Structured matrix algorithms for inverse scattering on the line

Mee, Cornelis van der; Seatzu, Sebastiano; Theis, Daniela

doi:10.1007/s10092-007-0129-9

Cited by 10 publications

(4 citation statements)

References 31 publications

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“…The governing equations of the 1-D CISP are given by (1) where and are the right-and left-propagating waves, respectively, is the local reflectivity function (LRF) at the position , and is the time variable. We try to find when and , called scattering data, are given.…”

Section: A Discretization Of the 1-d Cispmentioning

confidence: 99%

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The Schur Algorithm Applied to the One-Dimensional Continuous Inverse Scattering Problem

Choi

Chun

Kim

et al. 2013

IEEE Trans. Signal Process.

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The one-dimensional continuous inverse scattering problem can be solved by the Schur algorithm in the discrete-time domain using sampled scattering data. The sampling rate of the scattering data should be increased to reduce the discretization error, but the complexity of the Schur algorithm is proportional to the square of the sampling rate. To improve this tradeoff between the complexity and the accuracy, we propose a Schur algorithm with the Richardson extrapolation (SARE). The asymptotic expansion of the Schur algorithm, necessary for the Richardson extrapolation, is derived in powers of the discretization step, which shows that the accuracy order (with respect to the discretization step) of the Schur algorithm is 1. The accuracy order of the SARE with the -step Richardson extrapolation is increased to with comparable complexity to the Schur algorithm. Therefore, the discretization error of the Schur algorithm can be decreased in a computationally efficient manner by the SARE.

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Section: A Discretization Of the 1-d Cispmentioning

confidence: 99%

“…The Taylor series for at is given by 1 The normalization factor (NF) is important for energy conservation. However, this NF is a cofactor for all elements of a generator matrix.…”

Section: B Asymptotic Expansionmentioning

confidence: 99%

The Schur Algorithm Applied to the One-Dimensional Continuous Inverse Scattering Problem

Choi

Chun

Kim

et al. 2013

IEEE Trans. Signal Process.

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“…Different numerical approaches to this computationally difficult problem have been developed. We mention the papers of Aktosun and Sacks and Rundell and Sacks restricted to potentials not admitting bound state data, the paper of van der Mee et al in which a discretization scheme for the Gel'fand‐Levitan‐Marchenko equation was developed, and the paper of Altundağ et al where a brief review of existing approaches can be found.…”

Section: Introductionmentioning

confidence: 99%

On a method for solving the inverse scattering problem on the line

Kravchenko

2019

Math Methods in App Sciences

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“…In all of the applications mentioned above, accuracy of the numerical algorithms form a bottleneck either at higher powers in the non-Hermitian class or at reflectivities approaching unity in the Hermitian class. There is a vast amount literature on numerical methods for the solution of the Gelfand-Levitan-Marchenko (GLM) integral equations notable among them are the integral layer-peeling [7], Töplitz inner-bordering [9][10][11] and the Nyström method [12]. From a practical viewpoint, these algorithms work for a large class of problems; however, these methods cannot provide accuracies upto the machine precision with the exception of the method due to Trogdon and Olver [13].…”

Section: Introductionmentioning

confidence: 99%

Nonlinearly bandlimited signals

Vaibhav

2019

J. Phys. A: Math. Theor.

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In this paper, we study the inverse scattering problem for a class of signals that have a compactly supported reflection coefficient. The problem boils down to the solution of the Gelfand-Levitan-Marchenko (GLM) integral equations with a kernel that is bandlimited. By adopting a sampling theory approach to the associated Hankel operators in the Bernstein spaces, a constructive proof of existence of a solution of the GLM equations is obtained under various restrictions on the nonlinear impulse response (NIR). The formalism developed in this article also lends itself well to numerical computations yielding algorithms that are shown to have algebraic rates of convergence. In particular, the use Whittaker-Kotelnikov-Shannon sampling series yields an algorithm that converges as O(N −1/2 ) whereas the use of Helms and Thomas (HT) version of the sampling expansion yields an algorithm that converges as O(N −m−1/2 ) for any m > 0 provided the regularity conditions are fulfilled. The complexity of the algorithms depend on the linear solver used. The use of conjugate-gradient (CG) method yields an algorithm of complexity O(Niter.N 2 ) per sample of the signal where N is the number of sampling basis functions used and N iter. is the number of CG iterations involved. The HT version of the sampling expansions facilitates the development of algorithms of complexity O(Niter.N log N) (per sample of the signal) by exploiting the special structure as well as the (approximate) sparsity of the matrices involved. The algorithms are numerically validated using Schwartz class functions as NIRs that are either bandlimited or effectively bandlimited. The results suggest that the HT variant of our algorithm is spectrally convergent for an input of the aforementioned class.

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Structured matrix algorithms for inverse scattering on the line

Cited by 10 publications

References 31 publications

The Schur Algorithm Applied to the One-Dimensional Continuous Inverse Scattering Problem

The Schur Algorithm Applied to the One-Dimensional Continuous Inverse Scattering Problem

On a method for solving the inverse scattering problem on the line

Nonlinearly bandlimited signals

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