April  2017, 11(2): 277-304. doi: 10.3934/ipi.2017014

Applications of CGO solutions to coupled-physics inverse problems

1. 

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA

2. 

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA

3. 

Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, MN 55455, USA

4. 

Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

5. 

Department of Mathematics, Northeastern University, 360 Huntington Ave., Boston MA 02115, USA

Received  December 2015 Published  March 2017

Fund Project: R.-Y. L. was partly supported by the AMS-Simons Travel Grants. TZ. was supported by NSF grant DMS-1501049 and Alfred P. Sloan Research Fellowship FR-2015-65641.

This paper surveys inverse problems arising in several coupled-physics imaging modalities for both medical and geophysical purposes. These include Photo-acoustic Tomography (PAT), Thermo-acoustic Tomography (TAT), Electro-Seismic Conversion, Transient Elastrography (TE) and Acousto-Electric Tomography (AET). These inverse problems typically consists of multiple inverse steps, each of which corresponds to one of the wave propagations involved. The review focuses on those steps known as the inverse problems with internal data, in which the complex geometrical optics (CGO) solutions to the underlying equations turn out to be useful in showing the uniqueness and stability in determining the desired information.

Citation: Ilker Kocyigit, Ru-Yu Lai, Lingyun Qiu, Yang Yang, Ting Zhou. Applications of CGO solutions to coupled-physics inverse problems. Inverse Problems and Imaging, 2017, 11 (2) : 277-304. doi: 10.3934/ipi.2017014
References:
[1]

S. Acosta and C. Montalto, Multiwave imaging in an enclosure with variable wave speed, Inverse Problems, 31 (2015), 065009, 12pp.  doi: 10.1088/0266-5611/31/6/065009.

[2]

G. Alessandrini, Stable determination of conductivity by boundary measurements, App. Anal., 27 (1988), 153-172.  doi: 10.1080/00036818808839730.

[3]

H. AmmariE. BonnetierY. CapdeboscqM. Tanter and M. Fink, Electrical impedance tomography by elastic deformation, SIAM Journal on Applied Mathematics, 68 (2008), 1557-1573.  doi: 10.1137/070686408.

[4]

H. Ammari, E. Bossy, V. Jugnon and H. Kang, Mathematical models in photo-acoustic imaging of small absorbers, SIAM Review.

[5]

G. Bal, Hybrid inverse problems and internal functionals, Inside Out II, MSRI Publications, 60 (2013), 325-368. 

[6]

G. Bal, Cauchy problem and ultrasound modulated EIT, Analysis and PDE, 6 (2013), 751-775.  doi: 10.2140/apde.2013.6.751.

[7]

G. Bal, Hybrid inverse problems and systems of partial differential equations, Contemp. Math., 615 (2014), 15pp.  doi: 10.1090/conm/615/12289.

[8]

G. BalC. BellisS. Imperiale and F. Monard, Reconstruction of moduli in isotropic linear elasticity from full-field measurements, Inverse Problems, 30 (2014), 125004, 22pp.  doi: 10.1088/0266-5611/30/12/125004.

[9]

G. BalE. BonnetierF. Monard and F. Triki, Inverse diffusion from knowledge of power densities, Inverse Problems and Imaging, 7 (2013), 353-375.  doi: 10.3934/ipi.2013.7.353.

[10]

G. Bal and C. Guo, Imaging of complex-valued tensors for two-dimensional Maxwell's equations, accepted by Journal of Inverse and Ill-posed Problems.

[11]

G. Bal and C. Guo, Reconstruction of complex-valued tensors in the {M}axwell system from knowledge of internal magnetic fields, Inverse Problems and Imaging, 8 (2014), 1033-1051.  doi: 10.3934/ipi.2014.8.1033.

[12]

G. BalC. Guo and F. Monard, Imaging of anisotropic conductivities from current densities in two dimensions, SIAM J. Imaging Sci., 7 (2014), 2538-2557.  doi: 10.1137/140961754.

[13]

G. BalC. Guo and F. Monard, Inverse anisotropic conductivity from internal current densities, Inverse Problems, 30 (2014), 025001, 21pp.  doi: 10.1088/0266-5611/30/2/025001.

[14]

G. BalC. Guo and F. Monard, Linearized internal functional for anisotropic conductivities, Inverse Problems and Imaging, 8 (2014), 1-22.  doi: 10.3934/ipi.2014.8.1.

[15]

G. Bal and F. Monard, Inverse diffusion problems with redundant internal information, Inverse Problems and Imaging, 6 (2012), 289-313.  doi: 10.3934/ipi.2012.6.289.

[16]

G. BalF. Monard and G. Uhlmann, Reconstruction of a fully anisotropic elasticity tensor from knowledge of displacement fields, SIAM J. Applied Math., 75 (2015), 2214-2231.  doi: 10.1137/151005269.

[17]

G. Bal and K. Ren, Multi-source quantitative pat in diffusive regime, Inverse Problems, 27 075003.

[18]

G. Bal and K. Ren, On multi-spectral quantitative photoacoustic tomography, Inverse Problems, 28 025010.

[19]

G. BalK. RenG. Uhlmann and T. Zhou, Quantitative thermo-acoustics and related problems, Inverse Problems, 27 (2011), 055007, 15pp.  doi: 10.1088/0266-5611/27/5/055007.

[20]

G. Bal and G. Uhlmann, Inverse diffusion theory for photoacoustics Inverse Problems, 26 (2010), 085010. doi: 10.1088/0266-5611/26/8/085010.

[21]

G. Bal and G. Uhlmann, Reconstruction of coefficients in scalar second-order elliptic equations from knowledge of their solutions, Comm. on Pure and Applied Math, 66 (2013), 1629-1652.  doi: 10.1002/cpa.21453.

[22]

G. Bal and T. Zhou, Hybrid inverse problems for a system of Maxwell's equations, Inverse Problems, 30 (2014), 055013, 17pp.  doi: 10.1088/0266-5611/30/5/055013.

