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Multiwave tomography with reflectors: Landweber's iteration
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA |
We use the Landweber method for numerical simulations for the multiwave tomography problem with a reflecting boundary and compare it with the averaged time reversal method. We also analyze the rate of convergence and the dependence on the step size for the Landweber iterations on a Hilbert space.
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] |
S. Acosta and C. Montalto, Photoacoustic imaging taking into account thermodynamic attenuation
Inverse Problems, 32 (2016), arXiv: 1602.01872.
doi: 10.1088/0266-5611/32/11/115001. |
[3] |
N. Albin, O. P. Bruno, T. Y. Cheung and R. O. Cleveland,
Fourier continuation methods for high-fidelity simulation of nonlinear acoustic beams, The Journal of the Acoustical Society of America, 132 (2012), 2371-2387.
doi: 10.1121/1.4742722. |
[4] |
R. M. Alford, K. R. Kelly and D. M. Boore,
Accuracy of finite-difference modeling of the acoustic wave equation, Geophysics, 39 (1974), 834-842.
doi: 10.1190/1.1440470. |
[5] |
S. Arridge, B. Marta, B. Cox, F. Lucka and B. Treeby, On the adjoint operator in photoacoustic tomography
Inverse Problems, 32 (2016), 110512, 19pp, arXiv: 1602.02027.
doi: 10.1088/0266-5611/32/11/115012. |
[6] |
D. Auroux and J. Blum,
Back and forth nudging algorithm for data assimilation problems, Comptes Rendus Mathematique, 340 (2005), 873-878.
doi: 10.1016/j.crma.2005.05.006. |
[7] |
C. Bardos, G. Lebeau and J. Rauch,
Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.
doi: 10.1137/0330055. |
[8] |
Z. Belhachmi, T. Glatz and O. Scherzer, Photoacoustic tomography with spatially varying
compressibility and density, J. Inverse Ill-Posed Probl., 25 (2017), 119-133, arXiv:1512.07411 (2015).
doi: 10.1515/jiip-2015-0113. |
[9] |
______, A direct method for photoacoustic tomography with inhomogeneous sound speed Inverse Problems, 32 (2016), 045005, 25pp.
doi: 10.1088/0266-5611/32/4/045005. |
[10] |
C. Byrne,
Iterative Algorithms in Inverse Problems, http://faculty.uml.edu/cbyrne/ITER2.pdf, 2006. |
[11] |
O. Chervova and L. Oksanen, Time reversal method with stabilizing boundary conditions for Photoacoustic tomography Inverse Problems, 32 (2016), arXiv: 1605.07817 (2016).
doi: 10.1088/0266-5611/32/12/125004. |
[12] |
B. T. Cox, S. R. Arridge and P. C. Beard,
Photoacoustic tomography with a limited-aperture planar sensor and a reverberant cavity, Inverse Problems, 23 (2007), S95-S112.
doi: 10.1088/0266-5611/23/6/S08. |
[13] |
M. A. Dablain,
The application of high-order differencing to the scalar wave equation, Geophysics, 51 (1986), 54-66.
doi: 10.1190/1.1442040. |
[14] | |
[15] |
Finch, Patch and Rakesh,
Determining a function from its mean values over a family of spheres, SIAM J. Math. Anal., 35 (2004), 1213-1240 (electronic).
doi: 10.1137/S0036141002417814. |
[16] |
J. A. Goldstein and M. Wacker,
The energy space and norm growth for abstract wave equations, Applied Mathematics Letters, 16 (2003), 767-772.
doi: 10.1016/S0893-9659(03)00080-6. |
[17] |
M. Hanke,
Accelerated Landweber iterations for the solution of ill-posed equations, Numer. Math., 60 (1991), 341-373.
doi: 10.1007/BF01385727. |
[18] |
M. Hanke, A. Neubauer and O. Scherzer,
A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72 (1995), 21-37.
doi: 10.1007/s002110050158. |
[19] |
B. Holman and L. Kunyansky, Gradual time reversal in thermo-and photo-acoustic tomography within a resonant cavity Inverse Problems, 31 (2015), 035008, 25pp.
doi: 10.1088/0266-5611/31/3/035008. |
[20] |
A. Homan,
Multi-wave imaging in attenuating media, Inverse Probl. Imaging, 7 (2013), 1235-1250.
doi: 10.3934/ipi.2013.7.1235. |
[21] |
K. R. Kelly, R. W. Ward, S. Treitel and R. M. Alford,
Synthetic seismograms: A finite-difference approach, Geophysics, 41 (1976), 2-27.
