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We examine sparse grid quadrature on weighted tensor products (WTP) of reproducing kernel Hilbert spaces on products of the unit sphere, in the case of worst case quadrature error for rules with arbitrary quadrature weights. We describe a dimension adaptive quadrature algorithm based on an algorithm of Hegland (2003), and also formulate a version of Wasilkowski and Wozniakowskis WTP algorithm (1999), here called the WW algorithm. We prove that the dimension adaptive algorithm is optimal in the sense of Dantzig (1957) and therefore no greater in cost than the WW algorithm. Both algorithms therefore have the optimal asymptotic rate of convergence given by Theorem 3 of Wasilkowski and Wozniakowski (1999). A numerical example shows that, even though the asymptotic convergence rate is optimal, if the dimension weights decay slowly enough, and the dimensionality of the problem is large enough, the initial convergence of the dimension adaptive algorithm can be slow.
This article is dedicated to the anisotropic sparse grid quadrature for functions which are analytically extendable into an anisotropic tensor product domain. Taking into account this anisotropy, we end up with a dimension independent error versus co
This work is a follow-up to our previous contribution (Convergence of sparse collocation for functions of countably many Gaussian random variables (with application to elliptic PDEs), SIAM J. Numer. Anal., 2018), and contains further insights on some
We recover jump-sparse and sparse signals from blurred incomplete data corrupted by (possibly non-Gaussian) noise using inverse Potts energy functionals. We obtain analytical results (existence of minimizers, complexity) on inverse Potts functionals
We propose to compute a sparse approximate inverse Cholesky factor $L$ of a dense covariance matrix $Theta$ by minimizing the Kullback-Leibler divergence between the Gaussian distributions $mathcal{N}(0, Theta)$ and $mathcal{N}(0, L^{-top} L^{-1})$,
The randomized sparse Kaczmarz method was recently proposed to recover sparse solutions of linear systems. In this work, we introduce a greedy variant of the randomized sparse Kaczmarz method by employing the sampling Kaczmarz-Motzkin method, and pro