## A Lipschitz stable reconstruction formula for the inverse problem for the wave equation

HTML articles powered by AMS MathViewer

- by Shitao Liu and Lauri Oksanen PDF
- Trans. Amer. Math. Soc.
**368**(2016), 319-335 Request permission

## Abstract:

We consider the problem to reconstruct a wave speed $c \in C^\infty (M)$ in a domain $M \subset \mathbb {R}^n$ from acoustic boundary measurements modelled by the hyperbolic Dirichlet-to-Neumann map $\Lambda$. We introduce a reconstruction formula for $c$ that is based on the Boundary Control method and incorporates features also from the complex geometric optics solutions approach. Moreover, we show that the reconstruction formula is locally Lipschitz stable for a low frequency component of $c^{-2}$ under the assumption that the Riemannian manifold $(M, c^{-2} dx^2)$ has a strictly convex function with no critical points. That is, we show that for all bounded $C^2$ neighborhoods $U$ of $c$, there is a $C^1$ neighborhood $V$ of $c$ and constants $C, R > 0$ such that \begin{align*} |\mathcal {F}\left (\widetilde c^{-2} - c^{-2}\right )(\xi )| \le C e^{2R|\xi |}\left \|\widetilde \Lambda - \Lambda \right \|_*, \quad \xi \in \mathbb {R}^n, \end{align*} for all $\widetilde c \in U \cap V$, where $\widetilde \Lambda$ is the Dirichlet-to-Neumann map corresponding to the wave speed $\widetilde c$ and $\left |\cdot \right |_*$ is a norm capturing certain regularity properties of the Dirichlet-to-Neumann maps.## References

