Analysis & PDE

  • Anal. PDE
  • Volume 6, Number 4 (2013), 751-775.

Cauchy problem for ultrasound-modulated EIT

Guillaume Bal

Full-text: Open access


Ultrasound modulation of electrical or optical properties of materials offers the possibility of devising hybrid imaging techniques that combine the high electrical or optical contrast observed in many settings of interest with the high resolution of ultrasound. Mathematically, these modalities require that we reconstruct a diffusion coefficient σ(x) for xX, a bounded domain in n, from knowledge of σ(x)|u|2(x) for xX, where u is the solution to the elliptic equation σu=0 in X with u=f on X.

This inverse problem may be recast as a nonlinear equation, which formally takes the form of a 0-Laplacian. Whereas p-Laplacians with p>1 are well-studied variational elliptic nonlinear equations, p=1 is a limiting case with a convex but not strictly convex functional, and the case p<1 admits a variational formulation with a functional that is not convex. In this paper, we augment the equation for the 0-Laplacian with Cauchy data at the domain’s boundary, which results in a formally overdetermined, nonlinear hyperbolic equation.

This paper presents existence, uniqueness, and stability results for the Cauchy problem of the 0-Laplacian. In general, the diffusion coefficient σ(x) can be stably reconstructed only on a subset of X described as the domain of influence of the space-like part of the boundary X for an appropriate Lorentzian metric. Global reconstructions for specific geometries or based on the construction of appropriate complex geometric optics solutions are also analyzed.

Article information

Anal. PDE, Volume 6, Number 4 (2013), 751-775.

Received: 3 April 2011
Revised: 18 December 2011
Accepted: 26 June 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B38: Critical points 35R30: Inverse problems 35CXX 35L72: Quasilinear second-order hyperbolic equations

hybrid inverse problems nonlinear hyperbolic equations power density internal functional


Bal, Guillaume. Cauchy problem for ultrasound-modulated EIT. Anal. PDE 6 (2013), no. 4, 751--775. doi:10.2140/apde.2013.6.751.

