Analytic Continuation by the Fast Fourier Transform

Franklin, Joel

doi:10.1137/0911007

Cited by 25 publications

(18 citation statements)

References 14 publications

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“…More information on analytic continuation can be found in [1], [4], [23], [25], [35], [41]- [42], [50], [54]- [55], [69], [70], [74], [75], [81], [88], [124], [125], [130], [126], [149]- [150], [152], [187], [192], [194], [211], [215], [221], [223], [226]. 3.3.…”

Section: 2mentioning

confidence: 99%

On Complex-valued 2D Eikonals. Part Four: Continuation Past a Caustic

Magnanini

Talenti

2009

Milan J. Math.

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Abstract. Theories of monochromatic high-frequency electromagnetic fields have been designed by Felsen, Kravtsov, Ludwig and others with a view to portraying features that are ignored by geometrical optics. These theories have recourse to eikonals that encode information on both phase and amplitude -in other words, are complex-valued. The following mathematical principle is ultimately behind the scenes: any geometric optical eikonal, which conventional rays engender in some light region, can be consistently continued in the shadow region beyond the relevant caustic, provided an alternative eikonal, endowed with a non-zero imaginary part, comes on stage.In the present paper we explore such a principle in dimension 2. We investigate a partial differential system that governs the real and the imaginary parts of complex-valued two-dimensional eikonals, and an initial value problem germane to it. In physical terms, the problem in hand amounts to detecting waves that rise beside, but on the dark side of, a given caustic. In mathematical terms, such a problem shows two main peculiarities: on the one hand, degeneracy near the initial curve; on the other hand, ill-posedness in the sense of Hadamard. We benefit from using a number of technical devices: hodograph transforms, artificial viscosity, and a suitable discretization. Approximate differentiation and a parody of the quasi-reversibility method are also involved. We offer an algorithm that restrains instability and produces effective approximate solutions. Mathematics Subject Classification (2000). Primary 35F25, 35Q60, 78A05; Secondary 65D25.

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Section: 2mentioning

confidence: 99%

On Complex-valued 2D Eikonals. Part Four: Continuation Past a Caustic

Magnanini

Talenti

2009

Milan J. Math.

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“…, m). Note that to (numerically) compute (3.13), the regularization technique developed by Franklin in [13] can be used.…”

Section: The Svd Of the Msr Matrixmentioning

confidence: 99%

Two Numerical Methods for Recovering Small Inclusions from the Scattering Amplitude at a Fixed Frequency

Ammari¹,

Яковлева²,

Lesselier³

2005

SIAM J. Sci. Comput.

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“…The analytic continuation is a classical problem in complex analysis, which is frequently encountered in many practical applications [6,15,19,22], while the stable numerical analytic continuation is a rather difficult problem. In general, this problem is severely ill-posed.…”

Section: Introductionmentioning

confidence: 99%

“…However, it seems that there are few applications of modern theory of regularization methods which have been developed intensively in the last few decades. A simple computer algorithm was given in [6] that is based on the fast Fourier transform. In [2,24], a hypergeometric summation method was used to reconstruct some analytic functions from exponentially spaced samples, where the approximation errors and stability estimates were obtained.…”

Section: Introductionmentioning

confidence: 99%