2012/07/21 by Mark Hubenthal, Hubenthal, Mark
Engineering · Medicine · #35Q20 #35R09 #35R30 (Primary) #35S05 #92C55 #Analysis of PDEs (math.AP) #Calibration and Measurement Techniques #FOS: Mathematics #Numerical Analysis (math.NA) #Optical Imaging and Spectroscopy Techniques #Photoacoustic and Ultrasonic Imaging
paper · pdf · doi:10.48550/arxiv.1207.5141
openalex publication_date 2012/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The inverse source problem for the radiative transfer equation is considered, with partial data. Here we demonstrate numerical computation of the normal operator XV*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 F. Monard and G. Bal. 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 XVf. In particular, we use a truncated Neumann series to estimate XV and XV*, which provides a good approximation of XV*XV with an error of higher Sobolev regularity. This paper provides a visual demonstration of the authors' previous work 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.