Well-Separated Spherical Designs

Bondarenko, A. A.; Radchenko, Danylo; Viazovska, Maryna

doi:10.1007/s00365-014-9238-2

Cited by 55 publications

(60 citation statements)

References 20 publications

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“…Bondarenko, Radchenko and Viazovska [9] have shown that on S d well-separated spherical t-designs exist for N ≥ c d t d . This combined with Yudin's result on the covering radius of spherical designs mean that there exist spherical t-designs with N = O(t d ) points and uniformly bounded mesh ratio.…”

Section: Geometric Qualitymentioning

confidence: 99%

“…2. These tables also list both the Delsarte, Goethals and Seidel lower bounds N * (2,t) and the Yudin lower bound N + (2,t), plus the actual number of points N. The number of points N = N(2,t), apart from t = 3, 5, 7,9,11,13,15 when N = N(2,t) − 1. There may well be spherical t-designs with smaller values of N and special symmetries, see [35] for example.…”

Section: Structure Of Point Setsmentioning

confidence: 99%

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Efficient Spherical Designs with Good Geometric Properties

Womersley

2018

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

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Dedicated to Ian H. Sloan on the occasion of his 80th birthday in acknowledgement of his many fruitful ideas and generosity.Abstract Spherical t-designs on S d ⊂ R d+1 provide N nodes for an equal weight numerical integration rule which is exact for all spherical polynomials of degree at most t. This paper considers the generation of efficient, where N is comparable to (1 + t) d /d, spherical t-designs with good geometric properties as measured by their mesh ratio, the ratio of the covering radius to the packing radius. Results for S 2 include computed spherical t-designs for t = 1, ..., 180 and symmetric (antipodal) t-designs for degrees up to 325, all with low mesh ratios. These point sets provide excellent points for numerical integration on the sphere. The methods can also be used to computationally explore spherical t-designs for d = 3 and higher.

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Section: Geometric Qualitymentioning

confidence: 99%

Section: Structure Of Point Setsmentioning

confidence: 99%

Efficient Spherical Designs with Good Geometric Properties

Womersley

2018

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

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“…The construction is not limited to S 2 (because [11] is done in full generality on S d ). The same argument yields, for all d ≥ 2, all n ≥ 2 and for all 0 < t ≤ c d n −2/d a function f t ∈ C ∞ (S d ) such that f is orthogonal to the first n eigenfunctions of −∆ S d−1 and…”

Section: Proof Of the Propositionmentioning

confidence: 99%

A metric Sturm–Liouville theory in two dimensions

Steinerberger

2019

Calc. Var.

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A central result of Sturm-Liouville theory (also called the Sturm-Hurwitz Theorem) states that if φ k is a sequence of eigenfunctions of a second order differential operator on the interval I ⊂ R, then any linear combination satisfies a uniform bound on the rootsWe provide a sharp (up to logarithmic factors) generalization to two dimensions: let (M, g) be a compact two-dimensional manifold (with or without boundary), let (φ k ) denote the sequence of eigenfunctions of a uniformly elliptic operator −div(a(·)∇) (with Dirichlet or Neumann boundary conditions). Then, for any linear combination of eigenfunctions above a certain index n,2010 Mathematics Subject Classification. 28A75, 34B24, 35B05, 35P20, 49Q20.

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“…For many cardinalities close (but not equal) to D(n, τ ) existence of spherical τ -designs is still an open problem. On the other hand, existence of designs with asymptotically optimal cardinalities was proved by Bondarenko, Radchenko, and Viazovska [6,7].…”

Section: Delsarte-goethals-seidel Bound and Polynomialsmentioning

confidence: 99%

Upper Energy Bounds for Spherical Designs of Relatively Small Cardinalities

Boyvalenkov

Delchev

Jourdain

2019

Discrete Comput Geom

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We derive upper bounds for the potential energy of spherical designs of cardinality close to the Delsarte-Goethals-Seidel bound. These bounds are obtained by linear programming with the use of the Hermite interpolating polynomial of the potential function in suitable nodes. Numerical computations show that the results are quite close to certain lower energy bounds confirming that spherical designs are, in a sense, energy efficient.

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Well-Separated Spherical Designs

Abstract: For each N ≥ C d t d we prove the existence of a well-separated spherical t-design in the sphere S d consisting of N points, where C d is a constant depending only on d.

Cited by 55 publications

References 20 publications

Efficient Spherical Designs with Good Geometric Properties

Efficient Spherical Designs with Good Geometric Properties

A metric Sturm–Liouville theory in two dimensions

Upper Energy Bounds for Spherical Designs of Relatively Small Cardinalities

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