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Fast Fourier transforms for nonequispaced data: A tutorial
, 2000
"... In this section, we consider approximative methods for the fast computation of multivariate discrete Fourier transforms for nonequispaced data (NDFT) in the time domain and in the frequency domain. In particular, we are interested in the approximation error as function of the arithmetic complexity o ..."
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Cited by 124 (35 self)
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In this section, we consider approximative methods for the fast computation of multivariate discrete Fourier transforms for nonequispaced data (NDFT) in the time domain and in the frequency domain. In particular, we are interested in the approximation error as function of the arithmetic complexity of the algorithm. We discuss the robustness of NDFTalgorithms with respect to roundoff errors and apply NDFTalgorithms for the fast computation of Bessel transforms.
Stability Results for Scattered Data Interpolation by Trigonometric Polynomials
 SIAM J. Sci. Comput
, 2007
"... A fast and reliable algorithm for the optimal interpolation of scattered data on the torus Td by multivariate trigonometric polynomials is presented. The algorithm is based on a variant of the conjugate gradient method in combination with the fast Fourier transforms for nonequispaced nodes. The main ..."
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Cited by 33 (17 self)
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A fast and reliable algorithm for the optimal interpolation of scattered data on the torus Td by multivariate trigonometric polynomials is presented. The algorithm is based on a variant of the conjugate gradient method in combination with the fast Fourier transforms for nonequispaced nodes. The main result is that under mild assumptions the total complexity for solving the interpolation problem at M arbitrary nodes is of order O(M log M). This result is obtained by the use of localised trigonometric kernels where the localisation is chosen in accordance to the spatial dimension d. Numerical examples show the efficiency of the new algorithm.
A MultiLevel Algorithm for the Solution of Moment Problems
, 1997
"... We study numerical methods for the solution of general linear moment problems, where the solution belongs to a family of nested subspaces of a Hilbert space. Multilevel algorithms, based on the conjugate gradient method and the LandweberRichardson method are proposed, that determines the "op ..."
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Cited by 3 (1 self)
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We study numerical methods for the solution of general linear moment problems, where the solution belongs to a family of nested subspaces of a Hilbert space. Multilevel algorithms, based on the conjugate gradient method and the LandweberRichardson method are proposed, that determines the "optimal" reconstruction level a posteriori from quantities that arise during the numerical calculations. As an important example we discuss the reconstruction of bandlimited signals from irregularly spaced noisy samples, when the actual bandwidth of the signal is not available. Numerical examples show the usefulness of the proposed algorithms. Keywords and Phrases: Moment problems, multilevel algorithms, Landweber Richardson method, conjugate gradient method, sampling theory. AMS Subject Classification: 44A60, 65J10, 65J20, 62D05, 42A15. 1 Introduction We study numerical methods for the solution of a family of general linear moment problems hx; g N j i = j j 2 I (1.1) where fg N j g j2I is...
A LEVINSONGALERKIN ALGORITHM FOR REGULARIZED TRIGONOMETRIC APPROXIMATION ∗
, 1999
"... Abstract. Trigonometric polynomials are widely used for the approximation of a smooth function f from a set of nonuniformly spaced samples {f(xj)} N−1 j=0. If the samples are perturbed by noise, controlling the smoothness of the trigonometric approximation becomes an essential issue to avoid overfit ..."
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Cited by 1 (0 self)
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Abstract. Trigonometric polynomials are widely used for the approximation of a smooth function f from a set of nonuniformly spaced samples {f(xj)} N−1 j=0. If the samples are perturbed by noise, controlling the smoothness of the trigonometric approximation becomes an essential issue to avoid overfitting and underfitting of the data. Using the polynomial degree as regularization parameter we derive a multilevel algorithm that iteratively adapts to the least squares solution of optimal smoothness. The proposed algorithm computes the solution in at most O(NM +M 2) operations (M being the polynomial degree of the approximation) by solving a family of nested Toeplitz systems. It is shown how the presented method can be extended to multivariate trigonometric approximation. We demonstrate the performance of the algorithm by applying it in echocardiography to the recovery of the boundary of the Left Ventricle.
Reconstruction of the Earth Surface Potential Distribution using Radial Basis Functions
"... Abstract – A credible reconstruction of a continuous twodimensional earth surface potential distribution from a discrete set of noisy samples is essential for correct identification of regions with high values of touch and step voltages, as a part of grounding grid safety assessment procedure. This ..."
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Abstract – A credible reconstruction of a continuous twodimensional earth surface potential distribution from a discrete set of noisy samples is essential for correct identification of regions with high values of touch and step voltages, as a part of grounding grid safety assessment procedure. This research provides some new and useful insights on the earth surface potential distribution reconstruction quality, focused mainly on the most commonly used radial basis function (RBF) interpolation kernels. The RBF approach to reconstruction was chosen because of the underlaying algorithm simplicity and the fact that it provides natural mathematical framework for smooth interpolation and approximation of potential fields from irregular sampling sets. Since the quality of potential distribution reconstruction is heavily influenced by the choice of interpolation method, sampling layout and a priori knowledge of the grounding grid structure under test, this research also elaborates some heuristic guidelines for planning proper sampling procedure and illustrates some examples of reconstructed functional shape distortions due to poor sampling layouts.