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Register a new userSparse grid quadrature on products of spheres
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Added by
Paul Leopardi
Publication date
2012
fields
Informatics Engineering
and research's language is
English
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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.
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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 cost estimate of the proposed quadrature. In addition, we provide a novel and improved estimate for the cardinality of the underlying anisotropic index set. To validate the theoretical findings, we present several examples ranging from simple quadrature problems to diffusion problems on random domains. These examples demonstrate the remarkable convergence behaviour of the anisotropic sparse grid quadrature in applications.
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 aspects of the solution of elliptic PDEs with lognormal diffusion coefficients using sparse grids. Specifically, we first focus on the choice of univariate interpolation rules, advocating the use of Gaussian Leja points as introduced by Narayan and Jakeman (Adaptive Leja sparse grid constructions for stochastic collocation and high-dimensional approximation, SIAM J. Sci. Comput., 2014) and then discuss the possible computational advantages of replacing the standard Karhunen-Lo`eve expansion of the diffusion coefficient with the Levy-Ciesielski expansion, motivated by theoretical work of Bachmayr, Cohen, DeVore, and Migliorati (Sparse polynomial approximation of parametric elliptic PDEs. part II: lognormal coefficients, ESAIM: M2AN, 2016). Our numerical results indicate that, for the problem under consideration, Gaussian Leja collocation points outperform Gauss-Hermite and Genz-Keister nodes for the sparse grid approximation and that the Karhunen-Lo`eve expansion of the log diffusion coefficient is more appropriate than its Levy-Ciesielski expansion for purpose of sparse grid collocation.
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 and provide relations to sparsity problems. We then propose a new optimization method for these functionals which is based on dynamic programming and the alternating direction method of multipliers (ADMM). A series of experiments shows that the proposed method yields very satisfactory jump-sparse and sparse reconstructions, respectively. We highlight the capability of the method by comparing it with classical and recent approaches such as TV minimization (jump-sparse signals), orthogonal matching pursuit, iterative hard thresholding, and iteratively reweighted $ell^1$ minimization (sparse signals).
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 $mathcal{N}(0, L^{-top} L^{-1})$, subject to a sparsity constraint. Surprisingly, this problem has a closed-form solution that can be computed efficiently, recovering the popular Vecchia approximation in spatial statistics. Based on recent results on the approximate sparsity of inverse Cholesky factors of $Theta$ obtained from pairwise evaluation of Greens functions of elliptic boundary-value problems at points ${x_{i}}_{1 leq i leq N} subset mathbb{R}^{d}$, we propose an elimination ordering and sparsity pattern that allows us to compute $epsilon$-approximate inverse Cholesky factors of such $Theta$ in computational complexity $mathcal{O}(N log(N/epsilon)^d)$ in space and $mathcal{O}(N log(N/epsilon)^{2d})$ in time. To the best of our knowledge, this is the best asymptotic complexity for this class of problems. Furthermore, our method is embarrassingly parallel, automatically exploits low-dimensional structure in the data, and can perform Gaussian-process regression in linear (in $N$) space complexity. Motivated by the optimality properties of our methods, we propose methods for applying it to the joint covariance of training and prediction points in Gaussian-process regression, greatly improving stability and computational cost. Finally, we show how to apply our method to the important setting of Gaussian processes with additive noise, sacrificing neither accuracy nor computational complexity.
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 prove its linear convergence in expectation with respect to the Bregman distance in the noiseless and noisy cases. This greedy variant can be viewed as a unification of the sampling Kaczmarz-Motzkin method and the randomized sparse Kaczmarz method, and hence inherits the merits of these two methods. Numerically, we report a couple of experimental results to demonstrate its superiority
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