[Paper Review] Global Lipschitz stability for inverse problems for radiative transport equations

Manabu Machida, Masahiro Yamamoto|arXiv (Cornell University)|Sep 9, 2020

Numerical methods in inverse problems22 references4 citations

TL;DR

This paper establishes global Lipschitz stability for inverse problems in radiative transport equations by developing a novel Carleman estimate with a piecewise linear weight function adapted to velocity domain partitioning. It removes restrictive assumptions on velocity direction found in prior work, requiring only strict positivity of initial data and source factors, enabling stability without solution extension or velocity cone constraints.

ABSTRACT

We consider inverse problems of determining coefficients or time independent factors of source terms in radiative transport equations by means of Carleman estimate. We establish global Lipschitz stability results with an additional condition which requires some strict positivity for initial value or given factor of source, but we need not any extra conditions on domains of velocities, which is the main achievement of this article compared with the existing work by Machida and Yamamoto ({\it Inverse Problems} {\bf 30} 035010, 2014). The proof relies on a Carleman estimate with a piecewise linear weight function according to the partition of the velocity domain.

Motivation & Objective

• To establish global Lipschitz stability for inverse coefficient problems in radiative transport equations under minimal assumptions.
• To eliminate the need for velocity domain restrictions such as (v·γ) > 0, which limited prior applications.
• To develop a Carleman estimate with a piecewise linear weight function tailored to partitioned velocity domains.
• To prove stability results using only the positivity of initial data and source factors, without requiring symmetry or extension of the solution to (−T, T).
• To generalize previous results by Machida and Yamamoto (2014) by removing the need for extra conditions on σ or velocity structure.

Proposed method

• Derive a new Carleman estimate using a piecewise linear weight function φ_i(x,t) = (γ_j·x) − βt, where γ_j are fixed vectors partitioning the velocity domain V.
• Partition the velocity domain V into m subdomains V_i, each associated with a distinct weight function φ_i to handle directional dependence.
• Apply the Carleman estimate to the radiative transport equation with unknown coefficient σ and time-independent source factor.
• Use the positivity of initial data a(x,v) and the factor R(x,v,0) > 0 to control lower-order terms and derive stability estimates.
• Employ energy estimates and integration over time and space to bound the L² norm of the difference in coefficients via boundary measurements.
• Utilize the structure of the albedo-type measurement data on Γ₊ to control the error in coefficient recovery.

Experimental results

Research questions

• RQ1 Can global Lipschitz stability be established for inverse radiative transport problems without requiring the velocity domain to be confined to a cone?
• RQ2 Is it possible to eliminate the need for extending the solution to (−T, T) and the symmetry condition (1.4) in Carleman-based stability proofs?
• RQ3 Can a piecewise linear weight function in the Carleman estimate effectively handle general bounded velocity domains V with 0 ∉ V̄?
• RQ4 How does the positivity of initial data and source factors influence the stability of coefficient recovery?
• RQ5 What is the role of velocity domain partitioning in constructing a valid Carleman estimate for non-stationary radiative transport equations?

Key findings

• Global Lipschitz stability is established for the inverse problem of recovering σ and time-independent source factors in radiative transport equations.
• The stability result holds without requiring the solution to be extended to (−T, T), unlike previous works.
• The method removes the restrictive condition (v·γ) > 0 on the velocity domain, allowing any bounded V with 0 ∉ V̄.
• The key Carleman estimate is constructed using a piecewise linear weight function based on partitioning the velocity domain into m subdomains.
• The proof relies on the strict positivity of initial data a(x,v) and the factor R(x,v,0) on Ω̄ × V̄, which is essential for the stability estimate.
• The final stability estimate bounds the L² norm of the difference in coefficients by a multiple of the measurement error on Γ₊, with exponential dependence on the Carleman parameter s.