An inverse kinematic problem with internal sources. (May 2015)

• Record Type: Journal Article
• Authors: Pestov, Leonid; Uhlmann, Gunther; Zhou, Hanming
• Is Part Of: Inverse problems. Volume 31:Number 5(2015:May)
• Page Start: 155
• Page End: 94
• Publication Date: 2015-05
• Subjects: Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/0266-5611/31/5/055006
• Languages: English
• ISSNs: 0266-5611

Abstract

<abstract> <title>Abstract</title> <p>Given a bounded domain <italic>M</italic> in <inline-formula><tex-math><?CDATA ${{\mathbb{R}}^{n}}$?></tex-math><?MML <mml:math> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> </mml:math>?><inline-graphic xlink:href="ark:/27927/pgj14jj1tm3" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /></inline-formula> with a conformally Euclidean metric <inline-formula><tex-math><?CDATA $g=\rho \;d{{x}^{2}}$?></tex-math><?MML <mml:math> <mml:mi>g</mml:mi> <mml:mo>=</mml:mo> <mml:mi>&rgr;</mml:mi> <mml:mspace width="0.25em"/> <mml:msup> <mml:mrow> <mml:mi mathvariant="italic">dx</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math>?><inline-graphic xlink:href="ark:/27927/pgj14jj1rqq" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /></inline-formula>, we consider the inverse problem of recovering a semigeodesic neighborhood of a domain <inline-formula><tex-math><?CDATA $\Gamma \subset \partial M$?></tex-math><?MML <mml:math> <mml:mi>Γ</mml:mi> <mml:mo>⊂</mml:mo> <mml:mo>∂</mml:mo> <mml:mi>M</mml:mi> </mml:math>?><inline-graphic xlink:href="ark:/27927/pgj14jj1rm2" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /></inline-formula> and the conformal factor <italic>ρ</italic> in the neighborhood from the travel time data (defined below) and the Cartesian coordinates of <italic>Γ</italic>. We develop an explicit reconstruction procedure for this problem. The key ingredient is the relation between the reconstruction procedure and a Cauchy problem of the conformal Killing equation.</p> </abstract>