Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Transport models for wave propagation in scattering media with nonlinear absorption

Joseph Kraisler, Wei Li, Kui Ren, John C. Schotland, Yimin Zhong

TL;DR

The paper addresses wave propagation in highly scattering media with nonlinear absorption and the corresponding inverse problem of recovering absorption coefficients from internal data. It develops semilinear radiative transport models via the Wigner transform and multiscale analysis, and analyzes their diffusion limits under strong scattering. Key contributions include the formal derivation of RTEs with quadratic and higher-order absorption, diffusion-limit results, and a rigorous uniqueness result for identifying nonlinear absorption from internal data using a fixed-point framework. The work provides a solid theoretical foundation for semilinear transport models applicable to multi-photon imaging and related technologies.

Abstract

This work considers the propagation of high-frequency waves in highly-scattering media where physical absorption of a nonlinear nature occurs. Using the classical tools of the Wigner transform and multiscale analysis, we derive semilinear radiative transport models for the phase-space intensity and the diffusive limits of such transport models. As an application, we consider an inverse problem for the semilinear transport equation, where we reconstruct the absorption coefficients of the equation from a functional of its solution. We obtain a uniqueness result on the inverse problem.

Transport models for wave propagation in scattering media with nonlinear absorption

TL;DR

The paper addresses wave propagation in highly scattering media with nonlinear absorption and the corresponding inverse problem of recovering absorption coefficients from internal data. It develops semilinear radiative transport models via the Wigner transform and multiscale analysis, and analyzes their diffusion limits under strong scattering. Key contributions include the formal derivation of RTEs with quadratic and higher-order absorption, diffusion-limit results, and a rigorous uniqueness result for identifying nonlinear absorption from internal data using a fixed-point framework. The work provides a solid theoretical foundation for semilinear transport models applicable to multi-photon imaging and related technologies.

Abstract

This work considers the propagation of high-frequency waves in highly-scattering media where physical absorption of a nonlinear nature occurs. Using the classical tools of the Wigner transform and multiscale analysis, we derive semilinear radiative transport models for the phase-space intensity and the diffusive limits of such transport models. As an application, we consider an inverse problem for the semilinear transport equation, where we reconstruct the absorption coefficients of the equation from a functional of its solution. We obtain a uniqueness result on the inverse problem.
Paper Structure (11 sections, 6 theorems, 98 equations)

This paper contains 11 sections, 6 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Derivation of the transport equation
  3. Multiscale expansion
  4. Extension to higher order absorption
  5. The nonlinear RTE in the standard form
  6. The diffusion limit
  7. Asymptotic expansion
  8. The case of degenerate coefficients
  9. An application in inverse problems
  10. Concluding remarks
  11. Proof of inequality

Key Result

Lemma 3.1

Let $\mathcal{M}$ be a bounded convex open subset of a real Banach space, and $F : \overline{ \mathcal{M}} \to \overline{\mathcal{M}}$ is a compact continuous map which is continuously Fréchet differentiable on $\mathcal{M}$. If (i) for each $m\in \mathcal{M}$, $1$ is not an eigenvalue of $F'(m)$, a

Theorems & Definitions (11)

  • Lemma 3.1: Kellogg Kellogg-PAMS76
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • Theorem 3.4
  • proof
  • Remark 3.5
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • ...and 1 more