[23]

G. Bal and F. Monard, Inverse anisotropic diffusion from power density measurements in two dimensions, Inverse Problems, 28 (2012), 084001, 20pp.  doi: 10.1088/0266-5611/28/8/084001.

[24]

G. Bal and F. Monard, Inverse anisotropic conductivity from power density measurements in dimensions $ n≥q3$, Comm. Partial Differential Equations, 38 (2013), 1183-1207.  doi: 10.1080/03605302.2013.787089.

[25]

M.A. Biot, Theory of propagation of elastic waves in a fluid-saturated porous solid, i. low-frequency range, Journal of the Acoustical Society of America, 28 (1956), 168-178.  doi: 10.1121/1.1908239.

[26]

M.A. Biot, Theory of propagation of elastic waves in a fluid-saturated porous solid. ii. high-frequency range, Journal of the Acoustical Society of America, 28 (1956), 179-191.  doi: 10.1121/1.1908241.

[27]

A.L. Bukhgeim and G. Uhlmann, Recovering a potential from partial cauchy data, Comm. in PDE, 27 (2002), 653-668.  doi: 10.1081/PDE-120002868.

[28]

K.E. ButlerR.D. RussellA.W. Kepic and M. Maxwell, Measurement of the seismoelectric response from a shallow boundary, Geophysics, 61 (1996), 1769-1778.  doi: 10.1190/1.1444093.

[29]

A.P. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Río de Janeiro), Soc. Brasil. Mat., Río de Janeiro, (1980), 65-73. 

[30]

Y. CapdeboscqJ. FehrenbachF. de Gournay and O. Kavian, Imaging by modification: Numerical reconstruction of local conductivities from corresponding power density measurements, SIAM J. Imaging Sci., 2 (2009), 1003-1030.  doi: 10.1137/080723521.

[31]

P. CaroP. Ola and M. Salo, Inverse boundary value problem for {M}axwell equations with local data, Comm. in PDE, 34 (2009), 1425-1464.  doi: 10.1080/03605300903296272.

[32]

P. Caro and K.M. Rogers, Global uniqueness for the calderón problem with lipschitz conductivities, Forum of Mathematics, 4 (2016), e2, 28pp.  doi: 10.1017/fmp.2015.9.

[33]

J. Chen and M. de Hoop, The inverse problem for electroseismic conversion: Stable recovery of the conductivity and the electrokinetic mobility parameter, Inverse Problems and Imaging, 10 (2016), 641-658.  doi: 10.3934/ipi.2016015.

[34]

J. Chen and Y. Yang, Quantitative photo-acoustic tomography with partial data, Inverse Problems, 28 (2012), 115014, 15pp.  doi: 10.1088/0266-5611/28/11/115014.

[35]

J. Chen and Y. Yang, Inverse problem of electro-seismic conversion, Inverse Problems, 29 (2013), 115006, 15pp.  doi: 10.1088/0266-5611/29/11/115006.

[36]

P. G. Ciarlet, Mathematical elasticity, Studies in Math. and its Appl.

[37]

D. Colton and L. Päivärinta, The uniqueness of a solution to an inverse scattering problem for electromagnetic waves, Arch. Rational Mech. Anal., 119 (1992), 59-70.  doi: 10.1007/BF00376010.

[38]

B.T. CoxS.R. Arridge and P.C. Beard, Estimating chromophore distributions from multiwavelength photoacoustic images, J. Opt. Soc. Am. A, 26 (2009), 443-455.  doi: 10.1364/JOSAA.26.000443.

[39]

B.T. CoxJ.G. Laufer and P.C. Beard, The challenges for quantitative photoacoustic imaging, Proc. of SPIE, 777 (2009), 717713.  doi: 10.1117/12.806788.

[40]

A. Douglis and L. Nirenberg, Interior estimates for elliptic systems of partial differential equations, Comm. Pure. Appl. Math., 8 (1955), 503-508.  doi: 10.1002/cpa.3160080406.

[41]

G. Eskin and J. Ralston, On the inverse boundary value problem for linear isotropic elasticity, Inverse Problems, 18 (2002), 907-921.  doi: 10.1088/0266-5611/18/3/324.

[42]

D.D.S. FerreiraC. KenigM. Salo and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Invent. Math., 178 (2009), 119-171.  doi: 10.1007/s00222-009-0196-4.

[43]

S. K. FinchD. Patch and D. Rakesh, Determining a function from its mean values over a family of spheres, SIAM J. Math. Anal., 35 (2004), 1213-1240.  doi: 10.1137/S0036141002417814.

[44]

A. R. FisherA. J. Schissler and J. C. Schotland, Photoacoustic effect for multiply scattered light, Phys. Rev. E., 76 (2007), 036604-1652.  doi: 10.1103/PhysRevE.76.036604.

[45]

B. Gebauer and O. Scherzer, Impedance-acoustic tomography, SIAM Journal of Applied Mathematics, 69 (2008), 565-576.  doi: 10.1137/080715123.

[46]

B. Haberman, Unique determination of a magnetic Schrödinger operator with unbounded magnetic potential from boundary data, preprint. doi: 10.1093/imrn/rnw263.

[47]

B. Haberman and D. Tataru, Uniqueness in Calderon's problem with lipschitz conductivities, Duke Math. J., 162 (2013), 497-516.  doi: 10.1215/00127094-2019591.

[48]

M. HaltmeierO. ScherzerP. Burgholzer and G. Paltauf, Thermoacoustic computed tomography with large planar receivers, Inverse Problems, 20 (2004), 1663-1673.  doi: 10.1088/0266-5611/20/5/021.

[49]

S. C. Hornbostel and A. H. Thompson, Waveform design for electroseismic exploration, SEG Technical Program Expanded Abstracts, (2005), 557-560.  doi: 10.1190/1.2144380.

[50]

Y. HristovaP. Kuchment and L. Nguyen, Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media, Inverse Problems, 24 (2008), 055006, 25pp.  doi: 10.1088/0266-5611/24/5/055006.

