger, Florian and Svane, Anne},
journal = {Journal of Complexity},
number = {6},
pages = {773 -- 797},
publisher = {Academic Press},
title = {{A Koksma-Hlawka inequality for general discrepancy systems}},
doi = {10.1016/j.jco.2015.06.002},
volume = {31},
year = {2015},
}
@article{1793,
abstract = {We present a software platform for reconstructing and analyzing the growth of a plant root system from a time-series of 3D voxelized shapes. It aligns the shapes with each other, constructs a geometric graph representation together with the function that records the time of growth, and organizes the branches into a hierarchy that reflects the order of creation. The software includes the automatic computation of structural and dynamic traits for each root in the system enabling the quantification of growth on fine-scale. These are important advances in plant phenotyping with applications to the study of genetic and environmental influences on growth.},
author = {Symonova, Olga and Topp, Christopher and Edelsbrunner, Herbert},
journal = {PLoS One},
number = {6},
publisher = {Public Library of Science},
title = {{DynamicRoots: A software platform for the reconstruction and analysis of growing plant roots}},
doi = {10.1371/journal.pone.0127657},
volume = {10},
year = {2015},
}
@article{1805,
abstract = {We consider the problem of deciding whether the persistent homology group of a simplicial pair (K,L) can be realized as the homology H∗(X) of some complex X with L ⊂ X ⊂ K. We show that this problem is NP-complete even if K is embedded in double-struck R3. As a consequence, we show that it is NP-hard to simplify level and sublevel sets of scalar functions on double-struck S3 within a given tolerance constraint. This problem has relevance to the visualization of medical images by isosurfaces. We also show an implication to the theory of well groups of scalar functions: not every well group can be realized by some level set, and deciding whether a well group can be realized is NP-hard.},
author = {Attali, Dominique and Bauer, Ulrich and Devillers, Olivier and Glisse, Marc and Lieutier, André},
journal = {Computational Geometry: Theory and Applications},
number = {8},
pages = {606 -- 621},
publisher = {Elsevier},
title = {{Homological reconstruction and simplification in R3}},
doi = {10.1016/j.comgeo.2014.08.010},
volume = {48},
year = {2015},
}
@article{1828,
abstract = {We construct a non-linear Markov process connected with a biological model of a bacterial genome recombination. The description of invariant measures of this process gives us the solution of one problem in elementary probability theory.},
author = {Akopyan, Arseniy and Pirogov, Sergey and Rybko, Aleksandr},
journal = {Journal of Statistical Physics},
number = {1},
pages = {163 -- 167},
publisher = {Springer},
title = {{Invariant measures of genetic recombination process}},
doi = {10.1007/s10955-015-1238-5},
volume = {160},
year = {2015},
}
@article{1938,
abstract = {We numerically investigate the distribution of extrema of 'chaotic' Laplacian eigenfunctions on two-dimensional manifolds. Our contribution is two-fold: (a) we count extrema on grid graphs with a small number of randomly added edges and show the behavior to coincide with the 1957 prediction of Longuet-Higgins for the continuous case and (b) we compute the regularity of their spatial distribution using discrepancy, which is a classical measure from the theory of Monte Carlo integration. The first part suggests that grid graphs with randomly added edges should behave like two-dimensional surfaces with ergodic geodesic flow; in the second part we show that the extrema are more regularly distributed in space than the grid Z2.},
author = {Pausinger, Florian and Steinerberger, Stefan},
journal = {Physics Letters, Section A},
number = {6},
pages = {535 -- 541},
publisher = {Elsevier},
title = {{On the distribution of local extrema in quantum chaos}},
doi = {10.1016/j.physleta.2014.12.010},
volume = {379},
year = {2015},
}
@article{2035,
abstract = {Considering a continuous self-map and the induced endomorphism on homology, we study the eigenvalues and eigenspaces of the latter. Taking a filtration of representations, we define the persistence of the eigenspaces, effectively introducing a hierarchical organization of the map. The algorithm that computes this information for a finite sample is proved to be stable, and to give the correct answer for a sufficiently dense sample. Results computed with an implementation of the algorithm provide evidence of its practical utility.