doi: 10.1190/1.1440605. |
[22] |
A. Kirsch,
An Introduction to the Mathematical Theory of Inverse Problems, second ed., Applied Mathematical Sciences, vol. 120, Springer, New York, 2011.
doi: 10.1007/978-1-4419-8474-6. |
[23] |
R.A. Kruger, W. L. Kiser, D. R. Reinecke and G. A. Kruger,
Thermoacoustic computed tomography using a conventional linear transducer array, Med Phys, 30 (2003), 856-860.
doi: 10.1118/1.1565340. |
[24] |
R. A. Kruger, D. R. Reinecke and G. A. Kruger,
Thermoacoustic computed tomography--technical considerations, Med Phys, 26 (1999), 1832-1837.
doi: 10.1118/1.598688. |
[25] |
P. Kuchment and L. Kunyansky, Mathematics of photoacoustic and thermoacoustic tomography, Handbook of Mathematical Methods in Imaging (Otmar Scherzer, ed.), Springer New
York, 2011, pp. 817-865. |
[26] |
L. Kunyansky, B. 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. |
[27] |
L. V. Nguyen and L. A. Kunyansky,
A dissipative time reversal technique for photoacoustic tomography in a cavity, SIAM J. Imaging Sci., 9 (2016), 748-769.
doi: 10.1137/15M1049683. |
[28] |
J. Qian, P. Stefanov, G. Uhlmann and H. Zhao,
An efficient Neumann series-based algorithm for thermoacoustic and photoacoustic tomography with variable sound speed, SIAM J. Imaging Sci., 4 (2011), 850-883.
doi: 10.1137/100817280. |
[29] |
M. Reed and B. Simon,
Methods of Modern Mathematical Physics. III, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979, Scattering theory. |
[30] |
______,
Methods of Modern Mathematical Physics. I, second ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980, Functional analysis. |
[31] |
P. Stefanov and G. Uhlmann,
Linearizing non-linear inverse problems and an application to inverse backscattering, J. Funct. Anal., 256 (2009), 2842-2866.
doi: 10.1016/j.jfa.2008.10.017. |
[32] |
______, Thermoacoustic tomography with variable sound speed,
Inverse Problems, 25 (2009), 075011, 16pp. |
[33] |
______, Thermoacoustic tomography arising in brain imaging,
Inverse Problems, 27 (2011), 045004, 26pp. |
[34] |
______, Multi-wave methods via ultrasound, Inside Out, MSRI Publications, 60 (2012), 271-324. . |
[35] |
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. |
[36] |
D. Tataru,
Unique continuation for solutions to PDE's; between Hörmander's theorem and Holmgren's theorem, Comm. Partial Differential Equations, 20 (1995), 855-884.
doi: 10.1080/03605309508821117. |
[37] |
______, On the regularity of boundary traces for the wave equation, Ann. Scuola Norm. Sup.
Pisa Cl. Sci., (4), 26 (1998), 185-206 |
[38] |
G. M. Vainikko,
Error estimates of the successive approximation method for ill-posed problems, Automat. Remote Control, (1980), 84-92.
|
[39] |
G. M. Vainikko and A. Yu. Veretennikov,
Iteration Procedures in Ill-Posed Problems (in russian)
"Nauka", Moscow, 1986. |
[40] |
M. Xu and L. V. Wang, Photoacoustic imaging in biomedicine Review of Scientific Instruments, 77 (2006), 041101.
doi: 10.1063/1.2195024. |
[41] |
Y. Xu and L. V. Wang,
Rhesus monkey brain imaging through intact skull with thermoacoustic tomography, IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 53 (2006), 542-548.
|
[42] |
X. Yang and L. V. Wang, Monkey brain cortex imaging by photoacoustic tomography Proc. SPIE, 6856 (2008), 762396.
doi: 10.1117/12.762396. |
[43] |
E. Zuazua,
Numerics for the control of partial differential equations, Reference Work Entry: Encyclopedia of Applied and Computational Mathematics, (2015), 1076-1080.
doi: 10.1007/978-3-540-70529-1_362. |
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] |
S. Acosta and C. Montalto, Photoacoustic imaging taking into account thermodynamic attenuation
Inverse Problems, 32 (2016), arXiv: 1602.01872.
doi: 10.1088/0266-5611/32/11/115001. |
[3] |
N. Albin, O. P. Bruno, T. Y. Cheung and R. O. Cleveland,
Fourier continuation methods for high-fidelity simulation of nonlinear acoustic beams, The Journal of the Acoustical Society of America, 132 (2012), 2371-2387.