[51]

M. Ikehata, A remark on an inverse boundary value problem arising in elasticity, Preprint.

[52]

C. KenigJ. Sjöstrand and G. Uhlmann, The Calderón problem with partial data, Ann. Math., 165 (2007), 567-591.  doi: 10.4007/annals.2007.165.567.

[53]

K. KnudsenM. LassasJ. L. Mueller and S. Siltanen, Regularized d-bar method for the inverse conductivity problem, Inverse Problems and Imaging, 3 (2007), 599-624.  doi: 10.3934/ipi.2009.3.599.

[54]

I. Kocyigit, Acousto-electric tomography and CGO solutions with internal data, Inverse Problems, 28 (2012), 125004, 20pp.  doi: 10.1088/0266-5611/28/12/125004.

[55]

P. Kuchment and L. Kunyansky, Mathematics of thermoacoustic tomography, Euro. J. Appl. Math., 19 (2008), 191-224.  doi: 10.1017/S0956792508007353.

[56]

P. Kuchment and L. Kunyansky, Synthetic focusing in ultrasound modulated tomography, Inverse Problems and Imaging, 4 (2010), 665-673.  doi: 10.3934/ipi.2010.4.665.

[57]

P. Kuchment and L. Kunyansky, 2D and 3D reconstructions in acousto-electric tomography, Inverse Problems, 27 (2011), 055013, 21pp.  doi: 10.1088/0266-5611/27/5/055013.

[58]

L. KunyanskyB. Holman and B. T. Cox, Photoacoustic tomography in a rectangular reflecting cavity, Inverse Problems, 29 (2013), 125010, 20pp.  doi: 10.1088/0266-5611/29/12/125010.

[59]

L. Kunyansky and L. Nguyen, A dissipative time reversal technique for photo-acoustic tomography in a cavity, SIAM J. Imaging Sciences, 9 (2016), 748-769.  doi: 10.1137/15M1049683.

[60]

R.-Y. Lai, Uniqueness and stability of lamé parameters in elastography, Journal of Spectral Theory, 4 (2014), 841-877.  doi: 10.4171/JST/88.

[61]

C. H. LiM. PramanikG. Ku and L. V. Wang, Image distortion in thermoacoustic tomography caused by microwave diffraction, Phys. Rev. E., 77 (2008), 031923.  doi: 10.1103/PhysRevE.77.031923.

[62]

Y. B. Lopatinskii, On a method of reducing boundary problems for a system of differential equations of elliptic type to regular equations, Ukrain. Mat. u'Z., 5 (1953), 123-151. 

[63]

J. R. McLaughlinN. Zhang and A. Manduca, Calculating tissue shear modulus and pressure by 2d log-elastographic methods, Inverse Problems, 26 (2010), 085007, 25pp.  doi: 10.1088/0266-5611/26/8/085007.

[64]

O. V. MikhailovM. W. Haartsen and N. Toksoz, Electroseismic investigation of the shallow subsurface: Field measurements and numerical modeling, Geophysics, 62 (1997), 97-105.  doi: 10.1190/1.1444150.

[65]

O. V. MikhailovJ. Queen and N. Toksoz, Using borehole electroseismic measurements to detect and characterize fractured (permeable) zones, SEG Technical Program Expanded Abstracts, (1997), 1981-1984.  doi: 10.1190/1.1885835.

[66]

A. I. Nachman, Reconstructions from boundary measurements, The Annals of Mathematics, 128 (1988), 531-576.  doi: 10.2307/1971435.

[67]

G. Nakamura and G. Uhlmann, Global uniqueness for an inverse boundary problem arising in elasticity, Invent. Math., 118 (1994), 457-474.  doi: 10.1007/BF01231541.

[68]

G. Nakamura and G. Uhlmann, Erratum: Global uniqueness for an inverse boundary problem arising in elasticity, Invent. Math., 152 (), 205-207. 

[69]

P. OlaL. Päivärinta and E. Somersalo, An inverse boundary value problem in electrodynamics, Duke Math. J., 70 (1993), 617-653.  doi: 10.1215/S0012-7094-93-07014-7.

[70]

P. Ola and E. Somersalo, Electromagnetic inverse problems and generalized sommerfeld potentials, SIAM J. Appl. Math., 56 (1996), 1129-1145.  doi: 10.1137/S0036139995283948.

[71]

S. Patch and O. Scherzer, Photo-and thermo-acoustic imaging, Inverse Problems, 23 (2007), 1-10.  doi: 10.1088/0266-5611/23/6/S01.

[72]

S. R. Pride, Governing equations for the coupled electro-magnetics and acoustics of porous media, Phys. Rev. B,, 50 (), 5678-1569, 16. 

[73]

S. R. Pride and M. W. Haartsen, Electroseismic wave properties, J. Acoust. Soc. Am., 100 (1996), 1301-1315.  doi: 10.1121/1.416018.

[74]

J. Ripoll and V. Ntziachristos, Quantitative point source photoacoustic inversion formulas for scattering and absorbing medium, Phys. Rev. E,, 71 (), 031912. 

[75]

J. E. SantosF. I. Zyserman and P. M. Gauzellino, Numerical electroseismic modeling: A finite element approach, Applied Mathematics and Computation, 218 (2012), 6351-6374.  doi: 10.1016/j.amc.2011.12.003.

[76]

V. Serov, Inverse fixed energy scattering problem for the generalized nonlinear Schrödinger operator, Inverse Problems, 28 (2012), 025002, 11pp.  doi: 10.1088/0266-5611/28/2/025002.

[77]

V. A. Solonnikov, Overdetermined elliptic boundary-value problems, J. Math. Sci., 1 (1973), 477-512.  doi: 10.1007/BF01084589.

[78]

P. Stefanov and G. Uhlmann, Thermoacoustic tomography with variable sound speed, Inverse Problems, 31 (2015), 075011, 16pp.  doi: 10.1088/0266-5611/25/7/075011.

[79]

P. Stefanov and Y. Yang, Multiwave tomography in a closed domain: Averaged sharp time reversal, Inverse Problems, 31 (2015), 065007, 23pp.  doi: 10.1088/0266-5611/31/6/065007.

