Lutz Kämmerer

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A straightforward discretisation of problems in high dimensions often leads to an exponential growth in the number of degrees of freedom. Sparse grid approximations allow for a severe decrease in the number of used Fourier coefficients to represent functions with bounded mixed derivatives and the fast Fourier transform (FFT) has been adapted to this thin(More)
Sparse grid discretisations allow for a severe decrease in the number of degrees of freedom for high dimensional problems. Recently, the corresponding hyperbolic cross fast Fourier transform has been shown to exhibit numerical instabilities already for moderate problem sizes. In contrast to standard sparse grids as spatial discretisation, we propose the use(More)
In this paper, we present algorithms for the approximation of multivariate functions by trigonometric polynomials. The approximation is based on sampling of multivariate functions on rank-1 lattices. To this end, we study the approximation of functions in periodic Sobolev spaces of dominating mixed smoothness. Recently an algorithm for the trigonometric(More)
The evaluation of multivariate trigonometric polynomials at the nodes of a rank-1 lattice leads to a onedimensional discrete Fourier transform. Often, one is also interested in the reconstruction of the Fourier coefficients from their samples. We present necessary and sufficient conditions on rank-1 lattices allowing a stable reconstruction of trigonometric(More)
In this paper, we present algorithms for the approximation of multivariate periodic functions by trigonometric polynomials. The approximation is based on sampling of multivariate functions on rank-1 lattices. To this end, we study the approximation of periodic functions of a certain smoothness. Our considerations include functions from periodic Sobolev(More)
In this paper, we present error estimates for the approximation of multivariate periodic functions in periodic Hilbert spaces of isotropic and dominating mixed smoothness by trigonometric polynomials. The approximation is based on sampling of the multivariate functions on rank-1 lattices. We use reconstructing rank-1 lattices with generating vectors of(More)
A straightforward discretisation of high-dimensional problems often leads to a curse of dimensions and thus the use of sparsity has become a popular tool. Efficient algorithms like the fast Fourier transform (FFT) have to be customised to these thinner discretisations and we focus on two major topics regarding the Fourier analysis of high-dimensional(More)
The approximation of problems in d spatial dimensions by sparse trigono-metric polynomials supported on known or unknown frequency index sets I ⊂ Z d is an important task with a variety of applications. The use of a generalization of rank-1 lattices as spatial discretizations offers a suitable possibility for sampling such sparse trigonometric polynomials.(More)
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