Identification of the collision kernel in the linear Boltzmann equation by a finite number of measurements on the boundary

Cipolatti, Rolci

doi:10.1590/s0101-82052006000200012

Cited by 3 publications

(2 citation statements)

References 17 publications

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“…That the reconstruction of the optical parameters is greatly improved when increases was demonstrated in the numerical simulations performed in e.g., [15,16]; see also [2] for a similar behavior in the diffusive regime. See also [10] for an approach to the (time-dependent) inverse problem using highly-oscillatory solutions.…”

Section: Introductionmentioning

confidence: 99%

Angular Average of Time-Harmonic Transport Solutions

Bal¹,

Jollivet²,

Langmore³

et al. 2011

Communications in Partial Differential Equations

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We consider the angular averaging of solutions to time-harmonic transport equations. Such quantities model measurements obtained for instance in optical tomography, a medical imaging technique, with frequency-modulated sources. Frequency modulated sources are useful to separate ballistic photons from photons that undergo scattering with the underlying medium. This paper presents a precise asymptotic description of the angularly averaged transport solutions as the modulation frequency tends to . Provided that scattering vanishes in the vicinity of measurements, we show that the ballistic contribution is asymptotically larger than the contribution corresponding to single scattering. Similarly, we show that singly scattered photons also have a much larger contribution to the measurements than multiply scattered photons. This decomposition is a necessary step toward the reconstruction of the optical coefficients from available measurements.

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Section: Introductionmentioning

confidence: 99%

Angular Average of Time-Harmonic Transport Solutions

Bal¹,

Jollivet²,

Langmore³

et al. 2011

Communications in Partial Differential Equations

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show abstract

“…In the case of the dynamic transport equation or linear Boltzmann equation (1.1) with time independent coefficients, unique recovery of the pair pa, kq from knowledge of the albedo operator (1.5) was shown by Choulli and Stefanov in [12], and stable recovery of the absorption coefficient a was proved by Cipolatti, Motta and Roberty in [15] and Cipolatti [14]. This latter result is extended to stable determination of the time-independent pair pa, kq from the albedo map by Bal and Jollivet in [4].…”

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confidence: 99%

An inverse problem for the linear Boltzmann equation with a time-dependent coefficient

Bellassoued

2019

Inverse Problems

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In this paper, we study the stability in the inverse problem of determining the time dependent absorption coefficient appearing in the linear Boltzmann equation, from boundary observations. We prove in dimension n ě 2, that the absorption coefficient can be uniquely determined in a precise subset of the domain, from the albedo operator. We derive a logarithm type stability estimate in the determination of the absorption coefficient from the albedo operator, in a subset of our domain assuming that it is known outside this subset. Moreover, we prove that we can extend this result to the determination of the coefficient in a larger region, and then in the whole domain provided that we have much more data. We prove also an identification result for the scattering coefficient appearing in the linear Boltzmann equation.

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Recovery of coefficients in the linear Boltzmann equation

Bellassoued

2019

Journal of Mathematical Physics

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In this paper, we treat the inverse problem of determining the scattering coefficient and the absorption coefficient appearing in the linear Boltzmann equation via boundary measurements. We show that the gauge-equivalent of the coefficients yields the same albedo operator. The albedo operator is defined as the mapping from the incoming boundary conditions to the outgoing transport solution at the boundary of a bounded and convex domain. We study the stability of the absorption coefficient up to a gauge transformation from the albedo operator, and we prove that the scattering time-dependent coefficient can be uniquely determined in a precise subset of domain, from the albedo operator.

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Identification of the collision kernel in the linear Boltzmann equation by a finite number of measurements on the boundary

Cited by 3 publications

References 17 publications

Angular Average of Time-Harmonic Transport Solutions

Angular Average of Time-Harmonic Transport Solutions

An inverse problem for the linear Boltzmann equation with a time-dependent coefficient

Recovery of coefficients in the linear Boltzmann equation

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