QMC Designs – Cubature on the Sphere Without Polynomial Exactness

Abstract QMC designs were introduced in a 2014 paper by Brauchart, Saff and the present authors ( Math. Comp. 83 :2821–2851). They represent a novel approach to cubature on the sphere $$\mathbb {S}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> , in which instead of requiring cubature rules to be exact for polynomials up to a certain degree, a sequence of cubature rules is a QMC design sequence if the worst-case error for functions in a Sobolev space $$\mathbb {H}^{s}(\mathbb {S}^{d})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is of order $$\mathcal {O}(N^{-s/d})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>N</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>s</mml:mi> <mml:mo>/</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where N is the number of cubature points. The original paper considered only equal cubature weights, but the present paper allows positive weights that sum to 1. After reviewing the known results, the present paper presents necessary and sufficient conditions for QMC design sequences, one of which requires only a finite sum for each tested value of N . The paper also gives experimental results that aim to give insight into the design of QMC designs, and extends the known QMC designs to a 3-dimensional example.