Conjecture on monotonicity of worst-case error for spherical t-designs

From papers

Let Q[XN]Q[\mathcal{X}_N] denote the quadrature rule associated with a spherical tt-design consisting of NN points, and let wce(Q[XN])\mathrm{wce}(Q[\mathcal{X}_N]) be its worst-case error.

Worst-case error monotonicity conjecture. For fixed tt, if N>NN'>N, then

wce(Q[XN])wce(Q[XN]).\mathrm{wce}(Q[\mathcal{X}_{N'}])\leq \mathrm{wce}(Q[\mathcal{X}_N]).

The conjecture is motivated by numerical experiments comparing well-conditioned and efficient spherical tt-designs. It predicts that increasing the number of points does not increase the worst-case error, but the source provides no proof or resolution.

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Sources & referencesView supporting material

Primary source

Congpei An and Siyong Chen, “Numerical Integration over the Unit Sphere by using spherical t-design”, arXiv:1611.02785 (2016).

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