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W2093529512 abstract "We study equal weight numerical integration, or Quasi Monte Carlo (QMC) rules, for functions in a Sobolev space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper H Superscript s Baseline left-parenthesis double-struck upper S Superscript d Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>H</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>S</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {H}^s( mathbb {S}^d)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with smoothness parameter <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s greater-than d slash 2> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>></mml:mo> <mml:mi>d</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>s > d/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> defined over the unit sphere <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper S Superscript d> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>S</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {S}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R Superscript d plus 1> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {R}^{d+1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Focusing on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding=application/x-tex>N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-point configurations that achieve optimal order QMC error bounds (as is the case for efficient spherical designs), we are led to introduce the concept of QMC designs: these are sequences of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding=application/x-tex>N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-point configurations <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X Subscript upper N> <mml:semantics> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>X_N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper S Superscript d> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>S</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {S}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that the worst-case error satisfies <disp-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sup Underscript StartLayout 1st Row f element-of double-struck upper H Superscript s Baseline left-parenthesis double-struck upper S Superscript d Baseline right-parenthesis comma 2nd Row double-vertical-bar f double-vertical-bar Subscript double-struck upper H Sub Superscript s Subscript Baseline less-than-or-equal-to 1 EndLayout Endscripts StartAbsoluteValue StartFraction 1 Over upper N EndFraction sigma-summation Underscript bold x element-of upper X Subscript upper N Baseline Endscripts f left-parenthesis bold x right-parenthesis minus integral Underscript double-struck upper S Superscript d Baseline Endscripts f left-parenthesis bold x right-parenthesis normal d sigma Subscript d Baseline left-parenthesis bold x right-parenthesis EndAbsoluteValue equals script upper O left-parenthesis upper N Superscript negative s slash d Baseline right-parenthesis comma upper N right-arrow normal infinity comma> <mml:semantics> <mml:mrow> <mml:munder> <mml:mo movablelimits=true form=prefix>sup</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mstyle scriptlevel=1> <mml:mtable rowspacing=0.1em columnspacing=0em displaystyle=false> <mml:mtr> <mml:mtd> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>H</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>S</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mo fence=false stretchy=false>‖</mml:mo> <mml:mi>f</mml:mi> <mml:msub> <mml:mo fence=false stretchy=false>‖</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>H</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msup> </mml:mrow> </mml:msub> <mml:mo>≤</mml:mo> <mml:mn>1</mml:mn> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mstyle> </mml:mrow> </mml:munder> <mml:mstyle scriptlevel=0> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo maxsize=2.470em minsize=2.470em>|</mml:mo> </mml:mrow> </mml:mstyle> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>N</mml:mi> </mml:mfrac> <mml:munder> <mml:mo>∑</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>x</mml:mi> </mml:mrow> <mml:mo>∈</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:mrow> </mml:munder> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>x</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> <mml:mo>−</mml:mo> <mml:msub> <mml:mo>∫</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>S</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> </mml:msub> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>x</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> <mml:mspace width=thinmathspace /> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>d</mml:mi> </mml:mrow> <mml:msub> <mml:mi>σ</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>x</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> <mml:mstyle scriptlevel=0> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo maxsize=2.470em minsize=2.470em>|</mml:mo> </mml:mrow> </mml:mstyle> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>O</mml:mi> </mml:mrow> <mml:mstyle scriptlevel=0> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo maxsize=1.2em minsize=1.2em>(</mml:mo> </mml:mrow> </mml:mstyle> <mml:msup> <mml:mi>N</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−</mml:mo> <mml:mi>s</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> <mml:mstyle scriptlevel=0> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo maxsize=1.2em minsize=1.2em>)</mml:mo> </mml:mrow> </mml:mstyle> <mml:mo>,</mml:mo> <mml:mspace width=2em /> <mml:mi>N</mml:mi> <mml:mo stretchy=false>→</mml:mo> <mml:mi mathvariant=normal>∞</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>begin{equation*} sup _{substack {f in mathbb {H}^s( mathbb {S}^d ), | f |_{mathbb {H}^s} leq 1}} Bigg | frac {1}{N} sum _{mathbf {x} in X_N} f( mathbf {x} ) - int _{mathbb {S}^d} f( mathbf {x} ) , mathrm {d} sigma _d( mathbf {x} ) Bigg | = mathcal {O}big ( N^{-s/d} big ), qquad N to infty , end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> with an implied constant that depends on the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper H Superscript s Baseline left-parenthesis double-struck upper S Superscript d Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>H</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>S</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {H}^s( mathbb {S}^d )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-norm, but is independent of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding=application/x-tex>N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Here <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma Subscript d> <mml:semantics> <mml:msub> <mml:mi>σ</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>sigma _d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the normalized surface measure on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper S Superscript d> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>S</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {S}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We provide methods for generation and numerical testing of QMC designs. An essential tool is an expression for the worst-case error in terms of a reproducing kernel for the space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper H Superscript s Baseline left-parenthesis double-struck upper S Superscript d Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>H</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>S</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {H}^s( mathbb {S}^d )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s greater-than d slash 2> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>></mml:mo> <mml:mi>d</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>s > d/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As a consequence of this and a recent result of Bondarenko et al. on the existence of spherical designs with appropriate number of points, we show that minimizers of the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding=application/x-tex>N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-point energy for this kernel form a sequence of QMC designs for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper H Superscript s Baseline left-parenthesis double-struck upper S Superscript d Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>H</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>S</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {H}^s( mathbb {S}^d )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Furthermore, without appealing to the Bondarenko et al. result, we prove that point sets that maximize the sum of suitable powers of the Euclidean distance between pairs of points form a sequence of QMC designs for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper H Superscript s Baseline left-parenthesis double-struck upper S Superscript d Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>H</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>S</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {H}^s( mathbb {S}^d )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding=application/x-tex>s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the interval <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis d slash 2 comma d slash 2 plus 1 right-parenthesis> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>(</mml:mo> <mml:mi>d</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>{(d/2,d/2+1)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For such spaces there exist reproducing kernels with simple closed forms that are useful for numerical testing of optimal order Quasi Monte Carlo integration. Numerical experiments suggest that many familiar sequences of point sets on the sphere (equal area points, spiral points, minimal [Coulomb or logarithmic] energy points, and Fekete points) are QMC designs for appropriate values of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding=application/x-tex>s</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For comparison purposes we show that configurations of random points that are independently and uniformly distributed on the sphere do not constitute QMC designs for any <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s greater-than d slash 2> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>></mml:mo> <mml:mi>d</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>s>d/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. If <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis upper X Subscript upper N Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(X_N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a sequence of QMC designs for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper H Superscript s Baseline left-parenthesis double-struck upper S Superscript d Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>H</mml:mi> </mml:mrow> <mml:mi>s</mml:mi> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>S</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {H}^s( mathbb {S}^d)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we prove that it is also a sequence of QMC designs for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper H Superscript s prime Baseline left-parenthesis double-struck upper S Superscript d Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>H</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>s</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>S</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {H}^{s’}( mathbb {S}^d)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s prime element-of left-parenthesis d slash 2 comma s right-parenthesis> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>s</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>∈</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>d</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>s’in (d/2,s)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This leads to the question of determining the supremum of such <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=s> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding=application/x-tex>s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (here called the QMC strength of the sequence), for which we provide estimates based on computations for the aforementioned sequences." @default.
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W2093529512 date "2014-04-21" @default.
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W2093529512 title "QMC designs: Optimal order Quasi Monte Carlo integration schemes on the sphere" @default.
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