[80]

P. Stefanov and Y. Yang, Multiwave tomography with reflectors: Landweber's iteration, arXiv: 1603.07045.

[81]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.  doi: 10.2307/1971291.

[82]

A. H. Thompson, Electromagnetic-to-seismic conversion: Successful developments suggest viable applications in exploration and production, SEG Technical Program Expanded Abstracts, (2005), 554-556.  doi: 10.1190/1.2144379.

[83]

A. H. Thompson and G. A. Gist, Geophysical applications of electrokinetic conversion, Leading Edge, 12 (1993), 1169-1173.  doi: 10.1190/1.1436931.

[84]

A. H. ThompsonS. C. HornbostelJ. S. BurnsT. J. MurrayR. A. RaschkeJ. C. WrideP. Z. McCammonJ. R. SumnerG. H. HaakeM. S. BixbyW. S. RossB. S. WhiteM. Zhou and P. K. Peczak, Field tests of electroseismic hydrocarbon detection, SEG Technical Program Expanded Abstracts, (2005), 565-568.  doi: 10.1190/1.2144382.

[85]

R. R. Thompson, The seismic electric effect, Geophysics, 1 (1936), 327-335.  doi: 10.1190/1.1437119.

[86]

G. Uhlmann, Developments in inverse problems since Calderón's foundational paper, Harmonic Analysis and Partial Differential Equations, The University of Chicago Press, Chicago, (1999), 295-345. 

[87]

G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 123011, 39pp.  doi: 10.1088/0266-5611/25/12/123011.

[88]

G. Uhlmann and J.-N. Wang, Complex spherical waves for the elasticity system and probing of inclusions, SIAM J. Math. Anal., 38 (2007), 1967-1980.  doi: 10.1137/060651434.

[89]

G. Uhlmann and J.-N. Wang, Reconstruction of discontinuities in systems, Journal of Physics: Conference Series,, 73 (2007), 012024.  doi: 10.1088/1742-6596/73/1/012024.

[90]

B. S. White, Asymptotic theory of electroseismic prospecting, SIAM J. Appl. Math., 65 (2005), 1443-1462.  doi: 10.1137/040604108.

[91]

B. S. White and M. Zhou, Electroseismic prospecting in layered media, SIAM J. Appl. Math., 67 (2006), 69-98.  doi: 10.1137/050633603.

[92]

M. Xu and L. V. Wang, Photoacoustic imaging in biomedicine, Rev. Sci. Instr., 77 (2006), 041101.  doi: 10.1063/1.2195024.

[93]

R. J. Zemp, Quantitative photoacoustic tomography with multiple optical sources, Applied Optics, 49 (2010), 3566-3572.  doi: 10.1364/AO.49.003566.

[94]

H. Zhang and L. V. Wang, Acousto-electric tomography, SPIE, 5320 (2004), 145-149.  doi: 10.1117/12.532610.

show all references

References:
[1]

S. Acosta and C. Montalto, Multiwave imaging in an enclosure with variable wave speed, Inverse Problems, 31 (2015), 065009, 12pp.  doi: 10.1088/0266-5611/31/6/065009.

[2]

G. Alessandrini, Stable determination of conductivity by boundary measurements, App. Anal., 27 (1988), 153-172.  doi: 10.1080/00036818808839730.

[3]

H. AmmariE. BonnetierY. CapdeboscqM. Tanter and M. Fink, Electrical impedance tomography by elastic deformation, SIAM Journal on Applied Mathematics, 68 (2008), 1557-1573.  doi: 10.1137/070686408.

[4]

H. Ammari, E. Bossy, V. Jugnon and H. Kang, Mathematical models in photo-acoustic imaging of small absorbers, SIAM Review.

[5]

G. Bal, Hybrid inverse problems and internal functionals, Inside Out II, MSRI Publications, 60 (2013), 325-368. 

[6]

G. Bal, Cauchy problem and ultrasound modulated EIT, Analysis and PDE, 6 (2013), 751-775.  doi: 10.2140/apde.2013.6.751.

[7]

G. Bal, Hybrid inverse problems and systems of partial differential equations, Contemp. Math., 615 (2014), 15pp.  doi: 10.1090/conm/615/12289.

[8]

G. BalC. BellisS. Imperiale and F. Monard, Reconstruction of moduli in isotropic linear elasticity from full-field measurements, Inverse Problems, 30 (2014), 125004, 22pp.  doi: 10.1088/0266-5611/30/12/125004.

[9]

G. BalE. BonnetierF. Monard and F. Triki, Inverse diffusion from knowledge of power densities, Inverse Problems and Imaging, 7 (2013), 353-375.  doi: 10.3934/ipi.2013.7.353.

[10]

G. Bal and C. Guo, Imaging of complex-valued tensors for two-dimensional Maxwell's equations, accepted by Journal of Inverse and Ill-posed Problems.

[11]

G. Bal and C. Guo, Reconstruction of complex-valued tensors in the {M}axwell system from knowledge of internal magnetic fields, Inverse Problems and Imaging, 8 (2014), 1033-1051.  doi: 10.3934/ipi.2014.8.1033.

[12]

G. BalC. Guo and F. Monard, Imaging of anisotropic conductivities from current densities in two dimensions, SIAM J. Imaging Sci., 7 (2014), 2538-2557.  doi: 10.1137/140961754.

[13]

G. BalC. Guo and F. Monard, Inverse anisotropic conductivity from internal current densities, Inverse Problems, 30 (2014), 025001, 21pp.  doi: 10.1088/0266-5611/30/2/025001.

[14]

G. BalC. Guo and F. Monard, Linearized internal functional for anisotropic conductivities, Inverse Problems and Imaging, 8 (2014), 1-22.  doi: 10.3934/ipi.2014.8.1.

[15]

G. Bal and F. Monard, Inverse diffusion problems with redundant internal information, Inverse Problems and Imaging, 6 (2012), 289-313.  doi: 10.3934/ipi.2012.6.289.

