TY - JOUR

T1 - Stabilizing inverse problems by internal data. II: non-local internal data and generic linearized uniqueness

AU - Kuchment, Peter

AU - Steinhauer, Dustin

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: The work of the first author was partly supported by the USNSF Grants DMS0908208 and DMS 1211463, as well as by the DHS Grant 2008-DN-077-ARI018-04. The work of both authors was partially supported by KAUST through IAMCS. Thanks also go to Y. Pinchover, P. Stefanov, G. Uhlmann, and T. Widlak for helpful comments and references.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

PY - 2015/5/10

Y1 - 2015/5/10

N2 - © 2015, Springer Basel. In the previous paper (Kuchment and Steinhauer in Inverse Probl 28(8):084007, 2012), the authors introduced a simple procedure that allows one to detect whether and explain why internal information arising in several novel coupled physics (hybrid) imaging modalities could turn extremely unstable techniques, such as optical tomography or electrical impedance tomography, into stable, good-resolution procedures. It was shown that in all cases of interest, the Fréchet derivative of the forward mapping is a pseudo-differential operator with an explicitly computable principal symbol. If one can set up the imaging procedure in such a way that the symbol is elliptic, this would indicate that the problem was stabilized. In the cases when the symbol is not elliptic, the technique suggests how to change the procedure (e.g., by adding extra measurements) to achieve ellipticity. In this article, we consider the situation arising in acousto-optical tomography (also called ultrasound modulated optical tomography), where the internal data available involves the Green’s function, and thus depends globally on the unknown parameter(s) of the equation and its solution. It is shown that the technique of (Kuchment and Steinhauer in Inverse Probl 28(8):084007, 2012) can be successfully adopted to this situation as well. A significant part of the article is devoted to results on generic uniqueness for the linearized problem in a variety of situations, including those arising in acousto-electric and quantitative photoacoustic tomography.

AB - © 2015, Springer Basel. In the previous paper (Kuchment and Steinhauer in Inverse Probl 28(8):084007, 2012), the authors introduced a simple procedure that allows one to detect whether and explain why internal information arising in several novel coupled physics (hybrid) imaging modalities could turn extremely unstable techniques, such as optical tomography or electrical impedance tomography, into stable, good-resolution procedures. It was shown that in all cases of interest, the Fréchet derivative of the forward mapping is a pseudo-differential operator with an explicitly computable principal symbol. If one can set up the imaging procedure in such a way that the symbol is elliptic, this would indicate that the problem was stabilized. In the cases when the symbol is not elliptic, the technique suggests how to change the procedure (e.g., by adding extra measurements) to achieve ellipticity. In this article, we consider the situation arising in acousto-optical tomography (also called ultrasound modulated optical tomography), where the internal data available involves the Green’s function, and thus depends globally on the unknown parameter(s) of the equation and its solution. It is shown that the technique of (Kuchment and Steinhauer in Inverse Probl 28(8):084007, 2012) can be successfully adopted to this situation as well. A significant part of the article is devoted to results on generic uniqueness for the linearized problem in a variety of situations, including those arising in acousto-electric and quantitative photoacoustic tomography.

UR - http://hdl.handle.net/10754/599716

UR - http://link.springer.com/10.1007/s13324-015-0104-6

UR - http://www.scopus.com/inward/record.url?scp=84944447646&partnerID=8YFLogxK

U2 - 10.1007/s13324-015-0104-6

DO - 10.1007/s13324-015-0104-6

M3 - Article

SN - 1664-2368

VL - 5

SP - 391

EP - 425

JO - Analysis and Mathematical Physics

JF - Analysis and Mathematical Physics

IS - 4

ER -