Numerical recovery of source singularities via the radiative transfer equation with partial data

Mark Hubenthal

Research output: Contribution to journalArticle

Abstract

The inverse source problem for the radiative transfer equation is considered, with partial data. Here we demonstrate numerical computation of the normal operator X*V XV, where XV is the partial data solution operator to the radiative transfer equation. The numerical scheme is based in part on a forward solver designed by Monard and Bal [J. Comput. Phys., 229 (2010), pp. 4952-4979]. We will see that one can detect quite well the visible singularities of an internal optical source f for generic anisotropic k and σ, with or without noise added to the accessible data XV f. In particular, we use a truncated Neumann series to estimate XV and X*V, which provides a good approximation of X*V XV with an error of higher Sobolev regularity. This paper provides a visual demonstration of the author's previous work [M. Hubenthal, Inverse Problems, 27 (2011), 125009] in recovering the microlocally visible singularities of an unknown source from partial data. We also give the theoretical analysis necessary to justify the smoothness of the remainder when approximating the normal operator.

Original languageEnglish (US)
Pages (from-to)1175-1198
Number of pages24
JournalSIAM Journal on Imaging Sciences
Volume6
Issue number3
DOIs
StatePublished - Oct 10 2013

Keywords

  • Inverse problems
  • Microlocal analysis
  • Optical molecular imaging
  • Partial data
  • Radiative transfer equation

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