[16]

G. BalF. Monard and G. Uhlmann, Reconstruction of a fully anisotropic elasticity tensor from knowledge of displacement fields, SIAM J. Applied Math., 75 (2015), 2214-2231.  doi: 10.1137/151005269.

[17]

G. Bal and K. Ren, Multi-source quantitative pat in diffusive regime, Inverse Problems, 27 075003.

[18]

G. Bal and K. Ren, On multi-spectral quantitative photoacoustic tomography, Inverse Problems, 28 025010.

[19]

G. BalK. RenG. Uhlmann and T. Zhou, Quantitative thermo-acoustics and related problems, Inverse Problems, 27 (2011), 055007, 15pp.  doi: 10.1088/0266-5611/27/5/055007.

[20]

G. Bal and G. Uhlmann, Inverse diffusion theory for photoacoustics Inverse Problems, 26 (2010), 085010. doi: 10.1088/0266-5611/26/8/085010.

[21]

G. Bal and G. Uhlmann, Reconstruction of coefficients in scalar second-order elliptic equations from knowledge of their solutions, Comm. on Pure and Applied Math, 66 (2013), 1629-1652.  doi: 10.1002/cpa.21453.

[22]

G. Bal and T. Zhou, Hybrid inverse problems for a system of Maxwell's equations, Inverse Problems, 30 (2014), 055013, 17pp.  doi: 10.1088/0266-5611/30/5/055013.

[23]

G. Bal and F. Monard, Inverse anisotropic diffusion from power density measurements in two dimensions, Inverse Problems, 28 (2012), 084001, 20pp.  doi: 10.1088/0266-5611/28/8/084001.

[24]

G. Bal and F. Monard, Inverse anisotropic conductivity from power density measurements in dimensions $ n≥q3$, Comm. Partial Differential Equations, 38 (2013), 1183-1207.  doi: 10.1080/03605302.2013.787089.

[25]

M.A. Biot, Theory of propagation of elastic waves in a fluid-saturated porous solid, i. low-frequency range, Journal of the Acoustical Society of America, 28 (1956), 168-178.  doi: 10.1121/1.1908239.

[26]

M.A. Biot, Theory of propagation of elastic waves in a fluid-saturated porous solid. ii. high-frequency range, Journal of the Acoustical Society of America, 28 (1956), 179-191.  doi: 10.1121/1.1908241.

[27]

A.L. Bukhgeim and G. Uhlmann, Recovering a potential from partial cauchy data, Comm. in PDE, 27 (2002), 653-668.  doi: 10.1081/PDE-120002868.

[28]

K.E. ButlerR.D. RussellA.W. Kepic and M. Maxwell, Measurement of the seismoelectric response from a shallow boundary, Geophysics, 61 (1996), 1769-1778.  doi: 10.1190/1.1444093.

[29]

A.P. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Río de Janeiro), Soc. Brasil. Mat., Río de Janeiro, (1980), 65-73. 

[30]

Y. CapdeboscqJ. FehrenbachF. de Gournay and O. Kavian, Imaging by modification: Numerical reconstruction of local conductivities from corresponding power density measurements, SIAM J. Imaging Sci., 2 (2009), 1003-1030.  doi: 10.1137/080723521.

[31]

P. CaroP. Ola and M. Salo, Inverse boundary value problem for {M}axwell equations with local data, Comm. in PDE, 34 (2009), 1425-1464.  doi: 10.1080/03605300903296272.

[32]

P. Caro and K.M. Rogers, Global uniqueness for the calderón problem with lipschitz conductivities, Forum of Mathematics, 4 (2016), e2, 28pp.  doi: 10.1017/fmp.2015.9.

[33]

J. Chen and M. de Hoop, The inverse problem for electroseismic conversion: Stable recovery of the conductivity and the electrokinetic mobility parameter, Inverse Problems and Imaging, 10 (2016), 641-658.  doi: 10.3934/ipi.2016015.

[34]

J. Chen and Y. Yang, Quantitative photo-acoustic tomography with partial data, Inverse Problems, 28 (2012), 115014, 15pp.  doi: 10.1088/0266-5611/28/11/115014.

[35]

J. Chen and Y. Yang, Inverse problem of electro-seismic conversion, Inverse Problems, 29 (2013), 115006, 15pp.  doi: 10.1088/0266-5611/29/11/115006.

[36]

P. G. Ciarlet, Mathematical elasticity, Studies in Math. and its Appl.

[37]

D. Colton and L. Päivärinta, The uniqueness of a solution to an inverse scattering problem for electromagnetic waves, Arch. Rational Mech. Anal., 119 (1992), 59-70.  doi: 10.1007/BF00376010.

[38]

B.T. CoxS.R. Arridge and P.C. Beard, Estimating chromophore distributions from multiwavelength photoacoustic images, J. Opt. Soc. Am. A, 26 (2009), 443-455.  doi: 10.1364/JOSAA.26.000443.

[39]

B.T. CoxJ.G. Laufer and P.C. Beard, The challenges for quantitative photoacoustic imaging, Proc. of SPIE, 777 (2009), 717713.  doi: 10.1117/12.806788.

[40]

A. Douglis and L. Nirenberg, Interior estimates for elliptic systems of partial differential equations, Comm. Pure. Appl. Math., 8 (1955), 503-508.  doi: 10.1002/cpa.3160080406.

[41]

G. Eskin and J. Ralston, On the inverse boundary value problem for linear isotropic elasticity, Inverse Problems, 18 (2002), 907-921.  doi: 10.1088/0266-5611/18/3/324.

[42]

D.D.S. FerreiraC. KenigM. Salo and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Invent. Math., 178 (2009), 119-171.  doi: 10.1007/s00222-009-0196-4.

[43]

S. K. FinchD. Patch and D. Rakesh, Determining a function from its mean values over a family of spheres, SIAM J. Math. Anal., 35 (2004), 1213-1240.  doi: 10.1137/S0036141002417814.

