Formulation of some Improperly Posed Problems of Mathematical Physics

Lavrentiev, Mikhail

doi:10.1007/978-3-642-88210-4_1

Cited by 34 publications

(35 citation statements)

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“…Ill-posedness is a typical drawback of initial value problems for partial differential equations and systems of elliptic type. It was observed by Hadamard [82]- [83], and deeply investigated by John [94]- [96], Lavrentiev jr. [120]- [122], [127], Miller [147]- [148], Payne [163]- [166] and [171]- [172], On Complex-valued 2D Eikonals IV Pucci [176]- [179], and Tikhonov [206]- [210]. Classical surveys on the subject have been authored by Lavrentiev jr., [123], [126], Payne [167], and Tikhonov [211].…”

Section: N)mentioning

confidence: 99%

“…Related information is in [19]- [20], [22], [15], [92], [93], [154], [159], [189] and [205]. A sample of more recent contributions includes [5], [6], [12], [17], [21], [16], [18], [29], [24], [30], [32], [36], [33], [34], [43], [46], [49], [62]- [63], [66]- [67], [84], [85], [89], [108], [109], [127], [132], [142]- [143], [153], [160]- [161], [168]- [170], [173], [183], [185], [186], [1...…”

Section: N)mentioning

confidence: 99%

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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: N)mentioning

confidence: 99%

Section: N)mentioning

confidence: 99%

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

Magnanini

Talenti

2009

Milan J. Math.

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“…Whatever the approach used, it should be remembered that the problem of estimating parameters in (2) is in general ill-posed in the sense of Hadamard (e.g., Lavrentiev, 1967). In our case, this means that parameter estimates are sensitive to certain types of model and measurement errors.…”

Section: Possible Approaches To Parameter Estimationmentioning

confidence: 99%

Identification of motion parameters from spatio-temporal atmospheric temperature patterns

Fradkin¹,

Knox²

1985

Boundary-Layer Meteorol

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In near-calm conditions it is difficult to make direct measurements of atmospheric advection reliably and cheaply, particularly at many points over a large area. An alternative indirect method is examined using time-series measurements of atmospheric temperature (or any other convenient conserved field variable) at points on a spatial grid.System identification methodology is applied to analyze atmospheric temperature data obtained in such near-calm conditions during an experiment with a low-flying helicopter. A three-parameter numerical model of atmospheric advection has been identified in the temperature data, the advection parameters being an eddy diffusion coefficient and horizontal components of the wind velocity. It has been demonstrated by analysis of all the assumptions, and using simulated data, that in this case the standard least-square procedure can be applied to recover sensible parameter estimates, even though the model is of the error-in-variables type and parameters appear to change abruptly at certain moments of time (but by a reasonably small amount). The parameter estimates have been validated against independent data.

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“…Wegmann 16] [12], Stefanescu [14], and Reichel [13]. Hadamard [7], Tikhonov and Arsenin [15], Lavrentiev [10], and Carasso and Stone [3]. Logarithmic convexity and ill-posed problems are discussed by Knops [9].…”

Section: Ix(r) <= T(1)l-tx(r)mentioning

confidence: 99%

Analytic Continuation by the Fast Fourier Transform

Franklin¹

1990

SIAM J. Sci. and Stat. Comput.

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Abstract. The ill-posed problem of analytic continuation is regularized by a prescribed bound. A simple computer algorithm is given that is based on the fast Fourier transform. The algorithm computes m complex values and a positive error bound with time complexity O(rn log rn). As a function of the data errors and the prescribed bound, the numerical error is shown to be consistent with that prescribed by the three-circles principle of Hadamard.

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Formulation of some Improperly Posed Problems of Mathematical Physics

Cited by 34 publications

References 0 publications

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

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

Identification of motion parameters from spatio-temporal atmospheric temperature patterns

Analytic Continuation by the Fast Fourier Transform

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