Numerical Algorithms

, Volume 40, Issue 4, pp 401–414 | Cite as

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

  • Piotr Gajda


We are interested in numerical algorithms for weighted L1 approximation of functions defined on \(\mathcal{D}=\mathbb{R}^{d}\) . We consider the space ℱr,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∈ℱr,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→∞.


Function Evaluation Weight Function Numerical Algorithm Polynomial Interpolation Interpolation Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Department of Mathematics, Informatics, and MechanicsUniversity of WarsawWarsawPoland

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