Transport equation and tomography
type: Item
instance of: scholarly article
title: Transport equation and tomography
zbMATH Open document ID: 1042.45009
author: D. S. Anikonov
author: I. V. Prokhorov
author: Andrey Egorovich Kovtanyuk
published in: Inverse and Ill-Posed Problems Series
publication date: 2003-11-06T00:00:00Z
review text: The authors are concerned with direct and inverse problems related to the stationary transport equation \[ \omega \cdot \nabla_rf(r,\omega,E) + \mu(r,E)f(r,\omega,E) \tag{1} \] \[ = {1\over 4\pi}\int_{E_1}^{E_2}dE' \int_{\Omega} k(r,\omega,\omega',E,E')f(r,\omega',E')d\omega' + J(r,\omega,E), \] \[ r\in G,\;\omega \in \Omega,\;E\in [E_1,E_2], \] where \(G\) is a convex bounded domain, \(\Omega=\{\omega \in {\mathbb R}^3: | \omega| =1\}\), \(0<E_1<E_2<+\infty\). In direct problems the attenuation coefficient \(\mu\), the scattering indicator \(k\) and the density of the internal radiation source \(J\) are assumed to be known. Instead in inverse problems the aim is to determine (completely or partially) all or some of functions \(\mu\), \(k\), \(J\) when the radiation \(f\) is measured on some part of the boundary of the medium \(G\). The book begins with the first section entirely devoted to the derivation of equation (1) on the basis of physical laws in order to highlight the physical effects to be taken into account or to be neglected. Then, after adding to (1) the \textit{input flux condition} \[ f(r,\omega,E)=h(r,\omega,E),\quad (r,\omega,E)\in \Gamma^-, \tag{2} \] \(\Gamma^- \subset \partial G\times \Omega\times [E_1,E_2]\) being related to the part \(\Gamma^-_\omega\) of \(\partial G\) where the radiation goes into the the medium, problem (1), (2) is shown to be well-posed. After establishing further properties of the solution \(f\), the first inverse problem consists of proving a uniqueness result for the coefficient \(\mu\) when the output flux condition \[ f(r,\omega,E)=h(r,\omega,E),\quad (r,\omega,E)\in \Gamma^+, \tag{3} \] is prescribed. A uniqueness result for \(\mu\) is proved when the scattering indicatrix admits the representation \(k(r,\omega,\omega',E,E')={\widetilde k}(r,\omega\cdot \omega',E,E')\) and condition (3) is replaced with the following one depending on a parameter \(\alpha\): \[ f(r,\omega,E,\alpha)=h(r,\omega,E,\alpha),\quad (r,\omega,E…
Mathematics Subject Classification ID: 45Q05
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 82C70
Mathematics Subject Classification ID: 45-02
Mathematics Subject Classification ID: 92C55
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 65R32
zbMATH DE Number: 2001458
zbMATH Keywords: Linear transport equations
zbMATH Keywords: well-posedness of direct problems
zbMATH Keywords: uniqueness for inverse problems
zbMATH Keywords: tomography problems
zbMATH Keywords: recovering unknown surfaces
zbMATH Keywords: uniqueness results
zbMATH Keywords: numerical algorithms
zbMATH Keywords: textbook
reviewed by: Alfredo Lorenzi
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2001458