2008, 2(4): 427-454. doi: 10.3934/ipi.2008.2.427

Stability estimates in stationary inverse transport

1. 

Department of Applied Physics and Applied Mathematics, Columbia University, New York NY, 10027

2. 

Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, United States

Received  April 2008 Revised  September 2008 Published  November 2008

We study the stability of the reconstruction of the scattering and absorption coefficients in a stationary linear transport equation from knowledge of the full albedo operator in dimension $n\geq3$. The albedo operator is defined as the mapping from the incoming boundary conditions to the outgoing transport solution at the boundary of a compact and convex domain. The uniqueness of the reconstruction was proved in [2, 3] and partial stability estimates were obtained in [12] for spatially independent scattering coefficients. We generalize these results and prove an $L^1$-stability estimate for spatially dependent scattering coefficients.
Citation: Guillaume Bal, Alexandre Jollivet. Stability estimates in stationary inverse transport. Inverse Problems & Imaging, 2008, 2 (4) : 427-454. doi: 10.3934/ipi.2008.2.427
[1]

Guillaume Bal, Alexandre Jollivet. Generalized stability estimates in inverse transport theory. Inverse Problems & Imaging, 2018, 12 (1) : 59-90. doi: 10.3934/ipi.2018003

[2]

Michele Di Cristo. Stability estimates in the inverse transmission scattering problem. Inverse Problems & Imaging, 2009, 3 (4) : 551-565. doi: 10.3934/ipi.2009.3.551

[3]

John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems & Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181

[4]

Valter Pohjola. An inverse problem for the magnetic Schrödinger operator on a half space with partial data. Inverse Problems & Imaging, 2014, 8 (4) : 1169-1189. doi: 10.3934/ipi.2014.8.1169

[5]

Ru-Yu Lai. Global uniqueness for an inverse problem for the magnetic Schrödinger operator. Inverse Problems & Imaging, 2011, 5 (1) : 59-73. doi: 10.3934/ipi.2011.5.59

[6]

Mehdi Badra, Fabien Caubet, Jérémi Dardé. Stability estimates for Navier-Stokes equations and application to inverse problems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2379-2407. doi: 10.3934/dcdsb.2016052

[7]

Frederic Weidling, Thorsten Hohage. Variational source conditions and stability estimates for inverse electromagnetic medium scattering problems. Inverse Problems & Imaging, 2017, 11 (1) : 203-220. doi: 10.3934/ipi.2017010

[8]

Pedro Caro. On an inverse problem in electromagnetism with local data: stability and uniqueness. Inverse Problems & Imaging, 2011, 5 (2) : 297-322. doi: 10.3934/ipi.2011.5.297

[9]

Yong-Kum Cho. A quadratic Fourier representation of the Boltzmann collision operator with an application to the stability problem. Kinetic & Related Models, 2012, 5 (3) : 441-458. doi: 10.3934/krm.2012.5.441

[10]

Sombuddha Bhattacharyya. An inverse problem for the magnetic Schrödinger operator on Riemannian manifolds from partial boundary data. Inverse Problems & Imaging, 2018, 12 (3) : 801-830. doi: 10.3934/ipi.2018034

[11]

Albert Clop, Daniel Faraco, Alberto Ruiz. Stability of Calderón's inverse conductivity problem in the plane for discontinuous conductivities. Inverse Problems & Imaging, 2010, 4 (1) : 49-91. doi: 10.3934/ipi.2010.4.49

[12]

Li Liang. Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data. Inverse Problems & Imaging, 2015, 9 (2) : 469-478. doi: 10.3934/ipi.2015.9.469

[13]

Bedr'Eddine Ainseba, Mostafa Bendahmane, Yuan He. Stability of conductivities in an inverse problem in the reaction-diffusion system in electrocardiology. Networks & Heterogeneous Media, 2015, 10 (2) : 369-385. doi: 10.3934/nhm.2015.10.369

[14]

Peijun Li, Ganghua Yuan. Increasing stability for the inverse source scattering problem with multi-frequencies. Inverse Problems & Imaging, 2017, 11 (4) : 745-759. doi: 10.3934/ipi.2017035

[15]

Guillaume Bal, Ian Langmore, François Monard. Inverse transport with isotropic sources and angularly averaged measurements. Inverse Problems & Imaging, 2008, 2 (1) : 23-42. doi: 10.3934/ipi.2008.2.23

[16]

Eemeli Blåsten, Oleg Yu. Imanuvilov, Masahiro Yamamoto. Stability and uniqueness for a two-dimensional inverse boundary value problem for less regular potentials. Inverse Problems & Imaging, 2015, 9 (3) : 709-723. doi: 10.3934/ipi.2015.9.709

[17]

Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems & Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014

[18]

Victor Isakov, Joseph Myers. On the inverse doping profile problem. Inverse Problems & Imaging, 2012, 6 (3) : 465-486. doi: 10.3934/ipi.2012.6.465

[19]

Laurent Amour, Jérémy Faupin. Inverse spectral results in Sobolev spaces for the AKNS operator with partial informations on the potentials. Inverse Problems & Imaging, 2013, 7 (4) : 1115-1122. doi: 10.3934/ipi.2013.7.1115

[20]

Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Inverse problems for evolution equations with time dependent operator-coefficients. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 737-744. doi: 10.3934/dcdss.2016025

2016 Impact Factor: 1.094

Metrics

  • PDF downloads (2)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]