[44]

A. R. FisherA. J. Schissler and J. C. Schotland, Photoacoustic effect for multiply scattered light, Phys. Rev. E., 76 (2007), 036604-1652.  doi: 10.1103/PhysRevE.76.036604.

[45]

B. Gebauer and O. Scherzer, Impedance-acoustic tomography, SIAM Journal of Applied Mathematics, 69 (2008), 565-576.  doi: 10.1137/080715123.

[46]

B. Haberman, Unique determination of a magnetic Schrödinger operator with unbounded magnetic potential from boundary data, preprint. doi: 10.1093/imrn/rnw263.

[47]

B. Haberman and D. Tataru, Uniqueness in Calderon's problem with lipschitz conductivities, Duke Math. J., 162 (2013), 497-516.  doi: 10.1215/00127094-2019591.

[48]

M. HaltmeierO. ScherzerP. Burgholzer and G. Paltauf, Thermoacoustic computed tomography with large planar receivers, Inverse Problems, 20 (2004), 1663-1673.  doi: 10.1088/0266-5611/20/5/021.

[49]

S. C. Hornbostel and A. H. Thompson, Waveform design for electroseismic exploration, SEG Technical Program Expanded Abstracts, (2005), 557-560.  doi: 10.1190/1.2144380.

[50]

Y. HristovaP. Kuchment and L. Nguyen, Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media, Inverse Problems, 24 (2008), 055006, 25pp.  doi: 10.1088/0266-5611/24/5/055006.

[51]

M. Ikehata, A remark on an inverse boundary value problem arising in elasticity, Preprint.

[52]

C. KenigJ. Sjöstrand and G. Uhlmann, The Calderón problem with partial data, Ann. Math., 165 (2007), 567-591.  doi: 10.4007/annals.2007.165.567.

[53]

K. KnudsenM. LassasJ. L. Mueller and S. Siltanen, Regularized d-bar method for the inverse conductivity problem, Inverse Problems and Imaging, 3 (2007), 599-624.  doi: 10.3934/ipi.2009.3.599.

[54]

I. Kocyigit, Acousto-electric tomography and CGO solutions with internal data, Inverse Problems, 28 (2012), 125004, 20pp.  doi: 10.1088/0266-5611/28/12/125004.

[55]

P. Kuchment and L. Kunyansky, Mathematics of thermoacoustic tomography, Euro. J. Appl. Math., 19 (2008), 191-224.  doi: 10.1017/S0956792508007353.

[56]

P. Kuchment and L. Kunyansky, Synthetic focusing in ultrasound modulated tomography, Inverse Problems and Imaging, 4 (2010), 665-673.  doi: 10.3934/ipi.2010.4.665.

[57]

P. Kuchment and L. Kunyansky, 2D and 3D reconstructions in acousto-electric tomography, Inverse Problems, 27 (2011), 055013, 21pp.  doi: 10.1088/0266-5611/27/5/055013.

[58]

L. KunyanskyB. Holman and B. T. Cox, Photoacoustic tomography in a rectangular reflecting cavity, Inverse Problems, 29 (2013), 125010, 20pp.  doi: 10.1088/0266-5611/29/12/125010.

[59]

L. Kunyansky and L. Nguyen, A dissipative time reversal technique for photo-acoustic tomography in a cavity, SIAM J. Imaging Sciences, 9 (2016), 748-769.  doi: 10.1137/15M1049683.

[60]

R.-Y. Lai, Uniqueness and stability of lamé parameters in elastography, Journal of Spectral Theory, 4 (2014), 841-877.  doi: 10.4171/JST/88.

[61]

C. H. LiM. PramanikG. Ku and L. V. Wang, Image distortion in thermoacoustic tomography caused by microwave diffraction, Phys. Rev. E., 77 (2008), 031923.  doi: 10.1103/PhysRevE.77.031923.

[62]

Y. B. Lopatinskii, On a method of reducing boundary problems for a system of differential equations of elliptic type to regular equations, Ukrain. Mat. u'Z., 5 (1953), 123-151. 

[63]

J. R. McLaughlinN. Zhang and A. Manduca, Calculating tissue shear modulus and pressure by 2d log-elastographic methods, Inverse Problems, 26 (2010), 085007, 25pp.  doi: 10.1088/0266-5611/26/8/085007.

[64]

O. V. MikhailovM. W. Haartsen and N. Toksoz, Electroseismic investigation of the shallow subsurface: Field measurements and numerical modeling, Geophysics, 62 (1997), 97-105.  doi: 10.1190/1.1444150.

[65]

O. V. MikhailovJ. Queen and N. Toksoz, Using borehole electroseismic measurements to detect and characterize fractured (permeable) zones, SEG Technical Program Expanded Abstracts, (1997), 1981-1984.  doi: 10.1190/1.1885835.

[66]

A. I. Nachman, Reconstructions from boundary measurements, The Annals of Mathematics, 128 (1988), 531-576.  doi: 10.2307/1971435.

[67]

G. Nakamura and G. Uhlmann, Global uniqueness for an inverse boundary problem arising in elasticity, Invent. Math., 118 (1994), 457-474.  doi: 10.1007/BF01231541.

[68]

G. Nakamura and G. Uhlmann, Erratum: Global uniqueness for an inverse boundary problem arising in elasticity, Invent. Math., 152 (), 205-207. 

[69]

P. OlaL. Päivärinta and E. Somersalo, An inverse boundary value problem in electrodynamics, Duke Math. J., 70 (1993), 617-653.  doi: 10.1215/S0012-7094-93-07014-7.

[70]

P. Ola and E. Somersalo, Electromagnetic inverse problems and generalized sommerfeld potentials, SIAM J. Appl. Math., 56 (1996), 1129-1145.  doi: 10.1137/S0036139995283948.

[71]

S. Patch and O. Scherzer, Photo-and thermo-acoustic imaging, Inverse Problems, 23 (2007), 1-10.  doi: 10.1088/0266-5611/23/6/S01.

[72]

S. R. Pride, Governing equations for the coupled electro-magnetics and acoustics of porous media, Phys. Rev. B,, 50 (), 5678-1569, 16. 

