Recovery of coefficients in the linear Boltzmann equation

Bellassoued, Mourad

doi:10.1063/1.5116899

Cited by 3 publications

(2 citation statements)

References 33 publications

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“…They also showed that an associated boundary operator, called the Albedo operator, determines the absorption and production parameters for radiative transfer equations in [11] and [12]. Other results for the recovery of the coefficients of a radiative transfer equation in Euclidean space from the Albedo operator have been proven by Tamasan [44], Tamasan and Stefanov [42], Stefanov and Uhlmann [43], Bellassoued and Boughanja [6], and Lai and Li [25]. Under certain curvature constraints on the metric, an inverse problem for a radiative transfer equation was studied in the Riemannian setting by McDowall [35], [36].…”

Section: Introductionmentioning

confidence: 99%

An Inverse Problem for the Relativistic Boltzmann Equation

Balehowsky

Kujanpää

Lassas

et al. 2022

Commun. Math. Phys.

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We consider an inverse problem for the Boltzmann equation on a globally hyperbolic Lorentzian spacetime (M, g) with an unknown metric g. We consider measurements done in a neighbourhood V ⊂ M of a timelike path µ that connects a point x − to a point x + . The measurements are modelled by a source-to-solution map, which maps a source supported in V to the restriction of the solution to the Boltzmann equation to the set V . We show that the source-to-solution map uniquely determines the Lorentzian spacetime, up to an isometry, in the set) is the intersection of the future of the point x − and the past of the point x + , and hence is the maximal set to where causal signals sent from x − can propagate and return to the point x + . The proof of the result is based on using the nonlinearity of the Boltzmann equation as a beneficial feature for solving the inverse problem. Contents 1. Introduction 1.1. Theorem 1.3 proof summary 2. Preliminaries 2.1. Notation 2.2. Lagrangian distributions and Fourier integral operators 3. Vlasov and Boltzmann Kinetic Models 3.1. The Vlasov model 3.2. The Boltzmann model 4. Microlocal analysis of particle interactions 4.1. Existence of transversal collisions 5. Proof of Theorem 1.3 5.1. Delta distribution of a submanifold 5.2. Nonlinear interaction in the inverse problem 5.3. Separation time functions 5.4. Source-to-Solution map determines earliest light observation sets

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Section: Introductionmentioning

confidence: 99%

An Inverse Problem for the Relativistic Boltzmann Equation

Balehowsky

Kujanpää

Lassas

et al. 2022

Commun. Math. Phys.

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show abstract

“…They also showed that an associated boundary operator, called the Albedo operator, determines the absorption and production parameters for radiative transfer equations in [11] and [12]. Other results for the recovery of the coefficients of a radiative transfer equation in Euclidean space from the Albedo operator have been proven by Tamasan [44], Tamasan and Stefanov [42], Stefanov and Uhlmann [43], Bellassoued and Boughanja [6], and Lai and Li [24]. Under certain curvature constraints on the metric, an inverse problem for a radiative transfer equation was studied in the Riemannian setting by McDowall [35], [36].…”

Section: Introductionmentioning

confidence: 99%

An Inverse Problem for the Relativistic Boltzmann Equation

Balehowsky¹,

Kujanpää²,

Lassas³

et al. 2020

Preprint

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We consider an inverse problem for the Boltzmann equation on a globally hyperbolic Lorentzian spacetime (M, g) with an unknown metric g. We consider measurements done in a neighbourhood V ⊂ M of a timelike path µ that connects a point x − to a point x + . The measurements are modelled by a source-to-solution map,which maps a source supported in V to the restriction of the solution of the Boltzmann equation to the set V . We show that the source-to-solution map uniquely determines the Lorentzian spacetime, up to an isometry, in the set) is the intersection of the future of the point x − and the past of the point x + , and hence is the maximal set to where causal signals sent from x − can propagate and return to the point x + . The proof of the result is based on using the nonlinearity of the Boltzmann equation as a beneficial feature for solving the inverse problem. Contents 1. Introduction 1.1. Theorem 1.3 proof summary 2. Preliminaries 2.1. Notation 2.2. Lagrangian distributions and Fourier integral operators 3. Vlasov and Boltzmann Kinetic Models 3.1. The Vlasov model 3.2. The Boltzmann model 4. Microlocal analysis of particle interactions 4.1. Existence of transversal collisions 5. Proof of Theorem 1.3 5.1. Delta distribution of a submanifold 5.2. Scattering of the sources 5.3. Separation time functions 5.4. Source-to-Solution map determines earliest light observation sets

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Stable Determination of Time-Dependent Collision Kernel in the Nonlinear Boltzmann Equation

Lai,

Yan

2024

SIAM J. Appl. Math.

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Recovery of coefficients in the linear Boltzmann equation

Cited by 3 publications

References 33 publications

An Inverse Problem for the Relativistic Boltzmann Equation

An Inverse Problem for the Relativistic Boltzmann Equation

An Inverse Problem for the Relativistic Boltzmann Equation

Stable Determination of Time-Dependent Collision Kernel in the Nonlinear Boltzmann Equation

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