Smolyak's algorithm for weighted L₁-approximation of multivariate functions with bounded rth mixed derivatives over ℝ^d

Abstract

We are interested in numerical algorithms for weighted L₁ approximation of functions defined on $\mathcal{D}=\mathbb{R}^{d}$ . We consider the space ℱ_r,d which consists of multivariate functions $f:\mathcal{D}\to \mathbb{R}$ whose all mixed derivatives of order r are bounded in L₁-norm. We approximate f∈ℱ_r,d by an algorithm which uses evaluations of the function. The error is measured in the weighted L₁-norm with a weight function ρ. We construct and analyze Smolyak's algorithm for solving this problem. The algorithm is based on one-dimensional piecewise polynomial interpolation of degree at most r−1, where the interpolation points are specially chosen dependently on the smoothness parameter r and the weight ρ. We show that, under some condition on the rate of decay of ρ, the error of the proposed algorithm asymptotically behaves as $\mathcal{O}((\ln n)^{(r+1)(d-1)}n^{-r})$ , where n denotes the number of function evaluations used. The asymptotic constant is known and it decreases to zero exponentially fast as d→∞.