### Abstract

We consider the radiative transfer equation (RTE) with two scalings in this paper: One is the diffusive scaling whose macroscopic limit is a diffusion equation, and the other is a highly forward peaked scaling, wherein the scattering term is approximated by a Fokker-Planck operator as a limit. In the inverse setting, we are concerned with reconstructing the scattering and absorption coefficients using boundary measurements. As the measurement is often polluted by errors, both experimental and computational, an important question is to quantify how the error is amplified or suppressed in the process of reconstruction. Since the solution to the forward RTE behaves differ-ently in different regimes, it is expected that stability of the inverse problem will vary accordingly. Particularly, we adopted the linearized approach and showed, in the former case, that the stability degrades when the limit is taken, following a similar approach as in [K. Chen, Q. Li and L. Wang, Inverse Problems, 34 (2018), 025004]. In the latter case, we showed that a full recovery of the scattering coefficient is less possible in the limit.

Original language | English (US) |
---|---|

Pages (from-to) | 2626-2647 |

Number of pages | 22 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 78 |

Issue number | 5 |

DOIs | |

State | Published - Jan 1 2018 |

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### Keywords

- Diffusion limit
- Fokker-Planck limit
- Inverse problem
- Stability
- Transport equation

### Cite this

*SIAM Journal on Applied Mathematics*,

*78*(5), 2626-2647. https://doi.org/10.1137/17M1157969

**Stability of inverse transport equation in diffusion scaling and fokker-planck limit.** / Chen, Ke; Li, Qin; Wang, Li.

Research output: Contribution to journal › Article

*SIAM Journal on Applied Mathematics*, vol. 78, no. 5, pp. 2626-2647. https://doi.org/10.1137/17M1157969

}

TY - JOUR

T1 - Stability of inverse transport equation in diffusion scaling and fokker-planck limit

AU - Chen, Ke

AU - Li, Qin

AU - Wang, Li

PY - 2018/1/1

Y1 - 2018/1/1

N2 - We consider the radiative transfer equation (RTE) with two scalings in this paper: One is the diffusive scaling whose macroscopic limit is a diffusion equation, and the other is a highly forward peaked scaling, wherein the scattering term is approximated by a Fokker-Planck operator as a limit. In the inverse setting, we are concerned with reconstructing the scattering and absorption coefficients using boundary measurements. As the measurement is often polluted by errors, both experimental and computational, an important question is to quantify how the error is amplified or suppressed in the process of reconstruction. Since the solution to the forward RTE behaves differ-ently in different regimes, it is expected that stability of the inverse problem will vary accordingly. Particularly, we adopted the linearized approach and showed, in the former case, that the stability degrades when the limit is taken, following a similar approach as in [K. Chen, Q. Li and L. Wang, Inverse Problems, 34 (2018), 025004]. In the latter case, we showed that a full recovery of the scattering coefficient is less possible in the limit.

AB - We consider the radiative transfer equation (RTE) with two scalings in this paper: One is the diffusive scaling whose macroscopic limit is a diffusion equation, and the other is a highly forward peaked scaling, wherein the scattering term is approximated by a Fokker-Planck operator as a limit. In the inverse setting, we are concerned with reconstructing the scattering and absorption coefficients using boundary measurements. As the measurement is often polluted by errors, both experimental and computational, an important question is to quantify how the error is amplified or suppressed in the process of reconstruction. Since the solution to the forward RTE behaves differ-ently in different regimes, it is expected that stability of the inverse problem will vary accordingly. Particularly, we adopted the linearized approach and showed, in the former case, that the stability degrades when the limit is taken, following a similar approach as in [K. Chen, Q. Li and L. Wang, Inverse Problems, 34 (2018), 025004]. In the latter case, we showed that a full recovery of the scattering coefficient is less possible in the limit.

KW - Diffusion limit

KW - Fokker-Planck limit

KW - Inverse problem

KW - Stability

KW - Transport equation

UR - http://www.scopus.com/inward/record.url?scp=85055788203&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85055788203&partnerID=8YFLogxK

U2 - 10.1137/17M1157969

DO - 10.1137/17M1157969

M3 - Article

VL - 78

SP - 2626

EP - 2647

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 5

ER -