Asymptotic behavior of the Eckhoff method for convergence acceleration of trigonometric interpolation

Poghosyan, Arnak

doi:10.1007/s10496-010-0236-3

Cited by 7 publications

(7 citation statements)

References 14 publications

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“…Note that the case of a-priori known nodes has been extensively treated in the literature (see e.g. [1,35] for the most recent results). Using the framework of finite difference calculus, one can easily prove the following result (see [8,Theorem 2.8]).…”

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confidence: 99%

On the Accuracy of Solving Confluent Prony Systems

Batenkov¹,

Yomdin²

2013

SIAM J. Appl. Math.

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In this paper we consider several nonlinear systems of algebraic equations which can be called "Prony-type". These systems arise in various reconstruction problems in several branches of theoretical and applied mathematics, such as frequency estimation and nonlinear Fourier inversion. Consequently, the question of stability of solution with respect to errors in the right-hand side becomes critical for the success of any particular application. We investigate the question of "maximal possible accuracy" of solving Prony-type systems, putting stress on the "local" behavior which approximates situations with low absolute measurement error. The accuracy estimates are formulated in very simple geometric terms, shedding some light on the structure of the problem. Numerical tests suggest that "global" solution techniques such as Prony's algorithm and ESPRIT method are suboptimal when compared to this theoretical "best local" behavior.

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confidence: 99%

On the Accuracy of Solving Confluent Prony Systems

Batenkov¹,

Yomdin²

2013

SIAM J. Appl. Math.

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“…Different approaches are known for convergence acceleration of the truncated Fourier series or trigonometric interpolation for non-periodic smooth functions. We refer to [4][5][6][7][8][9][10][11][12][13][14][15][16][17] with references therein for a detailed review of different methods.…”

Section: Introductionmentioning

confidence: 99%

On some quasi-periodic approximations

Poghosyan

Barkhudaryan

2022

Armen.J.Math.

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Trigonometric approximation or interpolation of a non-smooth function on a finite interval has poor convergence properties. This is especially true for discontinuous functions. The case of infinitely differentiable but non-periodic functions with discontinuous periodic extensions onto the real axis has attracted interest from many researchers. In a series of works, we discussed an approach based on quasi-periodic trigonometric basis functions whose periods are slightly bigger than the length of the approximation interval. We proved validness of the approach for trigonometric interpolations. In this paper, we apply those ideas to classical Fourier expansions.

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“…Methods of approximating piecewise smooth functions have been introduced by Krylov [54], Lanczos [55] and Eckhoff [25][26][27][28][29]61] in the context of Fourier methods (see also [1,11,12,50,63,64]) and by Lipman and Levin [56] in the context of finite-difference methods. These methods have used a polynomial correction term, written as an expansion in a monomial basis, with coefficients determined from suitable jump conditions.…”

Section: Introductionmentioning

confidence: 99%

Discontinuous collocation methods and gravitational self-force applications

Markakis¹,

O’Boyle²,

Brubeck³

et al. 2021

Class. Quantum Grav.

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Numerical simulations of extreme mass ratio inspirals, the most important sources for the LISA detector, face several computational challenges. We present a new approach to evolving partial differential equations occurring in black hole perturbation theory and calculations of the self-force acting on point particles orbiting supermassive black holes. Such equations are distributionally sourced, and standard numerical methods, such as finite-difference or spectral methods, face difficulties associated with approximating discontinuous functions. However, in the self-force problem we typically have access to full a priori information about the local structure of the discontinuity at the particle. Using this information, we show that high-order accuracy can be recovered by adding to the Lagrange interpolation formula a linear combination of certain jump amplitudes. We construct discontinuous spatial and temporal discretizations by operating on the corrected Lagrange formula. In a method-of-lines framework, this provides a simple and efficient method of solving time-dependent partial differential equations, without loss of accuracy near moving singularities or discontinuities. This method is well-suited for the problem of time-domain reconstruction of the metric perturbation via the Teukolsky or Regge–Wheeler–Zerilli formalisms. Parallel implementations on modern CPU and GPU architectures are discussed.

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Asymptotic behavior of the Eckhoff method for convergence acceleration of trigonometric interpolation

Cited by 7 publications

References 14 publications

On the Accuracy of Solving Confluent Prony Systems

On the Accuracy of Solving Confluent Prony Systems

On some quasi-periodic approximations

Discontinuous collocation methods and gravitational self-force applications

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