Approximation of multivariate periodic functions by trigonometric polynomials based on rank-1 lattice sampling
TLDR
Algorithms for the approximation of multivariate periodic functions by trigonometric polynomials and an algorithm for sampling multivariate functions on perturbed rank-1 lattices are presented and numerical stability of the suggested method is shown.About:
This article is published in Journal of Complexity.The article was published on 2015-08-01 and is currently open access. It has received 69 citations till now. The article focuses on the topics: Trigonometric polynomial & Trigonometric interpolation.read more
Citations
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Sparse high-dimensional FFT based on rank-1 lattice sampling
Daniel Potts,Toni Volkmer +1 more
TL;DR: This paper adaptively construct the index set of frequencies belonging to the non-zero Fourier coefficients in a dimension incremental way and discusses possibilities to reduce the number of samples and arithmetic operations by applying methods from compressed sensing and a version of Prony's method.
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Notes on ( s , t ) -weak tractability
Pawel Siedlecki,Markus Weimar +1 more
TL;DR: In this article, the authors proposed the notion of ( s, t ) -weak tractability, which allows to quantify the exact (sub-/super-) exponential dependence of n ( e, S d ) on both parameters e and d. In this new framework the parameters s and t are used to quantitatively refine the huge class of polynomially intractable problems.
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Tight error bounds for rank-1 lattice sampling in spaces of hybrid mixed smoothness
TL;DR: The main result is the fact that any (non-)linear reconstruction algorithm taking function values on any integration lattice of size M has a dimension-independent lower bound of 2-(α+1)/2M-α/2 when considering the optimal worst-case error with respect to function spaces of (hybrid) mixed smoothness.
Posted Content
Worst-case recovery guarantees for least squares approximation using random samples
TL;DR: It turns out that this simple method based on least squares regression can compete with the quasi-Monte Carlo methods in the literature which are based on lattices and digital nets.
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Efficient Spectral Estimation by MUSIC and ESPRIT with Application to Sparse FFT
TL;DR: For a trigonometric polynomial of large sparsity, a new sparse fast Fourier transform is presented by shifted sampling and using MUSIC resp.
References
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Topics in Matrix Analysis
TL;DR: The field of values as discussed by the authors is a generalization of the field of value of matrices and functions, and it includes singular value inequalities, matrix equations and Kronecker products, and Hadamard products.
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TL;DR: Theorems and statistical properties of least squares solutions are explained and basic numerical methods for solving least squares problems are described.
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Lattice Methods for Multiple Integration
TL;DR: In this paper, the use of lattice methods for the approximate integration of smooth periodic functions over the unit cube in any number of dimensions is discussed, and the authors show that the lattice method can be used to approximate any periodic function over a unit cube.
Journal ArticleDOI
High-dimensional integration: The quasi-Monte Carlo way
TL;DR: A survey of recent developments in lattice methods, digital nets, and related themes can be found in this paper, where the authors present a contemporary review of QMC (quasi-Monte Carlo) methods, that is, equalweight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0, 1] s, w heres may be large, or even infinite.