Export citation


  • G. Alessandrini, “An identification problem for an elliptic equation in two variables”, Ann. Mat. Pura Appl. $(4)$ 145 (1986), 265–295.
  • H. Ammari, E. Bonnetier, Y. Capdeboscq, M. Tanter, and M. Fink, “Electrical impedance tomography by elastic deformation”, SIAM J. Appl. Math. 68:6 (2008), 1557–1573.
  • S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems”, Inverse Problems 25 (2010), 123010.
  • M. Atlan, B. C. Forget, F. Ramaz, A. C. Boccara, and M. Gross, “Pulsed acousto-optic imaging in dynamic scattering media with heterodyne parallel speckle detection”, Optics Letters 30:11 (2005), 1360–1362.
  • G. Bal, “Inverse transport theory and applications”, Inverse Problems 25:5 (2009), 053001, 48.
  • G. Bal, “Hybrid inverse problems and internal functionals”, pp. 325–369 in Inside Out II, edited by G. Uhlmann, Math. Sci. Res. Inst. Publ. 60, Cambridge Univ. Press, 2013.
  • G. Bal and K. Ren, “Multi-source quantitative photoacoustic tomography in a diffusive regime”, Inverse Problems 27:7 (2011), 075003, 20.
  • G. Bal and J. C. Schotland, “Inverse scattering and acousto-optics imaging”, Phys. Rev. Letters 104 (2010), 043902.
  • G. Bal and G. Uhlmann, “Inverse diffusion theory of photoacoustics”, Inverse Problems 26:8 (2010), 085010, 20.
  • G. Bal, E. Bonnetier, F. Monard, and F. Triki, “Inverse diffusion from knowledge of power densities”, preprint, 2011.
  • G. Bal, K. Ren, G. Uhlmann, and T. Zhou, “Quantitative thermo-acoustics and related problems”, Inverse Problems 27:5 (2011), 055007, 15.
  • M. Briane, G. W. Milton, and V. Nesi, “Change of sign of the corrector's determinant for homogenization in three-dimensional conductivity”, Arch. Ration. Mech. Anal. 173:1 (2004), 133–150.
  • L. A. Caffarelli and A. Friedman, “Partial regularity of the zero-set of solutions of linear and superlinear elliptic equations”, J. Differential Equations 60:3 (1985), 420–433.
  • Y. Capdeboscq, J. Fehrenbach, F. de Gournay, and O. Kavian, “Imaging by modification: numerical reconstruction of local conductivities from corresponding power density measurements”, SIAM J. Imaging Sci. 2:4 (2009), 1003–1030.
  • L. C. Evans, Partial differential equations, Graduate Studies in Mathematics 19, American Mathematical Society, Providence, RI, 1998.
  • B. Gebauer and O. Scherzer, “Impedance-acoustic tomography”, SIAM J. Appl. Math. 69:2 (2008), 565–576.
  • D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften 224, Springer, Berlin, 1977.
  • R. Hardt, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and N. Nadirashvili, “Critical sets of solutions to elliptic equations”, J. Differential Geom. 51:2 (1999), 359–373.
  • L. H örmander, The analysis of linear partial differential operators, I: Distribution theory and Fourier analysis, Grundlehren der Mathematischen Wissenschaften 256, Springer, Berlin, 1983.
  • L. Hörmander, The analysis of linear partial differential operators, II: Differential operators with constant coefficients, Grundlehren der Mathematischen Wissenschaften 257, Springer, 1983.
  • L. Hörmander, Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications 26, Springer, 1997.
  • M. Kempe, M. Larionov, D. Zaslavsky, and A. Z. Genack, “Acousto-optic tomography with multiply scattered light”, J. Opt. Soc. Am. A 14:5 (1997).
  • P. Kuchment and L. Kunyansky, “2D and 3D reconstructions in acousto-electric tomography”, Inverse Problems 27:5 (2011), 055013, 21.
  • O. Kwon, E. J. Woo, J.-R. Yoon, and J. K. Seo, “Magnetic resonance electrical impedance tomography (MREIT): simulation study of J-substitution algorithm”, IEEE Trans. Biomed. Eng. 49 (2002), 160–167.
  • A. D. Melas, “An example of a harmonic map between Euclidean balls”, Proc. Amer. Math. Soc. 117:3 (1993), 857–859.
  • M. Morse and S. S. Cairns, Critical point theory in global analysis and differential topology: An introduction, Pure and Applied Mathematics 33, Academic Press, New York, 1969.
  • A. Nachman, A. Tamasan, and A. Timonov, “Conductivity imaging with a single measurement of boundary and interior data”, Inverse Problems 23:6 (2007), 2551–2563.
  • A. Nachman, A. Tamasan, and A. Timonov, “Recovering the conductivity from a single measurement of interior data”, Inverse Problems 25:3 (2009), 035014, 16.
  • O. Scherzer (editor), Handbook of Mathematical Methods in Imaging, Springer, 2011.
  • M. E. Taylor, Partial differential equations, I: Basic theory, Applied Mathematical Sciences 115, Springer, New York, 1996.
  • F. Triki, “Uniqueness and stability for the inverse medium problem with internal data”, Inverse Problems 26:9 (2010), 095014, 11.
  • G. Uhlmann, “Electrical impedance tomography and Calderón's problem”, Inverse Problems 25 (2009), 123011.
  • L. V. Wang, “Ultrasound-mediated biophotonic imaging: a review of acousto-optical tomography and photo-acoustic tomography”, Journal of Disease Markers 19:2–3 (2004), 123–138.
  • H. Zhang and L. V. Wang, “Acousto-electric tomography”, pp. 145–14 in Proc. SPIE, vol. 5320, edited by A. A. Oraevsky and L. V. Wang, 2004.