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17
Random sampling of sparse trigonometric polynomials
 Appl. Comput. Harm. Anal
, 2006
"... We investigate the problem of reconstructing sparse multivariate trigonometric polynomials from few randomly taken samples by Basis Pursuit and greedy algorithms such as Orthogonal Matching Pursuit (OMP) and Thresholding. While recovery by Basis Pursuit has recently been studied by several authors, ..."
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Cited by 41 (18 self)
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We investigate the problem of reconstructing sparse multivariate trigonometric polynomials from few randomly taken samples by Basis Pursuit and greedy algorithms such as Orthogonal Matching Pursuit (OMP) and Thresholding. While recovery by Basis Pursuit has recently been studied by several authors, we provide theoretical results on the success probability of reconstruction via Thresholding and OMP for both a continuous and a discrete probability model for the sampling points. We present numerical experiments, which indicate that usually Basis Pursuit is significantly slower than greedy algorithms, while the recovery rates are very similar.
D.X.: Shannon sampling and function reconstruction from point values
 Bull. Am. Math. Soc
, 2004
"... then came to the University of Chicago, where I was starting my job as instructor for the fall of 1956. He, Suzanne, Clara and I became good friends and saw much of each other for many decades, especially at IHES in Paris. Thom’s encouragement and support were important for me, especially in my firs ..."
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Cited by 32 (8 self)
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then came to the University of Chicago, where I was starting my job as instructor for the fall of 1956. He, Suzanne, Clara and I became good friends and saw much of each other for many decades, especially at IHES in Paris. Thom’s encouragement and support were important for me, especially in my first years after my Ph.D. I studied his work in cobordism, singularities of maps, and transversality, gaining many insights. I also enjoyed listening to his provocations, for example his disparaging remarks on complex analysis, 19th century mathematics, and Bourbaki. There was also a stormy side in our relationship. Neither of us could hide the pain that our public conflicts over “catastrophe theory ” caused. René Thom was a great mathematician, leaving his impact on a wide part of mathematics. I will always treasure my memories of him.
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 31 (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 randomized Kaczmarz algorithm with exponential convergence
"... The Kaczmarz method for solving linear systems of equations is an iterative algorithm that has found many applications ranging from computer tomography to digital signal processing. Despite the popularity of this method, useful theoretical estimates for its rate of convergence are still scarce. We i ..."
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Cited by 20 (1 self)
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The Kaczmarz method for solving linear systems of equations is an iterative algorithm that has found many applications ranging from computer tomography to digital signal processing. Despite the popularity of this method, useful theoretical estimates for its rate of convergence are still scarce. We introduce a randomized version of the Kaczmarz method for consistent, overdetermined linear systems and we prove that it converges with expected exponential rate. Furthermore, this is the first solver whose rate does not depend on the number of equations in the system. The solver does not even need to know the whole system, but only a small random part of it. It thus outperforms all previously known methods on general extremely overdetermined systems. Even for moderately overdetermined systems, numerical simulations as well as theoretical analysis reveal that our algorithm can converge faster than the celebrated conjugate gradient algorithm. Furthermore, our theory and numerical simulations confirm a prediction of Feichtinger et al. in the context of reconstructing bandlimited functions from nonuniform sampling. ∗ T.S. was supported by NSF DMS grant 0511461. R.V. was supported by the Alfred P.
Using NFFT 3  a software library for various nonequispaced fast Fourier transforms
, 2008
"... NFFT 3 is a software library that implements the nonequispaced fast Fourier transform (NFFT) and a number of related algorithms, e.g. nonequispaced fast Fourier transforms on the sphere and iterative schemes for inversion. This is to provide a survey on the mathematical concepts behind the NFFT and ..."
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Cited by 12 (8 self)
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NFFT 3 is a software library that implements the nonequispaced fast Fourier transform (NFFT) and a number of related algorithms, e.g. nonequispaced fast Fourier transforms on the sphere and iterative schemes for inversion. This is to provide a survey on the mathematical concepts behind the NFFT and its variants, as well as a general guideline for using the library. Numerical examples for a number of applications are given.
On the computation of the polar FFT
 Appl. Comput. Harmon. Anal
, 2007
"... We show that the polar as well as the pseudopolar FFT can be computed very accurately and efficiently by the well known nonequispaced FFT. Furthermore, we discuss the reconstruction of a 2d signal from its Fourier transform samples on a (pseudo)polar grid by means of the inverse nonequispaced FFT. ..."
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Cited by 9 (7 self)
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We show that the polar as well as the pseudopolar FFT can be computed very accurately and efficiently by the well known nonequispaced FFT. Furthermore, we discuss the reconstruction of a 2d signal from its Fourier transform samples on a (pseudo)polar grid by means of the inverse nonequispaced FFT.
Learning trigonometric polynomials from random samples and exponential inequalities for eigenvalues of random matrices
, 2007
"... Motivated by problems arising in random sampling of trigonometric polynomials, we derive exponential inequalities for the operator norm of the difference between the sample second moment matrix n −1 U ∗ U and its expectation where U is a complex random n × D matrix with independent rows. These resul ..."
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Cited by 8 (4 self)
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Motivated by problems arising in random sampling of trigonometric polynomials, we derive exponential inequalities for the operator norm of the difference between the sample second moment matrix n −1 U ∗ U and its expectation where U is a complex random n × D matrix with independent rows. These results immediately imply deviation inequalities for the largest (smallest) eigenvalues of the sample second moment matrix, which in turn lead to results on the condition number of the sample second moment matrix. We also show that trigonometric polynomials in several variables can be learned from const · D ln D random samples.
Nonlinear projection digital image inpainting and restoration methods
 Journal of Mathematical Imaging and Vision
, 2006
"... This paper concerns with nonuniform sampling and interpolation methods combined with variational models for the solution of a generalized image inpainting problem and the restoration of digital signals. In particular, we discuss the problem of reconstructing a digital signal/image from very few, spa ..."
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Cited by 3 (0 self)
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This paper concerns with nonuniform sampling and interpolation methods combined with variational models for the solution of a generalized image inpainting problem and the restoration of digital signals. In particular, we discuss the problem of reconstructing a digital signal/image from very few, sparse, and complete information and a substantial incomplete information, which will be assumed as the result of a nonlinear distortion. As a typical and inspiring example, we illustrate the concrete problem of the color restoration of a destroyed art fresco from its few known fragments and some gray picture taken prior to the damage. Numerical implementations are included together with several examples and numerical results to illustrate the proposed method. The numerical experience suggests furthermore that a particular system of coupled HamiltonJacobi equations is wellposed.
Numerical stability of nonequispaced fast Fourier transforms
"... Dedicated to Franz Locher in honor of his 65th birthday This paper presents some new results on numerical stability for multivariate fast Fourier transform of nonequispaced data (NFFT). In contrast to fast Fourier transform (of equispaced data), the NFFT is an approximate algorithm. In a worst case ..."
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Cited by 3 (1 self)
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Dedicated to Franz Locher in honor of his 65th birthday This paper presents some new results on numerical stability for multivariate fast Fourier transform of nonequispaced data (NFFT). In contrast to fast Fourier transform (of equispaced data), the NFFT is an approximate algorithm. In a worst case study, we show that both approximation error and roundoff error have a strong influence on the numerical stability of NFFT. Numerical tests confirm the theoretical estimates of numerical stability.
A probability argument in favor of ignoring small singular values
 Operators and Matrices, 1:31 – 43
, 2007
"... O perators a nd M atrices ..."