Reconstruction of a source domain from the Cauchy data

Ikehata, Masaru

doi:10.1088/0266-5611/15/2/019

Cited by 61 publications

(82 citation statements)

References 12 publications

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“…The unknown radii vector r = (r i ) i=1,N , characterising the star-shaped support Ω 2 , together with the unknown MFS coefficients a and b, giving the approximations of the solutions u 1 and u 2 , are simulateneously determined by imposing the transmission conditions (16), (17) and the Cauchy data (7), (8) at the collocating points (26) in a least-squares sense. This results into minimizing the following (regularized) least-squares nonlinear objective function:…”

Section: The Methods Of Fundamental Solutions (Mfs)mentioning

confidence: 99%

“…is an unknown function which is locally Holder continuous with exponent α ∈ (0, 1] in the neighbourhood of each vetex of ∂Ω 2 , then one can recover uniquely the convex hull [Ω 2 ] of Ω 2 in the problem (3)- (8), see Ikehata (1999).…”

Section: Mathematical Formulationmentioning

confidence: 99%

“…Using the decomposition (11) in fact we solve (9), (12), (16), (17) and (70) using the MFS expansions (21) and (22) to determine the solutions u 1 and u h 2 . This recasts to solving the following linear system of 3N equations with 3N unknowns a = (a j ) j=1,2N and b = (b j ) j=1,2N :…”

Section: Example 2 (Reconstructing a Bean Shape Source Domain)mentioning

confidence: 99%

“…the modified Helmholtz equation for the temperature in a fin, see Mamud et al (2014), or acoustics, i.e. the Helmholtz equation for the pressure, see Ikehata (1999).…”

Section: Introductionmentioning

confidence: 99%

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Reconstruction of a source domain from boundary measurements

Bin‐Mohsin

Lesnic

2017

Applied Mathematical Modelling

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Inverse and ill-posed problems which consist of reconstructing the unknown support of a source from a single pair of exterior boundary Cauchy data are investigated. The underlying dependent variable, e.g. potential, temperature or pressure, may satisfy the Laplace, Poisson, Helmholtz or modified Helmholtz partial differential equations (PDEs). For constant coefficients, the solutions of these elliptic PDEs are sought as linear combinations of explicitly available fundamental solutions (free-space Greens functions), as in the method of fundamental solutions (MFS). Prior to this application of the MFS, the free-term inhomogeneity represented by the intensity of the source is removed by the method of particular solutions. The resulting transmission problem then recasts as that of determining the interface between composite materials. In order to ensure a unique solution, the unknown source domain is assumed to be starshaped. This in turn enables its boundary to be parametrised by the radial coordinate, as a function of the polar or, spherical angles. The problem is nonlinear and the numerical solution which minimizes the gap between the measured and the computed data is achieved using the Matlab toolbox routine lsqnonlin which is designed to minimize a sum of squares starting from an initial guess and with no gradient required to be supplied by the user. Simple bounds on the variables can also be prescribed. Since the inverse problem is still ill-posed with respect to small errors in the data and possibly additional ill-conditioning introduced by the spectral feature of the MFS approximation, the least-squares functional which is minimized needs to be augmented with regularizing penalty terms on the MFS coefficients and on the radial function for a stable estimation of these couple of unknowns. Thorough numerical investigations are undertaken for retrieving regular and irregular shapes of the source support from both exact and noisy input data.

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Section: The Methods Of Fundamental Solutions (Mfs)mentioning

confidence: 99%

Section: Mathematical Formulationmentioning

confidence: 99%

Section: Example 2 (Reconstructing a Bean Shape Source Domain)mentioning

confidence: 99%

“…the modified Helmholtz equation for the temperature in a fin, see Mamud et al (2014), or acoustics, i.e. the Helmholtz equation for the pressure, see Ikehata (1999).…”

Section: Introductionmentioning

confidence: 99%

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Reconstruction of a source domain from boundary measurements

Bin‐Mohsin

Lesnic

2017

Applied Mathematical Modelling

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“…To overcome this difficulty, we assume that some a priori information on the source are available, depending on the underlying physical problem. In [12], the sources, in a Helmoltz equation were of either the form F = ρ(x)χ B (x) where B is an open subset of Ω , χ B is the characteristic function of B, or the form…”

Section: Statement and Modeling Of The Problemmentioning

confidence: 99%

Identification of multipolar sources in a bioluminescent tomography problem

2014

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Abstract. Bioluminescence Tomography (BLT) is a recently developed noninvasive imaging tool that allows a direct study of the molecular activity in small animal models. While the forward problem is reduced to a diffusion equation since the scattering phenomena are dominated by the absorption ones in biological tissues, the reconstruction of the distribution of the BLT source is an inverse source problem. In this paper, we concentrate on the reconstruction method where we present an algebraic method allowing to identify the number, the intensities and the location of monopolar sources. Finally, some numerical results are shown proving the robustness of the method.Résumé. Identification de sources multipolaire dans un problème de tomographie par bioluminescence La Tomographie par Bioluminescence (BLT) est un outil non invasive d'imagerie récemment développé dont l'objectif est d'étudier l'activité moléculaire dans de petits animaux.Étant donnée que dans les tissues biologiques, les phénomenes de dispersion sont dominé par l'absorption, le modèle mathématique sous adjacent est réduità uneéquation de diffusion. Le problème inverse en BLT consisteà déterminer de la distribution des sources bioluminescence au moyen de mesure de flux surfacique. Dans le cadre de ce travail, nous considérons des sources multipolaires pour lesquelles nous présentons une méthode d'identification algébrique nous permettant de déterminer leur nombre, leurs intensités et leurs positions. Des résultats numériques sont effectués montrant la robustesse de notre méthode.

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The Enclosure Method and its Applications

Ikehata

2001

Analytic Extension Formulas and Their Applications

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Reconstruction of a source domain from the Cauchy data

Cited by 61 publications

References 12 publications

Reconstruction of a source domain from boundary measurements

Reconstruction of a source domain from boundary measurements

Identification of multipolar sources in a bioluminescent tomography problem

The Enclosure Method and its Applications

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