Distributing many points on spheres: Minimal energy and designs

Brauchart, Johann S.; Grabner, Peter J.

doi:10.1016/j.jco.2015.02.003

Cited by 97 publications

(72 citation statements)

References 196 publications

(315 reference statements)

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“…Here x − y denotes the euclidean norm in R 3 . Moreover, see [13], the minimizer µ V is related to µ V by the following relation 5) where T #µ denotes the push-forward of the measure µ by the map T .…”

Section: Letmentioning

confidence: 99%

Renormalized Energy and Asymptotic Expansion of Optimal Logarithmic Energy on the Sphere

Bétermin

Sandier

2016

Constr Approx

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We study the Hamiltonian of a two-dimensional log-gas with a confining potential V satisfying the weak growth assumption -V is of the same order than 2 log x near infinity -considered by Hardy and Kuijlaars [J. Approx. Theory, 170(0):44-58, 2013]. We prove an asymptotic expansion, as the number n of points goes to infinity, for the minimum of this Hamiltonian using the Gamma-Convergence method of Sandier and Serfaty [24]. We show that the asymptotic expansion as n → +∞ of the minimal logarithmic energy of n points on the unit sphere in R 3 has a term of order n thus proving a long standing conjecture of Rakhmanov, Saff and Zhou [Math. Res. Letters, 1:647-662, 1994]. Finally we prove the equivalence between the conjecture of Brauchart, Hardin and Saff [Contemp. Math., 578:31-61,2012] about the value of this term and the conjecture of Sandier and Serfaty [Comm. Math. Phys., 313(3):635-743, 2012] about the minimality of the triangular lattice for a "renormalized energy" W among configurations of fixed asymptotic density.

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Section: Letmentioning

confidence: 99%

Renormalized Energy and Asymptotic Expansion of Optimal Logarithmic Energy on the Sphere

Bétermin

Sandier

2016

Constr Approx

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“…This line of question has been extended to general kernels k(x i , x j ) and studied in a very large number of settings, we refer to a recent survey of Brauchart & Grabner [4]. One expects minimizers of these functionals to be spread very evenly over the manifold and for this reason minimizing configurations of various energy functionals are often used as sampling points (see [4]).…”

mentioning

confidence: 99%

Spectral Limitations of Quadrature Rules and Generalized Spherical Designs

Steinerberger

2019

International Mathematics Research Notices

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We study manifolds M equipped with a quadrature ruleWe show that n−point quadrature rules with nonnegative weights on a compact d−dimensional manifold cannot integrate more than at most the first c d n + o(n) Laplacian eigenfunctions exactly. The constants c d are explicitly computed and c 2 = 4. The result is new even on S 2 where it generalizes results on spherical designs.2010 Mathematics Subject Classification. 35J05, 35J20, 35Q82, 52A37, 65D32.

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“…Step 2. The generation of uniform point distributions on spheres is a research topic on its own, see [33] for an overview. The method described in [12] is based on energy minimization, which is also used in the present work.…”

Section: Algorithmmentioning

confidence: 99%

Finite Strain Homogenization Using a Reduced Basis and Efficient Sampling

Kunc

Fritzen

2019

MCA

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The computational homogenization of hyperelastic solids in the geometrically nonlinear context has yet to be treated with sufficient efficiency in order to allow for real-world applications in true multiscale settings. This problem is addressed by a problem-specific surrogate model founded on a reduced basis approximation of the deformation gradient on the microscale. The setup phase is based upon a snapshot POD on deformation gradient fluctuations, in contrast to the widespread displacement-based approach. In order to reduce the computational offline costs, the space of relevant macroscopic stretch tensors is sampled efficiently by employing the Hencky strain. Numerical results show speed-up factors in the order of 5-100 and significantly improved robustness while retaining good accuracy. An open-source demonstrator tool with 50 lines of code emphasizes the simplicity and efficiency of the method.

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Distributing many points on spheres: Minimal energy and designs

Abstract: This survey discusses recent developments in the context of spherical designs and minimal energy point configurations on spheres. The recent solution of the long standing problem of the existence of spherical t-designs on

Cited by 97 publications

References 196 publications

Renormalized Energy and Asymptotic Expansion of Optimal Logarithmic Energy on the Sphere

Renormalized Energy and Asymptotic Expansion of Optimal Logarithmic Energy on the Sphere

Spectral Limitations of Quadrature Rules and Generalized Spherical Designs

Finite Strain Homogenization Using a Reduced Basis and Efficient Sampling

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