An intractability result for multiple integration

Abstract

We prove that the problem of multiple integration in the Korobov class <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Subscript alpha comma d"> <mml:semantics> <mml:msub> <mml:mi>E</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> α </mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">E_{\alpha ,d}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is intractable since the number of function evaluations required to achieve a worst case error less than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is exponential in the dimension.