Generalized stability estimates in inverse transport theory

Guillaume Bal; Alexandre Jollivet

doi:10.3934/ipi.2018003

Article Contents

2018, Volume 12, Issue 1: 59-90. Doi: 10.3934/ipi.2018003

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Generalized stability estimates in inverse transport theory

Guillaume Bal^1, and
Alexandre Jollivet^2, ,

1.
University of Chicago, 5747 S. Ellis Avenue, Jones 120B, Chicago, IL 60637, USA
2.
Laboratoire de Mathématiques Paul Painlevé, CNRS UMR 8524/Université Lille 1 Sciences et Technologies, 59655 Villeneuve d'Ascq Cedex, France

* Corresponding author: Alexandre Jollivet
* Corresponding author: Alexandre Jollivet

Received: March 2017

Published: February 2018

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Abstract

Inverse transport theory concerns the reconstruction of the absorption and scattering coefficients in a transport equation from knowledge of the albedo operator, which models all possible boundary measurements. Uniqueness and stability results are well known and are typically obtained for errors of the albedo operator measured in the $L^1$ sense. We claim that such error estimates are not always very informative. For instance, arbitrarily small blurring and misalignment of detectors result in $O(1)$ errors of the albedo operator and hence in $O(1)$ error predictions on the reconstruction of the coefficients, which are not useful.

This paper revisit such stability estimates by introducing a more forgiving metric on the measurements errors, namely the $1-$Wasserstein distances, which penalize blurring or misalignment by an amount proportional to the width of the blurring kernel or to the amount of misalignment. We obtain new stability estimates in this setting.

We also consider the effect of errors, still measured in the $1-$ Wasserstein distance, on the generation of the probing source. This models blurring and misalignment in the design of (laser) probes and allows us to consider discretized sources. Under appropriate assumptions on the coefficients, we quantify the effect of such errors on the reconstructions.

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Mathematics Subject Classification: Primary: 35R30; Secondary: 45Q05.

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