TY - JOUR
AU - Gajda, Piotr
AB - We are interested in numerical algorithms for weighted L
1 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 L
1-norm. We approximate f∈ℱr,d by an algorithm which uses evaluations of the function. The error is measured in the weighted L
1-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→∞.
TI - Smolyak's algorithm for weighted L 1-approximation of multivariate functions with bounded rth mixed derivatives over ℝd
JO - Numerical Algorithms
DO - 10.1007/s11075-005-0411-3
DA - 2005-06-30
UR - https://www.deepdyve.com/lp/springer-journals/smolyak-s-algorithm-for-weighted-l-1-approximation-of-multivariate-iUxRgedZJq
SP - 401
EP - 414
VL - 40
IS - 4
DP - DeepDyve
ER -