Approximation and convergence of the intrinsic volume

Edelsbrunner, Herbert; Pausinger, Florian

doi:10.1016/j.aim.2015.10.004

Cited by 6 publications

(3 citation statements)

References 25 publications

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“…The concept of discrepancy is a fundamental building block in the quantification of many point distributions problems. There is a list of interesting discrepancy measures, such as star discrepancy, extreme discrepancy, G−discrepancy, isotrope discrepancy, lattice discrepancy, and so on (see e.g., [21,22]). Among them, L 2 −discrepancy is the most widely studied.…”

Section: Introductionmentioning

confidence: 99%

Expected $L_2-$discrepancy bound for a class of new stratified sampling models

Xian¹,

Xu²

2022

Preprint

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We introduce a class of convex equivolume partitions. Expected L 2 −discrepancy are discussed under these partitions. There are two main results. First, under this kind of partitions, we generate random point sets with smaller expected L 2 −discrepancy than classical jittered sampling for the same sampling number. Second, an explicit expected L 2 −discrepancy upper bound under this kind of partitions is also given. Further, among these new partitions, there is optimal expected L 2 −discrepancy upper bound.

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Section: Introductionmentioning

confidence: 99%

Expected $L_2-$discrepancy bound for a class of new stratified sampling models

Xian¹,

Xu²

2022

Preprint

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“…For various applications the D−variation seems to be a more natural and suitable concept. A convincing example concerning an application to computational geometry is due to Pausinger & Edelsbrunner [11]. Pauisnger & Svane [25] considered the variation V K (f ) with respect to the class of convex sets.…”

Section: Introductionmentioning

confidence: 99%

Integral Equations, Quasi-Monte Carlo Methods and Risk Modeling

Preischl

Thonhauser

Tichy

2018

Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan

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We survey a QMC approach to integral equations and develop some new applications to risk modeling. In particular, a rigorous error bound derived from Koksma-Hlawka type inequalities is achieved for certain expectations related to the probability of ruin in Markovian models. The method is based on a new concept of isotropic discrepancy and its applications to numerical integration. The theoretical results are complemented by numerical examples and computations.

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“…(2) For a chosen field compute persistent homology, which roughly represents the size of holes in the space. While these two steps are up to some point theoretically justified in the general case by the Rips/ Čech interleaving and the Nerve theorem, there is nonetheless a lack of a precise formulation in any setting which would explain how the result depends on a choice of a complex or how the sizes of holes are measured precisely (with a notable exception [11]). Consequently there is a push in the research community to develop statistical framework for the interpretation of the obtained persistence diagrams (PDs), which would allow applications of the methods of artificial intelligence.…”

Section: Introductionmentioning

confidence: 99%

1-Dimensional Intrinsic Persistence of Geodesic Spaces

Virk

2017

Preprint

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Given a compact geodesic space X, we apply the fundamental group and alternatively the first homology group functor to the corresponding Rips or Čech filtration of X to obtain what we call a persistence. This paper contains the theory describing such persistence: properties of the set of critical points, their precise relationship to the size of holes, the structure of persistence and the relationship between open and close, Rips and Čech induced persistences. Amongst other results we prove that a Rips critical point c corresponds to an isometrically embedded circle of length 3c, that a homology persistence of a locally contractible space with coefficients in a field encodes the lengths of the lexicographically smallest base, and that Rips and Čech induced persistences are isomorphic up to a factor 3/4. The theory describes geometric properties of the underlying space encoded and extractable from persistence. ŽIGA VIRK 11.1. Persistence of compact geodesic spaces for filtrations by closed complexes 31 11.2. Application: metric graphs 33 11.3. Generalizations to length spaces and other relaxations 33 12. Discussion and future work 33 References 34

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Approximation and convergence of the intrinsic volume

Cited by 6 publications

References 25 publications

Expected $L_2-$discrepancy bound for a class of new stratified sampling models

Expected $L_2-$discrepancy bound for a class of new stratified sampling models

Integral Equations, Quasi-Monte Carlo Methods and Risk Modeling

1-Dimensional Intrinsic Persistence of Geodesic Spaces

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