Theoretical justification of computational algorithms for problems of analytic continuation

Федотов, Анатолий Михайлович

doi:10.1007/bf00970900

Cited by 2 publications

(7 citation statements)

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“…This approach is conceptually close to that for statistical decision functions [27]. In [7] it was developed for the analytic continuation problem with random errors. The analogous approach to problems with no errors in the data was described in [25] and extended to include the determined error model in [3].…”

Section: Solving Proceduresmentioning

confidence: 93%

“…It is not yet clear whether such correlation operators exist for the risk function (1.9). However, the following theorem (see [7]) holds for (1.9). Theorem 23.…”

Section: Error In Optimal Solving Procedures Random Error Modelmentioning

confidence: 99%

“…For the determined error model in the data, this was used by many authors (see, e.g. [3,9,17,25]); for the random error model, the function was constructed in [7]. According to the obvious inequalities…”

Section: Optimal Solving Proceduresmentioning

confidence: 99%

“…We According to [7], for the algorithm minimizing the functional (2.14) the optimal solving procedure is L a = (σ 2 Ε + Q)-1 g 0 (3.1)…”

Section: Continuation Of Functions In the Unit Diskmentioning

confidence: 99%

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Analytic continuation of functions from discrete sets

Федотов¹

1994

Journal of Inverse and Ill-Posed Problems

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The work deals with the continuation of analytic functions from discrete sets that lie inside the corresponding regularity sets when there are errors in the data. The problem of constructing the optimal continuation algorithms is considered. Estimates of accuracy and stability for the optimal continuation algorithms have been obtained. A new version of the uniqueness theorem for analytic functions, namely, the asymptotic uniqueness theorem, is proposed.

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Section: Solving Proceduresmentioning

confidence: 93%

“…It is not yet clear whether such correlation operators exist for the risk function (1.9). However, the following theorem (see [7]) holds for (1.9). Theorem 23.…”

Section: Error In Optimal Solving Procedures Random Error Modelmentioning

confidence: 99%

Section: Optimal Solving Proceduresmentioning

confidence: 99%

“…We According to [7], for the algorithm minimizing the functional (2.14) the optimal solving procedure is L a = (σ 2 Ε + Q)-1 g 0 (3.1)…”

Section: Continuation Of Functions In the Unit Diskmentioning

confidence: 99%

See 2 more Smart Citations

Analytic continuation of functions from discrete sets

Федотов¹

1994

Journal of Inverse and Ill-Posed Problems

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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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Theoretical justification of computational algorithms for problems of analytic continuation

Cited by 2 publications

References 3 publications

Analytic continuation of functions from discrete sets

Analytic continuation of functions from discrete sets

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

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