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Stability of the Non-abelian X-ray Transform in Dimension $$\ge 3$$

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Peer-reviewed

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Abstract

jats:titleAbstract</jats:title>jats:pNon-abelian jats:italicX</jats:italic>-ray tomography seeks to recover a matrix potential jats:inline-formulajats:alternativesjats:tex-math$$\Phi :M\rightarrow {\mathbb {C}}^{m\times m}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miΦ</mml:mi> mml:mo:</mml:mo> mml:miM</mml:mi> mml:mo→</mml:mo> mml:msup mml:mrow mml:miC</mml:mi> </mml:mrow> mml:mrow mml:mim</mml:mi> mml:mo×</mml:mo> mml:mim</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> in a domain jats:italicM</jats:italic> from measurements of its so-called scattering data jats:inline-formulajats:alternativesjats:tex-math$$C_\Phi $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msub mml:miC</mml:mi> mml:miΦ</mml:mi> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> at jats:inline-formulajats:alternativesjats:tex-math$$\partial M$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:mi∂</mml:mi> mml:miM</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>. For jats:inline-formulajats:alternativesjats:tex-math$$\dim M\ge 3$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:modim</mml:mo> mml:miM</mml:mi> mml:mo≥</mml:mo> mml:mn3</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> (and under appropriate convexity and regularity conditions), injectivity of the forward map jats:inline-formulajats:alternativesjats:tex-math$$\Phi \mapsto C_\Phi $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miΦ</mml:mi> mml:mo↦</mml:mo> mml:msub mml:miC</mml:mi> mml:miΦ</mml:mi> </mml:msub> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> was established in (Paternain et al. in Am J Math 141(6):1707–1750, 2019). The present article extends this result by proving a Hölder-type stability estimate. As an application, a statistical consistency result for jats:inline-formulajats:alternativesjats:tex-math$$\dim M =2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:modim</mml:mo> mml:miM</mml:mi> mml:mo=</mml:mo> mml:mn2</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> (Monard et al. in Commun Pure Appl Math, 2019) is generalised to higher dimensions. The injectivity proof in (Paternain et al. in Am J Math 141(6):1707–1750, 2019) relies on a novel method by Uhlmann and Vasy (Invent Math 205(1):83–120, 2016), which first establishes injectivity in a shallow layer below jats:inline-formulajats:alternativesjats:tex-math$$\partial M$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:mi∂</mml:mi> mml:miM</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> and then globalises this by a layer stripping argument. The main technical contribution of this paper is a more quantitative version of these arguments, in particular, proving uniform bounds on layer depth and stability constants.</jats:p>

Description

Funder: Munro-Greaves Bursary for Pure Mathematics


Funder: Engineering and Physical Sciences Research Council; doi: http://dx.doi.org/10.13039/501100000266

Keywords

4901 Applied Mathematics, 4904 Pure Mathematics, 49 Mathematical Sciences

Journal Title

The Journal of Geometric Analysis

Conference Name

Journal ISSN

1050-6926
1559-002X

Volume Title

31

Publisher

Springer Science and Business Media LLC