[73]

S. R. Pride and M. W. Haartsen, Electroseismic wave properties, J. Acoust. Soc. Am., 100 (1996), 1301-1315.  doi: 10.1121/1.416018.

[74]

J. Ripoll and V. Ntziachristos, Quantitative point source photoacoustic inversion formulas for scattering and absorbing medium, Phys. Rev. E,, 71 (), 031912. 

[75]

J. E. SantosF. I. Zyserman and P. M. Gauzellino, Numerical electroseismic modeling: A finite element approach, Applied Mathematics and Computation, 218 (2012), 6351-6374.  doi: 10.1016/j.amc.2011.12.003.

[76]

V. Serov, Inverse fixed energy scattering problem for the generalized nonlinear Schrödinger operator, Inverse Problems, 28 (2012), 025002, 11pp.  doi: 10.1088/0266-5611/28/2/025002.

[77]

V. A. Solonnikov, Overdetermined elliptic boundary-value problems, J. Math. Sci., 1 (1973), 477-512.  doi: 10.1007/BF01084589.

[78]

P. Stefanov and G. Uhlmann, Thermoacoustic tomography with variable sound speed, Inverse Problems, 31 (2015), 075011, 16pp.  doi: 10.1088/0266-5611/25/7/075011.

[79]

P. Stefanov and Y. Yang, Multiwave tomography in a closed domain: Averaged sharp time reversal, Inverse Problems, 31 (2015), 065007, 23pp.  doi: 10.1088/0266-5611/31/6/065007.

[80]

P. Stefanov and Y. Yang, Multiwave tomography with reflectors: Landweber's iteration, arXiv: 1603.07045.

[81]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.  doi: 10.2307/1971291.

[82]

A. H. Thompson, Electromagnetic-to-seismic conversion: Successful developments suggest viable applications in exploration and production, SEG Technical Program Expanded Abstracts, (2005), 554-556.  doi: 10.1190/1.2144379.

[83]

A. H. Thompson and G. A. Gist, Geophysical applications of electrokinetic conversion, Leading Edge, 12 (1993), 1169-1173.  doi: 10.1190/1.1436931.

[84]

A. H. ThompsonS. C. HornbostelJ. S. BurnsT. J. MurrayR. A. RaschkeJ. C. WrideP. Z. McCammonJ. R. SumnerG. H. HaakeM. S. BixbyW. S. RossB. S. WhiteM. Zhou and P. K. Peczak, Field tests of electroseismic hydrocarbon detection, SEG Technical Program Expanded Abstracts, (2005), 565-568.  doi: 10.1190/1.2144382.

[85]

R. R. Thompson, The seismic electric effect, Geophysics, 1 (1936), 327-335.  doi: 10.1190/1.1437119.

[86]

G. Uhlmann, Developments in inverse problems since Calderón's foundational paper, Harmonic Analysis and Partial Differential Equations, The University of Chicago Press, Chicago, (1999), 295-345. 

[87]

G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 123011, 39pp.  doi: 10.1088/0266-5611/25/12/123011.

[88]

G. Uhlmann and J.-N. Wang, Complex spherical waves for the elasticity system and probing of inclusions, SIAM J. Math. Anal., 38 (2007), 1967-1980.  doi: 10.1137/060651434.

[89]

G. Uhlmann and J.-N. Wang, Reconstruction of discontinuities in systems, Journal of Physics: Conference Series,, 73 (2007), 012024.  doi: 10.1088/1742-6596/73/1/012024.

[90]

B. S. White, Asymptotic theory of electroseismic prospecting, SIAM J. Appl. Math., 65 (2005), 1443-1462.  doi: 10.1137/040604108.

[91]

B. S. White and M. Zhou, Electroseismic prospecting in layered media, SIAM J. Appl. Math., 67 (2006), 69-98.  doi: 10.1137/050633603.

[92]

M. Xu and L. V. Wang, Photoacoustic imaging in biomedicine, Rev. Sci. Instr., 77 (2006), 041101.  doi: 10.1063/1.2195024.

[93]

R. J. Zemp, Quantitative photoacoustic tomography with multiple optical sources, Applied Optics, 49 (2010), 3566-3572.  doi: 10.1364/AO.49.003566.

[94]

