Transport equation and tomography
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 \(\alpha\): \[ f(r,\omega,E,\alpha)=h(r,\omega,E,\alpha),\quad (r,\omega,E)\in \Gamma^+,\;\alpha \in [0,1]. \] However, in this case all the functions \(\mu\), \(k\), \(J\) are assumed to be unknown. Chapter 1 ends with the description of two algorithms tested by some numerical experiments. The latter part of the book is devoted to the so-called mono-energetic transport problem, i.e. \[ \omega \cdot \nabla_rf(r,\omega) + \mu(r)f(r,\omega)= {1\over 4\pi} \int_{\Omega} k(r,\omega,\omega')f(r,\omega') \,d\omega' + J(r,\omega), \quad r\in G,\;\omega \in \Omega, \tag{4} \] \[ f(r,\omega)=h(r,\omega),\quad (r,\omega)\in \Gamma^-\subset \partial G\times \Omega. \tag{5} \] The domain \(G\) is assumed to contain some inhomogeneities characterized as the boundary \(\partial G_0\) of an open set \(G_0\subset G\) such that \({\overline G}_0={\overline G}\). In this chapter, more related to tomography, functions \(\mu\), \(J\) are assumed to belong to \(C_b(G_0)\) and \(C_b(G_0\times \Omega)\), respectively, while function \((r,\omega)\to k(r,\omega\cdot\omega')\) is assumed in \(C_b(G_0\times \Omega)\) for a.e. \(\omega'\in \Omega\). Under suitable assumption the direct problem (4), (5) is shown to be well-posed. After proving some preliminary results, the authors analyze the non-classical problem related to (4), (5) consisting of recovering the unknown surfaces \(\partial G_0\) accounting for the inhomogeneities in \(G\) when the output flux \[ f(r,\omega)=H(r,\omega),\quad (r,\omega)\in \Gamma^+, \tag{6} \] is prescribed. The introduction of an appropriate function -- the indicator of heterogeneity -- jumping to \(\infty\) only when approaching the inhomogeneity surfaces, allows to establish a uniqueness result for this non-classical problem, provided a so-called visibility condition is satisfied. The necessity of such a condition is investigated when the kernel \(k\) admits the following representation \[ k(r,\omega\cdot \omega)=\mu_s(r)S(\omega\cdot \omega). \tag{7} \] Further the theoretical results are tested by numerical experiments. The book concludes with the comparison of problem (4)--(7) with various approximating problems, such as the diffusion approximation, i.e. \[ -\text{ div}\,[D(r)\nabla u(r)]+\mu_a(r)u(r)=J(r), \] where \(D(r)=[3\mu_s(r)]^{-1}\) \(\mu_a=\mu-\mu_s\), \(u(r)=(4\pi)^{-1}\int_\Omega f(r,\omega')\,d\omega'\).
scientific article; zbMATH DE number 2001458