Lutz Kämmerer

On the stability of the hyperbolic cross discrete Fourier transform

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 discretisation. We show that this so called hyperbolic cross FFT suffers from an increase of its condition number for both increasing refinement and increasing spatial dimension.

Interpolation lattices for hyperbolic cross trigonometric polynomials

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 of oversampled lattice rules known from multivariate numerical integration. This allows for the highly efficient and perfectly stable evaluation and reconstruction of trigonometric polynomials using only one ordinary FFT. Moreover, we give numerical evidence that reasonable small lattices exist such that our new method outperforms the sparse grid based hyperbolic cross FFT for realistic problem sizes.

Reconstructing Hyperbolic Cross Trigonometric Polynomials by Sampling along Rank-1 Lattices

With given Fourier coefficients the evaluation of multivariate trigonometric polynomials at the nodes of a rank-1 lattice leads to a one-dimensional discrete Fourier transform. In many applications one is also interested in the reconstruction of the Fourier coefficients from samples in the spatial domain. We present necessary and sufficient conditions on rank-1 lattices allowing a stable reconstruction of trigonometric polynomials supported on hyperbolic crosses. In addition, we suggest approaches for determining suitable rank-1 lattices using a component-by-component algorithm. We present numerical results for reconstructing trigonometric polynomials up to spatial dimension 100.

Reconstructing Multivariate Trigonometric Polynomials from Samples Along Rank-1 Lattices

The approximation of problems in \(d\) spatial dimensions by trigonometric polynomials supported on known more or less sparse frequency index sets \(I\subset \mathbb {Z}^d\) is an important task with a variety of applications. The use of rank-1 lattices as spatial discretizations offers a suitable possibility for sampling such sparse trigonometric polynomials. Given an arbitrary index set of frequencies, we construct rank-1 lattices that allow a stable and unique discrete Fourier transform. We use a component-by-component method in order to determine the generating vector and the lattice size.

Approximation of multivariate periodic functions by trigonometric polynomials based on rank-1 lattice sampling

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 spaces of generalized mixed smoothness. Recently an algorithm for the trigonometric interpolation on generalized sparse grids for this class of functions was investigated by Griebel and Hamaekers (2014). The main advantage of our method is that the algorithm is based mainly on a single one-dimensional fast Fourier transform, and that the arithmetic complexity of the algorithm depends only on the cardinality of the support of the trigonometric polynomial in the frequency domain. Therefore, we investigate trigonometric polynomials with frequencies supported on hyperbolic crosses and energy norm based hyperbolic crosses in more detail. Furthermore, we present an algorithm for sampling multivariate functions on perturbed rank-1 lattices and show the numerical stability of the suggested method. Numerical results are presented up to dimension d = 10 , which confirm the theoretical findings.

Approximation of multivariate periodic functions by trigonometric polynomials based on sampling along rank-1 lattice with generating vector of Korobov form

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 Korobov form for the sampling and generalize the technique from Temlyakov (1986), in order to show that the aliasing error of that approximation is of the same order as the error of the approximation using the partial sum of the Fourier series. The main advantage of our method is that the computation of the Fourier coefficients of such a trigonometric polynomial, which we use as approximant, is based mainly on a one-dimensional fast Fourier transform, cf. Kammerer et?al. (2013), Kammerer (2014). This means that the arithmetic complexity of the computation depends only on the cardinality of the support of the trigonometric polynomial in the frequency domain. Numerical results are presented up to dimension d = 10 .

Reconstructing Multivariate Trigonometric Polynomials by Sampling Along Generated Sets

The approximation of problems in d spatial dimensions by sparse trigonometric polynomials supported on known or unknown frequency index sets \(I \subset {\mathbb{Z}}^{d}\) 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. Given an index set of frequencies, we construct corresponding sampling sets that allow a stable and unique discrete Fourier transform. Applying the one-dimensional non-equispaced fast Fourier transform (NFFT) enables the fast evaluation and reconstruction of the multivariate trigonometric polynomials.

Computational Methods for the Fourier Analysis of Sparse High-Dimensional Functions

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 functions: We present stable and effective algorithms for the fast evaluation and reconstruction of multivariate trigonometric polynomials with frequencies supported on an index set \(\mathcal{I}\subset \mathbb{Z}^{d}\).