Multiple Rank-1 Lattices as Sampling Schemes for Multivariate Trigonometric Polynomials

Kämmerer, Lutz

doi:10.1007/s00041-016-9520-8

Cited by 20 publications

(43 citation statements)

References 24 publications

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“…He furthermore proved that the upper bound on the lattice size improves with high probability to M ≤ C|I| log |I| for these particular reconstructing lattices. There are several other strategies in the literature to find appropriate reconstructing multiple rank-1 lattices, see [9,8].…”

Section: Multiple Transformed Rank-1 Latticesmentioning

confidence: 99%

“…For rank-1 lattices with M sampling points there are efficient algorithms for computing the discrete Fourier transforms [6], and furthermore these schemes provide good approximation properties, in particular for high-dimensional approximation problems, see [2]. We note that L. Kämmerer recently suggested to use multiple rank-1 lattices which are obtained by taking a union of several single rank-1 lattices, see [9,8]. Using this method one can overcome the limitations of the single rank-1 lattice approach.…”

Section: Introductionmentioning

confidence: 99%

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Transformed rank-1 lattices for high-dimensional approximation

Nasdala¹,

Potts²

2020

etna

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This paper describes an extension of Fourier approximation methods for multivariate functions defined on bounded domains to unbounded ones via a multivariate change of coordinate mapping. In this approach we adapt algorithms for the evaluation and reconstruction of multivariate trigonometric polynomials based on single and multiple reconstructing rank-1 lattices and make use of dimension incremental construction methods for sparse frequency sets. Various numerical tests confirm obtained theoretical results for the transformed methods.

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Section: Multiple Transformed Rank-1 Latticesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Transformed rank-1 lattices for high-dimensional approximation

Nasdala¹,

Potts²

2020

etna

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“…In this paper, we incorporate ideas from [1] into the algorithm that is developed in [10,Alg. 5] such that the construction of multiple rank-1 lattices will allow for estimates of the sizes and the number of single rank-1 lattices that are joined to a spatial discretization.…”

Section: Introductionmentioning

confidence: 99%

“…We reflect some basics on multivariate trigonometric polynomials and (multiple) rank-1 lattices from [9,10]. At first, we define the torus T ≃ [0, 1) and the multivariate trigonometric polynomial…”

Section: Introductionmentioning

confidence: 99%

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Constructing spatial discretizations for sparse multivariate trigonometric polynomials that allow for a fast discrete Fourier transform

Kämmerer

2019

Applied and Computational Harmonic Analysis

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The paper discusses the construction of high dimensional spatial discretizations for arbitrary multivariate trigonometric polynomials, where the frequency support of the trigonometric polynomial is known. We suggest a construction based on the union of several rank-1 lattices as sampling scheme. We call such schemes multiple rank-1 lattices. This approach automatically makes available a fast discrete Fourier transform (FFT) on the data.The key objective of the construction of spatial discretizations is the unique reconstruction of the trigonometric polynomial using the sampling values at the sampling nodes. We develop different construction methods for multiple rank-1 lattices that allow for this unique reconstruction. The symbol M denotes the total number of sampling nodes within the multiple rank-1 lattice. In addition, we assume that the multivariate trigonometric polynomial is a linear combination of T trigonometric monomials. The ratio of the number M of sampling points that are sufficient for the unique reconstruction to the number T of distinct monomials is called oversampling factor in this context. The presented construction methods for multiple rank-1 lattices allow for estimates of this number M . Roughly speaking, the oversampling factor M/T is independent of the spatial dimension and, with high probability, only logarithmic in T , which is much better than the oversampling factor that is expected for a sampling method that uses one single rank-1 lattice.The newly developed approaches for the construction of spatial discretizations are probabilistic methods. The arithmetic complexity of these algorithms depend only linearly on the spatial dimension and, with high probability, only linearly on T up to some logarithmic factors.Furthermore, we analyze the computational complexities of the resulting FFT algorithms, that exploits the structure of the suggested multiple rank-1 lattice spatial discretizations, in detail and obtain upper bounds in O (M log M ), where the constants depend only linearly on the spatial dimension. With high probability, we construct spatial discretizations where M/T ≤ C log T holds, which implies that the complexity of the corresponding FFT converts to O T log 2 T .

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