Simultaneously recover sound speed and initial pressure
Fix a ball $B \subset \mathbb{R}^3$. For $j = 1, 2$ let $c_j \in C^\infty(\mathbb{R}^3)$ be positive, equal to $1$ outside $B$, and nontrapping: every unit-speed geodesic of the metric $c_j^{-2}\,dx^2$ eventually leaves each bounded set. Let $f_j \in C_c^\infty(B)$ be real and not identically zero, and let $u_j$ solve $$\partial_t^2 u_j - c_j^2 \Delta u_j = 0, \qquad u_j(0, x) = f_j(x), \qquad \partial_t u_j(0, x) = 0.$$ Each pair $(c_j, f_j)$ thus gives a single measurement, $u_j$ on $(0, \infty) \times \partial B$. When the sound speed is known, Stefanov and Uhlmann reconstructed the initial pressure from such data by a Neumann series; they also showed that the linearization of the joint problem is unstable. Liu and Uhlmann proved uniqueness of both when the sound speed is constant in the region containing the source and the initial pressure is nonnegative and independent of one coordinate. Kian and Uhlmann proved it when $c_1 \le c_2$ or $c_2 \le c_1$ everywhere, for sound speeds with local energy decay, which holds for smooth nontrapping ones, and under local versions of this ordering, including piecewise constant sound speeds. In odd dimensions at least three, Finch and Hickmann proved that the thermoacoustic maps of two distinct radially symmetric nontrapping sound speeds have ranges intersecting only trivially.
Problem. Does $u_1|_{(0,\infty) \times \partial B} = u_2|_{(0,\infty) \times \partial B}$ imply $c_1 = c_2$ and $f_1 = f_2$? No ordering between $c_1$ and $c_2$, and no finite-dimensional model for the $f_j$, is assumed.
Source: Stated by AIM — Open Applied Problems, after Kian and Uhlmann (2025), abstract and §1, who describe recovering both the sound speed and the initial pressure from a single measurement as the general problem of thermoacoustic and photoacoustic tomography.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, photoacoustic tomography, sound speeds, wave equations
References
@article{kian2025,
author = {Kian, Yavar and Uhlmann, Gunther},
title = {Determination of the sound speed and an initial source in photoacoustic tomography},
journal = {Trans. Amer. Math. Soc.},
volume = {378},
number = {8},
year = {2025},
pages = {5329--5353},
doi = {10.1090/tran/9467},
eprint = {2302.03457},
archivePrefix = {arXiv},
}
@article{stefanov2009,
author = {Stefanov, Plamen and Uhlmann, Gunther},
title = {Thermoacoustic tomography with variable sound speed},
journal = {Inverse Problems},
volume = {25},
number = {7},
year = {2009},
pages = {075011},
doi = {10.1088/0266-5611/25/7/075011},
eprint = {0902.1973},
archivePrefix = {arXiv},
}
@article{stefanov2013,
author = {Stefanov, Plamen and Uhlmann, Gunther},
title = {Instability of the linearized problem in multiwave tomography of recovery both the source and the speed},
journal = {Inverse Probl. Imaging},
volume = {7},
number = {4},
year = {2013},
pages = {1367--1377},
doi = {10.3934/ipi.2013.7.1367},
}
@article{liu2015,
author = {Liu, Hongyu and Uhlmann, Gunther},
title = {Determining both sound speed and internal source in thermo- and photo-acoustic tomography},
journal = {Inverse Problems},
volume = {31},
number = {10},
year = {2015},
pages = {105005},
doi = {10.1088/0266-5611/31/10/105005},
eprint = {1502.01172},
archivePrefix = {arXiv},
}
@article{finch2013,
author = {Finch, David and Hickmann, Kyle S.},
title = {Transmission eigenvalues and thermoacoustic tomography},
journal = {Inverse Problems},
volume = {29},
number = {10},
year = {2013},
pages = {104016},
doi = {10.1088/0266-5611/29/10/104016},
eprint = {1306.5481},
archivePrefix = {arXiv},
}
@misc{aim-photoacoustic-joint-recovery,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Simultaneously recover sound speed and initial pressure}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/059-photoacoustic-joint-recovery.md},
note = {Entry 059 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00546, "Simultaneously recover sound speed and initial pressure", clopen.solutions.