H. Zhang and L. V. Wang, Acousto-electric tomography, SPIE, 5320 (2004), 145-149.  doi: 10.1117/12.532610.

Modality Equation and Data CGO solution Results
Section 2: Quantitative PAT (second step of PAT) $-\nabla\cdot\gamma\nabla u+\sigma u=0$
data: $u|_{\partial\Omega}\mapsto \sigma u|_{\Omega}$
(full boundary illuminations).
$u = e^{i \zeta\cdot x}(1+\psi_{\zeta})$
to the reduced equation $(\Delta+q)u=0$.
Uniqueness and stability in determining $(\gamma,\sigma)$ (see [20]).
data: $u|_{\Gamma}\mapsto\sigma u|_{\Omega}$
(partial boundary illuminations).
$u=e^{\frac{1}{h}(\varphi+i\psi)}\big(a+r\big)+z$
with $\text{supp}~u|_{\partial\Omega}\subset\Gamma$.
Uniqueness and stability in determining $(\gamma,\sigma)$ (see [34]).
Section 3: Electro Seismic Conversion Maxwell's equations:
$\begin{array}{l}\nabla\times E = i\omega\mu_{0} H,\\\nabla\times H = (\sigma - i\varepsilon\omega)E.\end{array}$
data:
$\nu\times E|_{\partial\Omega}\mapsto LE|_{\Omega}$
$E= e^{i\zeta\cdot x}(\eta + R_\zeta)$ Uniqueness and Stability in determining $(L,\sigma)$ (see [35]).
Section 4: Transient Elastography Elasticity system:
$\nabla\cdot (\lambda(\nabla\cdot u)I+2S(\nabla u)\mu)+k^2 u=0$
data:
$u|_{\partial\Omega}\mapsto u|_{\Omega}$.
$U = e^{i\zeta\cdot x} (C_0(x,\theta) p(\theta\cdot x)+O(\tau^{-1}))$
to the Schrödinger equation with external Yang-Mills potentials.
Uniqueness and Stability in determining the Lamé parameters $(\lambda,\mu)$(see [60]).
Section 5: Acousto Electric Tomography Step 1
Conductivity equation
$\nabla \cdot (\gamma \nabla u) = 0$.
data:
$m\mapsto(\Lambda_{\gamma_m}-\Lambda_{\gamma})(u|_{\partial\Omega})$
where $\Lambda_\gamma$ is the Dirichlet to Neumann map for $\gamma$ and $\gamma_m=(1+m)\gamma$.
$u={\gamma}^{-1/2}e^{i\zeta \cdot x} (1 + \psi_{\zeta})$
to the conductivity equation.
Reconstruction of $\sqrt{\gamma}\nabla u|_{\Omega}$ using CGOs (see [54]) or $\gamma |\nabla u|^2|_{\Omega}$ (see [9])
Step 2
data: $u|_{\partial\Omega}\mapsto \sqrt{\gamma}\nabla u|_{\Omega}$ or $u|_{\partial\Omega}\mapsto \gamma |\nabla u|^2 |_{\Omega}$
same as step 1 above Uniqueness and stability in determining $\gamma$(see [9,54]).
Section 6: Quantitative TAT Scalar Schrödinger
$(\Delta+q)u=0$
where $q=k^2+ik\sigma(x)$.
data: $u|_{\partial\Omega}\mapsto \sigma|u|^2_{\Omega}$
$u = e^{ i \zeta \cdot x}( 1 + \psi_\zeta)$ Uniqueness and Stability in determining $\sigma$(see [19]).
Maxwell system:
$-\nabla\times\nabla\times E+qE=0$
where $q=k^2n+ik\sigma$.
data: $\nu\times E|_{\partial\Omega}\mapsto \sigma|E|^2|_{\Omega}$
$E=\gamma_0^{-1/2} e^{i\zeta\cdot x}\big(\eta_\zeta+R_\zeta\big)$
where $\gamma_0=q/\kappa^2$.
Stability in determining $q$(see [22]).
Modality Equation and Data CGO solution Results
Section 2: Quantitative PAT (second step of PAT) $-\nabla\cdot\gamma\nabla u+\sigma u=0$
data: $u|_{\partial\Omega}\mapsto \sigma u|_{\Omega}$
(full boundary illuminations).
$u = e^{i \zeta\cdot x}(1+\psi_{\zeta})$
to the reduced equation $(\Delta+q)u=0$.
Uniqueness and stability in determining $(\gamma,\sigma)$ (see [20]).
data: $u|_{\Gamma}\mapsto\sigma u|_{\Omega}$
(partial boundary illuminations).
$u=e^{\frac{1}{h}(\varphi+i\psi)}\big(a+r\big)+z$
with $\text{supp}~u|_{\partial\Omega}\subset\Gamma$.
Uniqueness and stability in determining $(\gamma,\sigma)$ (see [34]).
Section 3: Electro Seismic Conversion Maxwell's equations:
$\begin{array}{l}\nabla\times E = i\omega\mu_{0} H,\\\nabla\times H = (\sigma - i\varepsilon\omega)E.\end{array}$
data:
$\nu\times E|_{\partial\Omega}\mapsto LE|_{\Omega}$
$E= e^{i\zeta\cdot x}(\eta + R_\zeta)$ Uniqueness and Stability in determining $(L,\sigma)$ (see [35]).
Section 4: Transient Elastography Elasticity system:
$\nabla\cdot (\lambda(\nabla\cdot u)I+2S(\nabla u)\mu)+k^2 u=0$
data:
$u|_{\partial\Omega}\mapsto u|_{\Omega}$.
$U = e^{i\zeta\cdot x} (C_0(x,\theta) p(\theta\cdot x)+O(\tau^{-1}))$
to the Schrödinger equation with external Yang-Mills potentials.
Uniqueness and Stability in determining the Lamé parameters $(\lambda,\mu)$(see [60]).
Section 5: Acousto Electric Tomography Step 1
Conductivity equation
$\nabla \cdot (\gamma \nabla u) = 0$.
data:
$m\mapsto(\Lambda_{\gamma_m}-\Lambda_{\gamma})(u|_{\partial\Omega})$
where $\Lambda_\gamma$ is the Dirichlet to Neumann map for $\gamma$ and $\gamma_m=(1+m)\gamma$.
$u={\gamma}^{-1/2}e^{i\zeta \cdot x} (1 + \psi_{\zeta})$
to the conductivity equation.
Reconstruction of $\sqrt{\gamma}\nabla u|_{\Omega}$ using CGOs (see [54]) or $\gamma |\nabla u|^2|_{\Omega}$ (see [9])
Step 2
data: $u|_{\partial\Omega}\mapsto \sqrt{\gamma}\nabla u|_{\Omega}$ or $u|_{\partial\Omega}\mapsto \gamma |\nabla u|^2 |_{\Omega}$
same as step 1 above Uniqueness and stability in determining $\gamma$(see [9,54]).
Section 6: Quantitative TAT Scalar Schrödinger
$(\Delta+q)u=0$
where $q=k^2+ik\sigma(x)$.
data: $u|_{\partial\Omega}\mapsto \sigma|u|^2_{\Omega}$
$u = e^{ i \zeta \cdot x}( 1 + \psi_\zeta)$ Uniqueness and Stability in determining $\sigma$(see [19]).
Maxwell system:
$-\nabla\times\nabla\times E+qE=0$
where $q=k^2n+ik\sigma$.
data: $\nu\times E|_{\partial\Omega}\mapsto \sigma|E|^2|_{\Omega}$
$E=\gamma_0^{-1/2} e^{i\zeta\cdot x}\big(\eta_\zeta+R_\zeta\big)$
where $\gamma_0=q/\kappa^2$.
Stability in determining $q$(see [22]).
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