Old And New Algorithms For Toeplitz Systems

Brent, Richard P.

doi:10.1117/12.948485

Cited by 9 publications

(5 citation statements)

References 47 publications

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“…Fortunately, the matrixM N is positive definite Hermitian Toeplitz matrix which can be inverted in O(N 2 ) or even O(N log 2 (N )) floating point operations in a numerically stable way, as opposed to O(N 3 ) for a general matrix. See, e.g., [44,42,4]. Hence the CD kernel can be accurately evaluated even for a very large number of moments (e.g., N 1000).…”

Section: Numerical Computation Of the CD Kernelmentioning

confidence: 99%

“…Since U is unitary, the spectrum of U , σ(U ), lies on the unit circle T in the complex plane and the spectral theorem [10, Part II, X.2.2, Theorem 1, p. 895] ensures the existence of a projection-valued 4 spectral measure E supported on σ(U ) such that…”

Section: Spectral Resolutionmentioning

confidence: 99%

“…The framework developed in our article is simple to implement and readily applicable to high-dimensional systems since the computational complexity is fully determined only by the number of the moments N and in particular is independent of the dimension of the underlying state-space. To be more precise, the complexity is governed by the inversion (or Cholesky factorization) of an N × N Hermitian positive-definite Toeplitz matrix which can be carried out with asymptotic complexity O(N 2 ) or even O(N log 2 (N )) as opposed to O(N 3 ) for a general matrix; see, e.g., [44,42,4].…”

Section: Introductionmentioning

confidence: 99%

“…The fact that f •T ∈ L 2 (ν) for any f ∈ L 2 (ν) follows from the assumption of T being measure preserving with respect to the measure ν 4. By a projection-valued measure, we mean a measure with values in the space of orthogonal projection operators on H.…”

mentioning

confidence: 99%

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Data-driven spectral analysis of the Koopman operator

Korda

Putinar

Mezić

2020

Applied and Computational Harmonic Analysis

105

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Starting from measured data, we develop a method to compute the fine structure of the spectrum of the Koopman operator with rigorous convergence guarantees. The method is based on the observation that, in the measure-preserving ergodic setting, the moments of the spectral measure associated to a given observable are computable from a single trajectory of this observable. Having finitely many moments available, we use the classical Christoffel-Darboux kernel to separate the atomic and absolutely continuous parts of the spectrum, supported by convergence guarantees as the number of moments tends to infinity. In addition, we propose a technique to detect the singular continuous part of the spectrum as well as two methods to approximate the spectral measure with guaranteed convergence in the weak topology, irrespective of whether the singular continuous part is present or not. The proposed method is simple to implement and readily applicable to large-scale systems since the computational complexity is dominated by inverting an N × N Hermitian positive-definite Toeplitz matrix, where N is the number of moments, for which efficient and numerically stable algorithms exist; in particular, the complexity of the approach is independent of the dimension of the underlying state-space. We also show how to compute, from measured data, the spectral projection on a given segment of the unit circle, allowing us to obtain a finite approximation of the operator that explicitly takes into account the point and continuous parts of the spectrum. Finally, we describe a relationship between the proposed method and the so-called Hankel Dynamic Mode Decomposition, providing new insights into the behavior of the eigenvalues of the Hankel DMD operator. A number of numerical examples illustrate the approach, including a study of the spectrum of the lid-driven two-dimensional cavity flow.

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Section: Numerical Computation Of the CD Kernelmentioning

confidence: 99%

Section: Spectral Resolutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Data-driven spectral analysis of the Koopman operator

Korda

Putinar

Mezić

2020

Applied and Computational Harmonic Analysis

105

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“…Therefore, ( 2 ) is required to ensure computational storage and ( 3 ) is required to solve the linear equation system using Gaussian elimination because the impedance matrix does not have useful properties. In contrast, the system matrix ( ) obtained by (15) is a symmetric Toeplitz matrix, which only requires ( ) for computational storage, and several efficient algorithms are available to solve the linear equation system [22]. We compare using PM-P and BM-P with = 0.47 in table 1a.…”

Section: Example 2: a Wideband Kernelmentioning

confidence: 99%

Novel computational approach to solve convolutional integral equations: method of sampling for one dimension [Online First]

Páez-Rueda

Bustamante-Miller

2019

IyU

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Objective: This paper proposes a new methodology to solve one-dimensional cases of integral equations with difference kernels using Fourier analysis. Methodology: In this study, it was proven that any Fredholm equation of the first kind can be expressed as an extended convolutional problem; consequently, a new approach to solve that problem, using the nonideal instantaneous sampling theory and Fourier analysis, can be developed. Results and Discussion: The proposal was extensively evaluated and compared with the method of moments by considering two benchmarks. The first was a narrowband problem related to a second-order differential equation with specific boundaries. The second was a standard wideband problem related to wire antenna radiation in electrodynamics, known as the Pocklington equation. In both cases, we derived new interpretations and different approaches to solve the problems efficiently. Conclusions: The new proposal generalized the method of moments via new interpretations, strategies and design rules. We found that the techniques based on the method of moments are point-matching procedures independent of the weighting functions; the basis functions can be designed as generalized interpolation functions with more information provided by the original domain; the weighting functions literally represent a sampled linear filter; the unknown continuous function can be approximated without using the classical variational approach; and several new strategies based on the Fourier transform can be used to reduce the computational cost

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Fast orthogonal decomposition of rank deficient Toeplitzmatrices

Hansen

Gesmar

1993

Numer Algor

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Old And New Algorithms For Toeplitz Systems

Cited by 9 publications

References 47 publications

Data-driven spectral analysis of the Koopman operator

Data-driven spectral analysis of the Koopman operator

Novel computational approach to solve convolutional integral equations: method of sampling for one dimension [Online First]

Fast orthogonal decomposition of rank deficient Toeplitzmatrices

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