Asymptotic Analysis of Numerical Steepest Descent with Path Approximations

Asheim, Andreas; Huybrechs, Daan

doi:10.1007/s10208-010-9068-y

Cited by 38 publications

(34 citation statements)

References 25 publications

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“…We first concentrate on the computation of the inverse Fourier transform, assuming thatq 0 can be computed accurately. Our approach is related to that of Asheim and Huybrechs [6]. To do so, we must truncate the infinite integral.…”

Section: A Steepest Descent-based Numerical Techniquementioning

confidence: 99%

Riemann–Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special Functions

Trogdon¹,

Olver²

2015

197

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The computation of special functions has important implications throughout engineering and the physical sciences. Nonlinear special functions include the solutions of integrable partial differential equations and the Painlevé transcendents. Many problems in water wave theory, nonlinear optics and statistical mechanics are reduced to the study of a nonlinear special function in particular limits. The universal object that these functions share is a Riemann-Hilbert representation: the nonlinear special function can be recovered from the solution of a Riemann-Hilbert problem (RHP). A RHP consists of finding a piecewiseanalytic function in the complex plane when the behavior of its discontinuities is specified. In this dissertation, the applied theory of Riemann-Hilbert problems, using both Hölder and Lebesgue spaces, is reviewed. The numerical solution of RHPs is discussed. Furthermore, the uniform approximation theory for the numerical solution of RHPs is presented, proving that in certain cases the convergence of the numerical method is uniform with respect to a parameter. This theory shares close relation to the method of nonlinear steepest descent for RHPs.The inverse scattering transform for the Korteweg-de Vries and Nonlinear Schrödinger equation is made effective by solving the associated RHPs numerically. This technique is extended to solve the Painlevé II equation numerically. Similar Riemann-Hilbert techniques are used to compute the so-called finite-genus solutions of the Korteweg-de Vries equation. This involves ideas from Riemann surface theory. Finally, the methodology is applied to compute orthogonal polynomials with exponential weights. This allows for the computation of statistical quantities stemming from random matrix ensembles. ACKNOWLEDGMENTS

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Section: A Steepest Descent-based Numerical Techniquementioning

confidence: 99%

Riemann–Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special Functions

Trogdon¹,

Olver²

2015

197

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“…Indeed, we can go a step further and forego an explicit asymptotic expansion of y(t) at the first place. Instead, we go back to the variation-of-constants representation (1.4) and compute the integral therein by any of the many modern quadrature methods for highly oscillatory integrals [6,7].…”

Section: Y(t)mentioning

confidence: 99%

Differential equations with general highly oscillatory forcing terms

Condon

Iserles

Nørsett³

2014

Proc. R. Soc. A.

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The concern of this paper is in expanding and computing initial-value problems of the form y = f (y) + h ω (t), where the function h ω oscillates rapidly for ω 1. Asymptotic expansions for such equations are well understood in the case of modulated Fourier oscillators h ω (t) = m a m (t) e imωt and they can be used as an organizing principle for very accurate and affordable numerical solvers. However, there is no similar theory for more general oscillators, and there are sound reasons to believe that approximations of this kind are unsuitable in that setting. We follow in this paper an alternative route, demonstrating that, for a much more general family of oscillators, e.g. linear combinations of functions of the form e iωg k (t) , it is possible to expand y(t) in a different manner. Each rth term in the expansion is O(ω −1/ς ) for some ς > 0 and it can be represented as an r-dimensional highly oscillatory integral. Because computation of multivariate highly oscillatory integrals is fairly well understood, this provides a powerful method for an effective discretization of a numerical solution for a large family of highly oscillatory ordinary differential equations.

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“…Pathak [18] and Borovikov [21] gave the asymptotic approximation leading terms for resonance points. From Equation (36) in [18], or Equation (70) …”

Section: The Triangular Patch Examplementioning

confidence: 99%

“…However, it generally leads to limited accuracy, especially when the object is not very large. These challenging PO type oscillatory integrals are extensively studied in [30][31][32][33][34][35][36][37]. Relevant mathematical theories and error analysis are developed to provide clearer pictures about their oscillatory behaviors.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Method for Computing Highly Oscillatory Physical Optics Integral

Jiang

Chew

2012

PIER

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Abstract-In this work, we use the numerical steepest descent path (numerical SDP) method in complex analysis theory to calculate the highly oscillatory physical optics (PO) integral with quadratic phase and amplitude variations on the triangular patch. The Stokes' phenomenon will occur due to various asymptotic behaviors on different domains. The stationary phase point contributions are carefully studied by the numerical SDP method and complex analysis using contour deformation. Its result agrees very well with the leading terms of the traditional asymptotic expansion. Furthermore, the resonance points and vertex points contributions from the PO integral are also extracted. Compared with traditional approximate asymptotic expansion approach, our method has significantly improved the PO integral accuracy by one to two digits (10 −1 to 10 −2 ) for evaluating the PO integral. Moreover, the computation effort for the highly oscillatory integral is frequency independent. Numerical results for PO integral on the triangular patch are given to verify the proposed numerical SDP theory.

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Asymptotic Analysis of Numerical Steepest Descent with Path Approximations

Cited by 38 publications

References 25 publications

Riemann–Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special Functions

Riemann–Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special Functions

Differential equations with general highly oscillatory forcing terms

An Efficient Method for Computing Highly Oscillatory Physical Optics Integral

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