},
author = {Edelsbrunner, Herbert and Jablonski, Grzegorz and Mrozek, Marian},
journal = {Foundations of Computational Mathematics},
number = {5},
pages = {1213 -- 1244},
publisher = {Springer},
title = {{The persistent homology of a self-map}},
doi = {10.1007/s10208-014-9223-y},
volume = {15},
year = {2015},
}
@phdthesis{1399,
abstract = {This thesis is concerned with the computation and approximation of intrinsic volumes. Given a smooth body M and a certain digital approximation of it, we develop algorithms to approximate various intrinsic volumes of M using only measurements taken from its digital approximations. The crucial idea behind our novel algorithms is to link the recent theory of persistent homology to the theory of intrinsic volumes via the Crofton formula from integral geometry and, in particular, via Euler characteristic computations. Our main contributions are a multigrid convergent digital algorithm to compute the first intrinsic volume of a solid body in R^n as well as an appropriate integration pipeline to approximate integral-geometric integrals defined over the Grassmannian manifold.},
author = {Pausinger, Florian},
pages = {144},
publisher = {IST Austria},
title = {{On the approximation of intrinsic volumes}},
year = {2015},
}
@inproceedings{1424,
abstract = {We consider the problem of statistical computations with persistence diagrams, a summary representation of topological features in data. These diagrams encode persistent homology, a widely used invariant in topological data analysis. While several avenues towards a statistical treatment of the diagrams have been explored recently, we follow an alternative route that is motivated by the success of methods based on the embedding of probability measures into reproducing kernel Hilbert spaces. In fact, a positive definite kernel on persistence diagrams has recently been proposed, connecting persistent homology to popular kernel-based learning techniques such as support vector machines. However, important properties of that kernel enabling a principled use in the context of probability measure embeddings remain to be explored. Our contribution is to close this gap by proving universality of a variant of the original kernel, and to demonstrate its effective use in twosample hypothesis testing on synthetic as well as real-world data.},
author = {Kwitt, Roland and Huber, Stefan and Niethammer, Marc and Lin, Weili and Bauer, Ulrich},
location = {Montreal, Canada},
pages = {3070 -- 3078},
publisher = {Neural Information Processing Systems},
title = {{Statistical topological data analysis-A kernel perspective}},
volume = {28},
year = {2015},
}
@inproceedings{1483,
abstract = {Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this work, we establish such a connection by designing a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data. We show that this kernel is positive definite and prove its stability with respect to the 1-Wasserstein distance. Experiments on two benchmark datasets for 3D shape classification/retrieval and texture recognition show considerable performance gains of the proposed method compared to an alternative approach that is based on the recently introduced persistence landscapes.},
author = {Reininghaus, Jan and Huber, Stefan and Bauer, Ulrich and Kwitt, Roland},
location = {Boston, MA, USA},
pages = {4741 -- 4748},
publisher = {IEEE},
title = {{A stable multi-scale kernel for topological machine learning}},
doi = {10.1109/CVPR.2015.7299106},
year = {2015},
}
@inproceedings{1495,
abstract = {Motivated by biological questions, we study configurations of equal-sized disks in the Euclidean plane that neither pack nor cover. Measuring the quality by the probability that a random point lies in exactly one disk, we show that the regular hexagonal grid gives the maximum among lattice configurations. },
author = {Edelsbrunner, Herbert and Iglesias Ham, Mabel and Kurlin, Vitaliy},
booktitle = {Proceedings of the 27th Canadian Conference on Computational Geometry},
location = {Ontario, Canada},
pages = {128--135},
publisher = {Queen's University},
title = {{Relaxed disk packing}},
volume = {2015-August},
year = {2015},
}
@inproceedings{1510,
abstract = {The concept of well group in a special but important case captures homological properties of the zero set of a continuous map f from K to R^n on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within L_infty distance r from f for a given r > 0. The main drawback of the approach is that the computability of well groups was shown only when dim K = n or n = 1. Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of R^n by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and dim K < 2n-2, our approximation of the (dim K-n)th well group is exact. For the second part, we find examples of maps f, f' from K to R^n with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status. },
author = {Franek, Peter and Krcál, Marek},
location = {Eindhoven, Netherlands},
pages = {842 -- 856},
publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
title = {{On computability and triviality of well groups}},
doi = {10.4230/LIPIcs.SOCG.2015.842},
volume = {34},
year = {2015},
}