Multivariate Anisotropic Interpolation on the Torus

Bergmann, Ronny; Prestin, Jürgen

doi:10.1007/978-3-319-06404-8_3

Cited by 5 publications

(10 citation statements)

References 17 publications

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“…De la Vallée Poussin means. A special case of M-invariant spaces are the ones defined via de la Vallée Poussin means, following the construction of [5]. We call a function g :…”

Section: Translation Invariant Spaces Of Periodic Functionsmentioning

confidence: 99%

FFT-based homogenization on periodic anisotropic translation invariant spaces

Bergmann

Merkert

2020

Applied and Computational Harmonic Analysis

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In this paper we derive a discretisation of the equation of quasi-static elasticity in homogenization in form of a variational formulation and the so-called Lippmann-Schwinger equation, in anisotropic spaces of translates of periodic functions. We unify and extend the truncated Fourier series approach, the constant finite element ansatz and the anisotropic lattice derivation. The resulting formulation of the Lippmann-Schwinger equation in anisotropic translation invariant spaces unifies and analyses for the first time both the Fourier methods and finite element approaches in a common mathematical framework. We further define and characterize the resulting periodised Green operator. This operator coincides in case of a Dirichlet kernel corresponding to a diagonal matrix with the operator derived for the Galerkin projection stemming from the truncated Fourier series approach and to the anisotropic lattice derivation for all other Dirichlet kernels. Additionally, we proof the boundedness of the periodised Green operator. The operator further constitutes a projection if and only if the space of translates is generated by a Dirichlet kernel. Numerical examples for both the de la Vallée Poussin means and Box splines illustrate the flexibility of this framework.

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“…De la Vallée Poussin means. A special case of M-invariant spaces are the ones defined via de la Vallée Poussin means, following the construction of [5]. We call a function g :…”

Section: Translation Invariant Spaces Of Periodic Functionsmentioning

confidence: 99%

FFT-based homogenization on periodic anisotropic translation invariant spaces

Bergmann

Merkert

2020

Applied and Computational Harmonic Analysis

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“…However, their corresponding generating sets G(M T n ) differ. This can be interpreted as choosing different directional sine and cosine functions that can be defined on the same set of points in order to analyze given discrete data or employing the periodicity of c M h (f ) with respect to M T that is implied by (6). This introduces the possibility of anisotropic analysis and interpretation even on a usual pixel grid.…”

Section: A Fast Fourier Transform On Patternsmentioning

confidence: 99%

“…Besides the theory of a discrete Fourier transform (DFT) on abelian groups, also known as generalized Fourier transform [1], the DFT has also been generalized to arbitrary sampling lattices, e.g. in order to derive periodic wavelets [16,6] and a corresponding fast Fourier and fast wavelet transform [2]. The computational complexity on these lattices even stays the same as on the usual rectangular or pixel grid.…”

Section: Introductionmentioning

confidence: 99%

A framework for FFT-based homogenization on anisotropic lattices

Bergmann¹,

Merkert²

2018

Computers & Mathematics with Applications

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In order to take structural anisotropies of a given composite and different shapes of its unit cell into account, we generalize the Basic Scheme in homogenization by Moulinec and Suquet to arbitrary sampling lattices and tilings of the d-dimensional Euclidean space. We employ a Fourier transform for these lattices by introducing the corresponding set of sample points, the so called pattern, and its frequency set, the generating set, both representing the anisotropy of both the shape of the unit cell and the chosen preferences in certain sampling directions. In several cases, this Fourier transform is of lower dimension than the space itself. For the so called rank-1-lattices it even reduces to a one-dimensional Fourier transform having the same leading coefficient as the fastest Fourier transform implementation available. We illustrate the generalized Basic Scheme on an anisotropic laminate and a generalized ellipsoidal Hashin structure. For both we give an analytical solution to the elasticity problem, in two-and three dimensions, respectively. We then illustrate the possibilities of choosing a pattern. Compared to classical grids this introduces both a reduction of computation time and a reduced error of the numerical method. It also allows for anisotropic subsampling, i.e. choosing a sub lattice of a pixel or voxel grid based on anisotropy information of the material at hand.

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“…The wavelet parts g N 1 ∈ V ψ 1 N 1 and g N 2 ∈ V M k , we obtain an approximation f M k in this space, cf. [4].…”

Section: Lemma 52mentioning

confidence: 99%

“…The scaling spaces obtained in this setting can also be used to construct and characterize anisotropic interpolation operators. Corresponding error estimates and function spaces are discussed in [4].…”

Section: Introductionmentioning

confidence: 99%

Multivariate Periodic Wavelets of de la Vallée Poussin Type

Bergmann¹,

Prestin²

2014

J Fourier Anal Appl

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In this paper we present a general approach to multivariate periodic wavelets generated by scaling functions of de la Vallée Poussin type. These scaling functions and their corresponding wavelets are determined by their Fourier coefficients, which are sample values of a function, that can be chosen arbitrarily smooth, even with different smoothness in each direction. This construction generalizes the one-dimensional de la Vallée Poussin means to the multivariate case and enables the construction of wavelet systems, where the set of dilation matrices for the two-scale relation of two spaces of the multiresolution analysis may contain shear and rotation matrices. It further enables the functions contained in each of the function spaces from the corresponding series of scaling spaces to have a certain direction or set of directions as their focus, which is illustrated by detecting jumps of certain directional derivatives of higher order.

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Multivariate Anisotropic Interpolation on the Torus

Cited by 5 publications

References 17 publications

FFT-based homogenization on periodic anisotropic translation invariant spaces

FFT-based homogenization on periodic anisotropic translation invariant spaces

A framework for FFT-based homogenization on anisotropic lattices

Multivariate Periodic Wavelets of de la Vallée Poussin Type

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