Smolyak's Algorithm for Integration and L1-Approximation of Multivariate Functions with Bounded Mixed Derivatives of Second Order

Abstract

We propose and analyze two algorithms for multiple integration and L1-approximation of functions \(f:[0,1]^s \to \mathbb{R}\) that have bounded mixed derivatives of order 2. The algorithms are obtained by applying Smolyak's construction (see [8]) to one-dimensional composite midpoint rules (for integration) and to one-dimensional piecewise linear interpolation algorithm (for L1-approximation). Denoting by n the number of function evaluations used, the worst case error of the obtained Smolyak's cubature is asymptotically bounded from above by $$\frac{{16\pi ^2 s}}{{3(s - 1)((s - 2)!)^3 }} \cdot \frac{{(\log _2 n)^{3(s - 1)} }}{{n^2 }} \cdot (1 + o(1))$$ as n→∞. The error of the corresponding algorithm for L1-approximation is bounded by the same expression multiplied by 4s−1.