On the orthogonality of the Chebyshev-Frolov lattice and applications

Kacwin, Christopher; Oettershagen, Jens; Ullrich, Tino

doi:10.48550/arxiv.1606.00492

Cited by 2 publications

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“…Following [6], we use scaled (and coordinate-permuted) Chebyshev-Frolov lattices as admissible lattices for Frolov's cubature formula. Let n ∈ N and let A n be defined as in (4).…”

Section: Sequential Enumerationmentioning

confidence: 99%

“…However the enumeration is a difficult task even in moderate dimensions. Recently, an efficient enumeration algorithm for the so-called Chebyshev-Frolov lattices up to d = 16 was suggested by Kacwin, Oettershagen and Ullrich [6]. Since the lattices are admissible when d = 2 n , it is possible to implement Frolov's cubature formula for d = 2 n , up to d = 16.…”

Section: Introductionmentioning

confidence: 99%

“…Since the lattices are admissible when d = 2 n , it is possible to implement Frolov's cubature formula for d = 2 n , up to d = 16. Numerical experiments based on the algorithm are given in [5] and will be given in the forthcoming paper by the authors of [6] and Mario Ullrich. (…”

Section: Introductionmentioning

confidence: 99%

“…An advantage of the Chebyshev-Frolov lattices is that the generating matrices are explicitly given. We now briefly recall results in [6]. The paper established an enumeration algorithm of the lattice points in [−a/2, a/2] d , for any orthogonal lattices.…”

Section: Introductionmentioning

confidence: 99%

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Enumeration of the Chebyshev-Frolov lattice points in axis-parallel boxes

Suzuki¹,

Yoshiki²

2019

Hiroshima Math. J.

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For a positive integer d, the d-dimensional Chebyshev-Frolov lattice is the Z-lattice in R d generated by the Vandermonde matrix associated to the roots of the d-dimensional Chebyshev polynomial. It is important to enumerate the points from the Chebyshev-Frolov lattices in axis-parallel boxes when d = 2 n for a non-negative integer n, since the points are used for the nodes of Frolov's cubature formula, which achieves the optimal rate of convergence for many spaces of functions with bounded mixed derivatives and compact support. The existing enumeration algorithm for such points by Kacwin, Oettershagen and Ullrich is efficient up to dimension d = 16. In this paper we suggest a new enumeration algorithm of such points for d = 2 n , efficient up to d = 32.

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“…Following [6], we use scaled (and coordinate-permuted) Chebyshev-Frolov lattices as admissible lattices for Frolov's cubature formula. Let n ∈ N and let A n be defined as in (4).…”

Section: Sequential Enumerationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Enumeration of the Chebyshev-Frolov lattice points in axis-parallel boxes

Suzuki¹,

Yoshiki²

2019

Hiroshima Math. J.

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show abstract

“…We begin with a definition. The following results, see for instance [357,IV.4] and [189], play a fundamental role in the construction of these formulas.…”

Section: The Frolov Cubature Formulasmentioning

confidence: 99%

Hyperbolic Cross Approximation

Dũng¹,

Temlyakov²,

Ullrich³

2016

Preprint

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Hyperbolic cross approximation is a special type of multivariate approximation. Recently, driven by applications in engineering, biology, medicine and other areas of science new challenging problems have appeared. The common feature of these problems is high dimensions. We present here a survey on classical methods developed in multivariate approximation theory, which are known to work very well for moderate dimensions and which have potential for applications in really high dimensions. The theory of hyperbolic cross approximation and related theory of functions with mixed smoothness are under detailed study for more than 50 years. It is now well understood that this theory is important both for theoretical study and for practical applications. It is also understood that both theoretical analysis and construction of practical algorithms are very difficult problems. This explains why many fundamental problems in this area are still unsolved. Only a few survey papers and monographs on the topic are published. This and recently discovered deep connections between the hyperbolic cross approximation (and related sparse grids) and other areas of mathematics such as probability, discrepancy, and numerical integration motivated us to write this survey. We try to put emphases on the development of ideas and methods rather than list all the known results in the area. We formulate many problems, which, to our knowledge, are open problems. We also include some very recent results on the topic, which sometimes highlight new interesting directions of research. We hope that this survey will stimulate further active research in this fascinating and challenging area of approximation theory and numerical analysis.

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On the orthogonality of the Chebyshev-Frolov lattice and applications

Cited by 2 publications

References 0 publications

Enumeration of the Chebyshev-Frolov lattice points in axis-parallel boxes

Enumeration of the Chebyshev-Frolov lattice points in axis-parallel boxes

Hyperbolic Cross Approximation

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