The L 2 Discrepancy of Two-Dimensional Lattices

Bilyk, Dmitriy; Temlyakov, Vladimir; Yu, Rui

doi:10.1007/978-1-4614-4565-4_9

Cited by 4 publications

(10 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Furthermore, it is well known that log F n n. This shows that all the L 2 discrepancies under study of the Fibonacci lattice are of optimal order of magnitude with respect to the corresponding Roth-type lower bounds. In fact it follows from [4,Lemma 7] that in the case of extreme and periodic L 2 discrepancy the same is true for all irrational α = [a 0 ; a 1 , a 2 , . .…”

Section: Fnmentioning

confidence: 82%

“…A few years later the constant involved on the right hand side of (8) was identified to have the explicit expression 4 15 √ 5 (see [5]). Furthermore, it is well known that log F n n. This shows that all the L 2 discrepancies under study of the Fibonacci lattice are of optimal order of magnitude with respect to the corresponding Roth-type lower bounds.…”

Section: Fnmentioning

confidence: 99%

“…The result follows by multiplying by 2 and taking the square root. For symmetric P we have equality in (4), because then N −1 n=0 (1/2 − x n ) equals 0. We now show that extreme L 2 discrepancy is indeed always smaller than standard L 2 discrepancy as conjectured in [23].…”

mentioning

confidence: 99%

“…The first formula for the L 2 discrepancy is [4,Theorem 6]. The proofs of the formulas for the extreme and periodic L 2 discrepancy will be given in Section 7.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Extreme and periodic $L_2$ discrepancy of plane point sets

Hinrichs

Kritzinger²,

Pillichshammer³

2021

Acta Arith.

View full text Add to dashboard Cite

We study several discrepancy notions of two wellknown instances of plane point sets, namely the Hammersley point set and rational lattices. The discrepancies are considered with respect to the L 2 norm and a variety of test sets. We define (standard) L 2 discrepancy, extreme L 2 discrepancy and periodic L 2 discrepancy. Let P = {x 0 , x 1 ,. .. , x N −1 } be an arbitrary N-element point set in the unit square [0, 1) 2. For any measurable subset B of [0, 1] 2 we define the counting function A(B, P) := n ∈ {0, 1,. .. , N − 1} : x n ∈ B , i.e., the number of elements from P that belong to the set B. By the local discrepancy of P with respect to a given measurable "test set" B one understands the expression A(B, P) − N λ(B), where λ denotes the Lebesgue measure of B. A global discrepancy measure is then obtained by considering a norm of the local discrepancy with respect to a fixed class of test sets. Here we restrict ourselves to the L 2 norm, but we vary the class of test sets. The (standard) L 2 discrepancy uses the class of axis-parallel squares anchored at the origin as test sets. The formal definition is L 2,N (P) :=

show abstract

Section: Fnmentioning

confidence: 82%

Section: Fnmentioning

confidence: 99%

mentioning

confidence: 99%

“…The first formula for the L 2 discrepancy is [4,Theorem 6]. The proofs of the formulas for the extreme and periodic L 2 discrepancy will be given in Section 7.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Extreme and periodic $L_2$ discrepancy of plane point sets

Hinrichs

Kritzinger²,

Pillichshammer³

2021

Acta Arith.

View full text Add to dashboard Cite

show abstract

“…Either solution, however, may be computationally expensive. 6. Appendix: separation properties of sequences {L n } and {M n } This Appendix deals with the results of our numerical experiments, set in the 3-dimensional space.…”

Section: 2mentioning

confidence: 99%

Fast high-dimensional node generation with variable density

Vlasiuk

Michaels

Flyer

et al. 2018

Computers & Mathematics with Applications

View full text Add to dashboard Cite

We present an algorithm for producing discrete distributions with a prescribed nearestneighbor distance function. Our approach is a combination of quasi-Monte Carlo (Q-MC) methods and weighted Riesz energy minimization: the initial distribution is a stratified Q-MC sequence with some modifications; a suitable energy functional on the configuration space is then minimized to ensure local regularity. The resulting node sets are good candidates for building meshless solvers and interpolants, as well as for other purposes where a point cloud with a controlled separation-covering ratio is required. Applications of a three-dimensional implementation of the algorithm, in particular to atmospheric modeling, are also given.where φ(·) is a radial function, and x k , k = 1, . . . , K, is a collection of pairwise distinct points in R d . A common choice of φ is the Gaussian φ(r) = e −( r) 2 , although one may also use 1/(1 + ( r) 2 ), r 2p log(r), p ∈ N, etc. In this discussion, we are not concerned with the distinctions between the

show abstract

Optimal and typical $$L^2$$ discrepancy of 2-dimensional lattices

Borda

2024

Annali di Matematica

View full text Add to dashboard Cite

We undertake a detailed study of the $$L^2$$ L 2 discrepancy of 2-dimensional Korobov lattices and their irrational analogues, either with or without symmetrization. We give a full characterization of such lattices with optimal $$L^2$$ L 2 discrepancy in terms of the continued fraction partial quotients, and compute the precise asymptotics whenever the continued fraction expansion is explicitly known, such as for quadratic irrationals or Euler’s number e. In the metric theory, we find the asymptotics of the $$L^2$$ L 2 discrepancy for almost every irrational, and the limit distribution for randomly chosen rational and irrational lattices.

show abstract

The L 2 Discrepancy of Two-Dimensional Lattices

Cited by 4 publications

References 21 publications

Extreme and periodic $L_2$ discrepancy of plane point sets

Extreme and periodic $L_2$ discrepancy of plane point sets

Fast high-dimensional node generation with variable density

Optimal and typical $$L^2$$ discrepancy of 2-dimensional lattices

Contact Info

Product

Resources

About