2016/03/15 by David Krieg, Krieg, David
Mathematics · #65C05 #65D30 #65Y20 #68Q25 #Advanced Harmonic Analysis Research #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical functions and polynomials #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1603.04637
openalex publication_date 2016/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We are concerned with the numerical integration of functions from the Sobolev space Hr,mix([0,1]d) of dominating mixed smoothness r∈ℕ over the d-dimensional unit cube. In 1976, K. K. Frolov introduced a deterministic quadrature rule whose worst case error has the order n-r (log n)(d-1)/2 with respect to the number n of function evaluations. This is known to be optimal. 39 years later, Erich Novak and me introduced a randomized version of this algorithm using d random dilations. We showed that its error is bounded above by a constant multiple of n-r-1/2 (log n)(d-1)/2 in expectation and by n-r (log n)(d-1)/2 almost surely. The main term n-r-1/2 is again optimal and it turns out that the very same algorithm is also optimal for the isotropic Sobolev space Hs([0,1]d) of smoothness s>d/2. We also added a random shift to this algorithm to make it unbiased. Just recently, Mario Ullrich proved that the expected error of the resulting algorithm on Hr,mix([0,1]d) is even bounded above by n-r-1/2. This thesis is a review of the mentioned upper bounds and their proofs.