- Giovanni Alessandrini,
*Stable determination of conductivity by boundary measurements*, Appl. Anal.**27**(1988), no. 1-3, 153–172. MR**922775**, DOI 10.1080/00036818808839730 - Giovanni Alessandrini and Sergio Vessella,
*Lipschitz stability for the inverse conductivity problem*, Adv. in Appl. Math.**35**(2005), no. 2, 207–241. MR**2152888**, DOI 10.1016/j.aam.2004.12.002 - Habib Ammari, Hajer Bahouri, David Dos Santos Ferreira, and Isabelle Gallagher,
*Stability estimates for an inverse scattering problem at high frequencies*, J. Math. Anal. Appl.**400**(2013), no. 2, 525–540. MR**3004984**, DOI 10.1016/j.jmaa.2012.10.066 - Michael Anderson, Atsushi Katsuda, Yaroslav Kurylev, Matti Lassas, and Michael Taylor,
*Boundary regularity for the Ricci equation, geometric convergence, and Gel′fand’s inverse boundary problem*, Invent. Math.**158**(2004), no. 2, 261–321. MR**2096795**, DOI 10.1007/s00222-004-0371-6 - Gang Bao and Kihyun Yun,
*On the stability of an inverse problem for the wave equation*, Inverse Problems**25**(2009), no. 4, 045003, 7. MR**2482154**, DOI 10.1088/0266-5611/25/4/045003 - Claude Bardos, Gilles Lebeau, and Jeffrey Rauch,
*Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary*, SIAM J. Control Optim.**30**(1992), no. 5, 1024–1065. MR**1178650**, DOI 10.1137/0330055 - M. I. Belishev,
*An approach to multidimensional inverse problems for the wave equation*, Dokl. Akad. Nauk SSSR**297**(1987), no. 3, 524–527 (Russian); English transl., Soviet Math. Dokl.**36**(1988), no. 3, 481–484. MR**924687** - M. I. Belishev,
*Recent progress in the boundary control method*, Inverse Problems**23**(2007), no. 5, R1–R67. MR**2353313**, DOI 10.1088/0266-5611/23/5/R01 - Mourad Bellassoued and David Dos Santos Ferreira,
*Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map*, Inverse Probl. Imaging**5**(2011), no. 4, 745–773. MR**2852371**, DOI 10.3934/ipi.2011.5.745 - Adi Ben-Israel and Thomas N. E. Greville,
*Generalized inverses*, 2nd ed., CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 15, Springer-Verlag, New York, 2003. Theory and applications. MR**1987382** - Kenrick Bingham, Yaroslav Kurylev, Matti Lassas, and Samuli Siltanen,
*Iterative time-reversal control for inverse problems*, Inverse Probl. Imaging**2**(2008), no. 1, 63–81. MR**2375323**, DOI 10.3934/ipi.2008.2.63 - A. S. Blagoveščenskiĭ,
*The inverse problem of the theory of seismic wave propagation*, Problems of mathematical physics, No. 1: Spectral theory and wave processes (Russian), Izdat. Leningrad. Univ., Leningrad, 1966, pp. 68–81. (errata insert) (Russian). MR**0371360** - A. L. Bukhgeĭm and M. V. Klibanov,
*Uniqueness in the large of a class of multidimensional inverse problems*, Dokl. Akad. Nauk SSSR**260**(1981), no. 2, 269–272 (Russian). MR**630135** - Nicolas Burq and Patrick Gérard,
*Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes*, C. R. Acad. Sci. Paris Sér. I Math.**325**(1997), no. 7, 749–752 (French, with English and French summaries). MR**1483711**, DOI 10.1016/S0764-4442(97)80053-5 - Maarten V. de Hoop, Lingyun Qiu, and Otmar Scherzer,
*Local analysis of inverse problems: Hölder stability and iterative reconstruction*, Inverse Problems**28**(2012), no. 4, 045001, 16. MR**2892888**, DOI 10.1088/0266-5611/28/4/045001 - Thomas Duyckaerts, Xu Zhang, and Enrique Zuazua,
*On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**25**(2008), no. 1, 1–41. MR**2383077**, DOI 10.1016/j.anihpc.2006.07.005 - Heinz W. Engl, Martin Hanke, and Andreas Neubauer,
*Regularization of inverse problems*, Mathematics and its Applications, vol. 375, Kluwer Academic Publishers Group, Dordrecht, 1996. MR**1408680**, DOI 10.1007/978-94-009-1740-8 - R. Gulliver, I. Lasiecka, W. Littman, and R. Triggiani,
*The case for differential geometry in the control of single and coupled PDEs: the structural acoustic chamber*, Geometric methods in inverse problems and PDE control, IMA Vol. Math. Appl., vol. 137, Springer, New York, 2004, pp. 73–181. MR**2169903**, DOI 10.1007/978-1-4684-9375-7_{5} - Robert Gulliver and Walter Littman,
*Chord uniqueness and controllability: the view from the boundary. I*, Differential geometric methods in the control of partial differential equations (Boulder, CO, 1999) Contemp. Math., vol. 268, Amer. Math. Soc., Providence, RI, 2000, pp. 145–175. MR**1804794**, DOI 10.1090/conm/268/04312 - Oleg Yu. Imanuvilov and Masahiro Yamamoto,
*Global uniqueness and stability in determining coefficients of wave equations*, Comm. Partial Differential Equations**26**(2001), no. 7-8, 1409–1425. MR**1855284**, DOI 10.1081/PDE-100106139 - Victor Isakov,
*Inverse problems for partial differential equations*, 2nd ed., Applied Mathematical Sciences, vol. 127, Springer, New York, 2006. MR**2193218** - Victor Isakov,
*Increasing stability for the Schrödinger potential from the Dirichlet-to-Neumann map*, Discrete Contin. Dyn. Syst. Ser. S**4**(2011), no. 3, 631–640. MR**2746425**, DOI 10.3934/dcdss.2011.4.631 - Saichi Izumino,
*Convergence of generalized inverses and spline projectors*, J. Approx. Theory**38**(1983), no. 3, 269–278. MR**705545**, DOI 10.1016/0021-9045(83)90133-8 - Alexander Katchalov, Yaroslav Kurylev, and Matti Lassas,
*Inverse boundary spectral problems*, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 123, Chapman & Hall/CRC, Boca Raton, FL, 2001. MR**1889089**, DOI 10.1201/9781420036220 - Mohammad A. Kazemi and Michael V. Klibanov,
