Inverse problems involving the one-way wave equation: medium function reconstruction

Wall, David J. N.; Lundstedt, Jonas

doi:10.1016/s0378-4754(99)00100-7

Cited by 4 publications

(7 citation statements)

References 23 publications

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“…This is achieved by constructing {J δ c 0 (t i )} N i=0 and then using central differencing to estimate the numerical derivatives. The extension and mollification of the data functions utilizes equations (11) and (12).…”

Section: Numerical Methods and Resultsmentioning

confidence: 99%

“…where H denotes the Heaviside function. If radial dependence in the downstream concentration is available, then reconstruction of the velocity profile is a simple well-posed problem [11]. However in applications radial concentration measurements are unavailable, and the concentration measurement at the station x = is assayed over the whole pipe.…”

Section: The Mass Transport Equationsmentioning

confidence: 99%

See 1 more Smart Citation

Fluid velocity profile reconstruction for non-Newtonian shear dispersive flow

Shorten

Wall

2001

Journal of Applied Mathematics and Decision Sciences

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An inverse problem associated with the mass transport of a material concentration down a pipe where the flowing non-Newtonian medium has a two-dimensional velocity profile is examined. The problem of determining the two-dimensional fluid velocity profile from temporally varying cross-sectional average concentration measurements at upstream and downstream locations is considered. The special case of a known input upstream concentration with a time zero step, and a strictly decreasing velocity profile is shown to be a well-posed problem. This inverse problem is in general ill-posed and mollification is used to obtain a well conditioned problem.

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Section: Numerical Methods and Resultsmentioning

confidence: 99%

Section: The Mass Transport Equationsmentioning

confidence: 99%

Fluid velocity profile reconstruction for non-Newtonian shear dispersive flow

Shorten

Wall

2001

Journal of Applied Mathematics and Decision Sciences

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“…Regularization of this operator, and hence T −1 , is necessary to restore continuity for this problem that is of mild ill-posedness. Mollification is one of the several techniques available and has been used for differentiation-type operators in various inverse problems [28,49,38].…”

Section: Mapping Properties Of the Inverse Stiffness Mapmentioning

confidence: 99%

On an Inverse Problem from Magnetic Resonance Elastic Imaging

Wall¹,

Olsson²,

Houten³

2011

SIAM J. Appl. Math.

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Abstract. The imaging problem of elastography is an inverse problem. The nature of an inverse problem is that it is ill-conditioned. We consider properties of the mathematical map which describes how the elastic properties of the tissue being reconstructed vary with the field measured by magnetic resonance imaging (MRI). This map is a nonlinear mapping, and our interest is in proving certain conditioning and regularity results for this operator which occurs implicitly in this problem of imaging in elastography. In this treatment we consider the tissue to be linearly elastic, isotropic, and spatially heterogeneous. We determine the conditioning of this problem of function reconstruction, in particular for the stiffness function. We further examine the conditioning when determining both stiffness and density. We examine the Fréchet derivative of the nonlinear mapping, which enables us to describe the properties of how the field affects the individual maps to the stiffness and density functions. We illustrate how use of the implicit function theorem can considerably simplify the analysis of Fréchet differentiability and regularity properties of this underlying operator. We present new results which show that the stiffness map is mildly ill-posed, whereas the density map suffers from medium ill-conditioning. Computational work has been done previously to study the sensitivity of these maps, but our work here is analytical. The validity of the Newton-Kantorovich and optimization methods for the computational solution of this inverse problem is directly linked to the Fréchet differentiability of the appropriate nonlinear operator, which we justify.

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“…It is assumed throughout this paper that the advective medium also allows diffusion. If this is not the case, the problem is considerably simpler and can be solved by the techniques illustrated in Connolly & Wall (1997) and Wall & Lundstedt (1998). In many other applications, it is required to estimate an input signal given a signal that has been modified by transmission through a distorting system (Eldén 1988;Kristensson & Wall 1998;Lundstedt & He 1994, 1997Murio & Roth 1988;Wall 1997;Weber 1981).…”

Section: Introductionmentioning

confidence: 99%

“…Whenv tends to zero, the reconstruction problem reverts to the one for a pure diffusion problem, as considered in (I). For non-zerov and small κ, the problem may be solved without regularization by techniques similar to those in Wall & Lundstedt (1998). † However, for moderate values of κ, techniques that restore the continuity of the solution on the data, like those discussed in this paper, must be used.…”

Section: Introductionmentioning

confidence: 99%

Signal restoration after transmission through an advective and diffusive medium

Shorten

Wall

2004

Proc. R. Soc. Lond. A

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ABSTRACT. Inverse problem, regularisation, singular perturbation, wave splitting, wave propagators, square root operator, inverse mass transport This paper considers an inverse problem associated with mass transport in a pipe. It illustrates how wave splitting techniques can be utilised for an inverse problem associated with one-dimensional mass transport processes. This is done by using a generalisation of Fick's law which introduces a relaxation parameter into the problem, so converting the parabolic partial differential equation by a singular perturbation into a hyperbolic one. This generalised law by ensuring finite mass flux propagation speeds, enables a stable equation to be utilised to reconstruct the interior boundary condition; so providing a regularised solution to the inverse problem. Theoretical results for the solution of the inverse problem are also developed. INTRODUCTIONThe perifusion apparatus is an experimental in vitro tool used by endocrinologists to model information transfer in endocrine systems (Mcintosh & Mcintosh, 1983;Mcintosh, Mcintosh, & Kean, 1984;Evans et al., 1985). The major drawback of the perifusion system derives from the dispersion, diffusion and mixing of the hormone within the apparatus which distort the original released hormone concentration profile. Recently mathematical techniques have been applied to improving this situation (Shorten & Wall, 2001). This paper presents another approach to the inverse problem of concentration signal restoration after signal transmission through an advective and diffusive medium, with applications to perifusion.The problem considered here has direct application to a related problem involving the estimation of secretion of adrenocorticotropic hormone (ACTH) from the pituitary gland. In this problem, assays of the blood flow are taken downstream from the pituitary in a horse, which is secreting ACTH, and it is then required to estimate the concentration of ACTH at the pituitary, (Alexander, Irvine, Liversey, & Donald, 1988). This in vivo experimental technique poses similar problems to the perifusion apparatus mentioned previously, but the flow situation is more complicated.When a mass concentration of a material is transported within a fluid of a different material, the estimation of the final temporal profile, downstream of the injection point, from the knowledge of the initial injection temporal profile, is a relatively straight-forward problem; it is required Date: September 9, 2002. 1991. to solve a well-posed parabolic partial differential equation. This is termed a direct problem. However, when a mass concentration of a material flowing and diffusing within an advecting fluid is measured downstream of the injection point, the estimation of the initial injection profile is a difficult problem. This is termed an inverse problem; in particular an inverse source problem. This problem is difficult because it is ill-posed and besides the difficulty of problem formulation, it must also be made well-posed.It is assumed throughout this paper ...

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Inverse problems involving the one-way wave equation: medium function reconstruction

Cited by 4 publications

References 23 publications

Fluid velocity profile reconstruction for non-Newtonian shear dispersive flow

Fluid velocity profile reconstruction for non-Newtonian shear dispersive flow

On an Inverse Problem from Magnetic Resonance Elastic Imaging

Signal restoration after transmission through an advective and diffusive medium

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