*Stability estimates for ill-posed Cauchy problems involving hyperbolic equations and inequalities*, Appl. Anal.**50**(1993), no. 1-2, 93–102. MR**1281205**, DOI 10.1080/00036819308840186 - I. Lasiecka, J.-L. Lions, and R. Triggiani,
*Nonhomogeneous boundary value problems for second order hyperbolic operators*, J. Math. Pures Appl. (9)**65**(1986), no. 2, 149–192. MR**867669** - M. M. Lavrent′ev, V. G. Romanov, and S. P. Shishat⋅skiĭ,
*Ill-posed problems of mathematical physics and analysis*, Translations of Mathematical Monographs, vol. 64, American Mathematical Society, Providence, RI, 1986. Translated from the Russian by J. R. Schulenberger; Translation edited by Lev J. Leifman. MR**847715**, DOI 10.1090/mmono/064 - J.-L. Lions,
*Exact controllability, stabilization and perturbations for distributed systems*, SIAM Rev.**30**(1988), no. 1, 1–68. MR**931277**, DOI 10.1137/1030001 - Shitao Liu and Roberto Triggiani,
*Global uniqueness and stability in determining the damping coefficient of an inverse hyperbolic problem with nonhomogeneous Neumann B.C. through an additional Dirichlet boundary trace*, SIAM J. Math. Anal.**43**(2011), no. 4, 1631–1666. MR**2821598**, DOI 10.1137/100808988 - Niculae Mandache,
*Exponential instability in an inverse problem for the Schrödinger equation*, Inverse Problems**17**(2001), no. 5, 1435–1444. MR**1862200**, DOI 10.1088/0266-5611/17/5/313 - Luc Miller,
*Escape function conditions for the observation, control, and stabilization of the wave equation*, SIAM J. Control Optim.**41**(2002), no. 5, 1554–1566. MR**1971962**, DOI 10.1137/S036301290139107X - Sei Nagayasu, Gunther Uhlmann, and Jenn-Nan Wang,
*Increasing stability in an inverse problem for the acoustic equation*, Inverse Problems**29**(2013), no. 2, 025012, 11. MR**3020433**, DOI 10.1088/0266-5611/29/2/025012 - Rakesh,
*Reconstruction for an inverse problem for the wave equation with constant velocity*, Inverse Problems**6**(1990), no. 1, 91–98. MR**1036380**, DOI 10.1088/0266-5611/6/1/009 - Luca Rondi,
*A remark on a paper by G. Alessandrini and S. Vessella: “Lipschitz stability for the inverse conductivity problem” [Adv. in Appl. Math. 35 (2005), no. 2, 207–241; MR2152888]*, Adv. in Appl. Math.**36**(2006), no. 1, 67–69. MR**2198854**, DOI 10.1016/j.aam.2004.12.003 - Plamen Stefanov and Gunther Uhlmann,
*Recovery of a source term or a speed with one measurement and applications*, Trans. Amer. Math. Soc.**365**(2013), no. 11, 5737–5758. MR**3091263**, DOI 10.1090/S0002-9947-2013-05703-0 - Plamen Stefanov and Gunther Uhlmann,
*Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media*, J. Funct. Anal.**154**(1998), no. 2, 330–358. MR**1612709**, DOI 10.1006/jfan.1997.3188 - Plamen Stefanov and Gunther Uhlmann,
*Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map*, Int. Math. Res. Not.**17**(2005), 1047–1061. MR**2145709**, DOI 10.1155/IMRN.2005.1047 - Plamen Stefanov and Gunther Uhlmann,
*Thermoacoustic tomography with variable sound speed*, Inverse Problems**25**(2009), no. 7, 075011, 16. MR**2519863**, DOI 10.1088/0266-5611/25/7/075011 - Zi Qi Sun,
*On continuous dependence for an inverse initial-boundary value problem for the wave equation*, J. Math. Anal. Appl.**150**(1990), no. 1, 188–204. MR**1059582**, DOI 10.1016/0022-247X(90)90207-V - John Sylvester and Gunther Uhlmann,
*A global uniqueness theorem for an inverse boundary value problem*, Ann. of Math. (2)**125**(1987), no. 1, 153–169. MR**873380**, DOI 10.2307/1971291 - Daniel Tataru,
*Unique continuation for solutions to PDE’s; between Hörmander’s theorem and Holmgren’s theorem*, Comm. Partial Differential Equations**20**(1995), no. 5-6, 855–884. MR**1326909**, DOI 10.1080/03605309508821117 - D. Tataru,
*Carleman estimates and unique continuation for solutions to boundary value problems*, J. Math. Pures Appl. (9)**75**(1996), no. 4, 367–408. MR**1411157** - Roberto Triggiani and P. F. Yao,
*Carleman estimates with no lower-order terms for general Riemann wave equations. Global uniqueness and observability in one shot*, Appl. Math. Optim.**46**(2002), no. 2-3, 331–375. Special issue dedicated to the memory of Jacques-Louis Lions. MR**1944764**, DOI 10.1007/s00245-002-0751-5 - G. Uhlmann. Electrical impedance tomography and calderon’s problem.
*Inverse Problems*, 25(12):123011, 2009.

## Additional Information

**Shitao Liu**- Affiliation: Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 FI-00014 Helsinki, Finland
- Address at time of publication: Department of Mathematical Sciences, Clemson University, Clemson, South Carolina 29634
- Email: shitao.liu@helsinki.fi, liul@clemson.edu
**Lauri Oksanen**- Affiliation: Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 FI-00014 Helsinki, Finland
- Address at time of publication: Department of Mathematics, University College London, Gower Street, WC1E 6BT, London, United Kingdom
- MR Author ID: 906909
- ORCID: 0000-0002-3228-7507
- Email: lauri.oksanen@helsinki.fi, l.oksanen@ucl.ac.uk
- Received by editor(s): October 3, 2012
- Received by editor(s) in revised form: November 2, 2013
- Published electronically: April 20, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**368**(2016), 319-335 - MSC (2010): Primary 35R30
- DOI: https://doi.org/10.1090/tran/6332
- MathSciNet review: 3413865