doi: 10.1121/1.4742722. |
[4] |
R. M. Alford, K. R. Kelly and D. M. Boore,
Accuracy of finite-difference modeling of the acoustic wave equation, Geophysics, 39 (1974), 834-842.
doi: 10.1190/1.1440470. |
[5] |
S. Arridge, B. Marta, B. Cox, F. Lucka and B. Treeby, On the adjoint operator in photoacoustic tomography
Inverse Problems, 32 (2016), 110512, 19pp, arXiv: 1602.02027.
doi: 10.1088/0266-5611/32/11/115012. |
[6] |
D. Auroux and J. Blum,
Back and forth nudging algorithm for data assimilation problems, Comptes Rendus Mathematique, 340 (2005), 873-878.
doi: 10.1016/j.crma.2005.05.006. |
[7] |
C. Bardos, G. Lebeau and J. Rauch,
Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.
doi: 10.1137/0330055. |
[8] |
Z. Belhachmi, T. Glatz and O. Scherzer, Photoacoustic tomography with spatially varying
compressibility and density, J. Inverse Ill-Posed Probl., 25 (2017), 119-133, arXiv:1512.07411 (2015).
doi: 10.1515/jiip-2015-0113. |
[9] |
______, A direct method for photoacoustic tomography with inhomogeneous sound speed Inverse Problems, 32 (2016), 045005, 25pp.
doi: 10.1088/0266-5611/32/4/045005. |
[10] |
C. Byrne,
Iterative Algorithms in Inverse Problems, http://faculty.uml.edu/cbyrne/ITER2.pdf, 2006. |
[11] |
O. Chervova and L. Oksanen, Time reversal method with stabilizing boundary conditions for Photoacoustic tomography Inverse Problems, 32 (2016), arXiv: 1605.07817 (2016).
doi: 10.1088/0266-5611/32/12/125004. |
[12] |
B. T. Cox, S. R. Arridge and P. C. Beard,
Photoacoustic tomography with a limited-aperture planar sensor and a reverberant cavity, Inverse Problems, 23 (2007), S95-S112.
doi: 10.1088/0266-5611/23/6/S08. |
[13] |
M. A. Dablain,
The application of high-order differencing to the scalar wave equation, Geophysics, 51 (1986), 54-66.
doi: 10.1190/1.1442040. |
[14] | |
[15] |
Finch, Patch and Rakesh,
Determining a function from its mean values over a family of spheres, SIAM J. Math. Anal., 35 (2004), 1213-1240 (electronic).
doi: 10.1137/S0036141002417814. |
[16] |
J. A. Goldstein and M. Wacker,
The energy space and norm growth for abstract wave equations, Applied Mathematics Letters, 16 (2003), 767-772.
doi: 10.1016/S0893-9659(03)00080-6. |
[17] |
M. Hanke,
Accelerated Landweber iterations for the solution of ill-posed equations, Numer. Math., 60 (1991), 341-373.
doi: 10.1007/BF01385727. |
[18] |
M. Hanke, A. Neubauer and O. Scherzer,
A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72 (1995), 21-37.
doi: 10.1007/s002110050158. |
[19] |
B. Holman and L. Kunyansky, Gradual time reversal in thermo-and photo-acoustic tomography within a resonant cavity Inverse Problems, 31 (2015), 035008, 25pp.
doi: 10.1088/0266-5611/31/3/035008. |
[20] |
A. Homan,
Multi-wave imaging in attenuating media, Inverse Probl. Imaging, 7 (2013), 1235-1250.
doi: 10.3934/ipi.2013.7.1235. |
[21] |
K. R. Kelly, R. W. Ward, S. Treitel and R. M. Alford,
Synthetic seismograms: A finite-difference approach, Geophysics, 41 (1976), 2-27.
doi: 10.1190/1.1440605. |
[22] |
A. Kirsch,
An Introduction to the Mathematical Theory of Inverse Problems, second ed., Applied Mathematical Sciences, vol. 120, Springer, New York, 2011.
doi: 10.1007/978-1-4419-8474-6. |
[23] |
R.A. Kruger, W. L. Kiser, D. R. Reinecke and G. A. Kruger,
Thermoacoustic computed tomography using a conventional linear transducer array, Med Phys, 30 (2003), 856-860.
doi: 10.1118/1.1565340. |
[24] |
R. A. Kruger, D. R. Reinecke and G. A. Kruger,
Thermoacoustic computed tomography--technical considerations, Med Phys, 26 (1999), 1832-1837.
doi: 10.1118/1.598688. |
[25] |
P. Kuchment and L. Kunyansky, Mathematics of photoacoustic and thermoacoustic tomography, Handbook of Mathematical Methods in Imaging (Otmar Scherzer, ed.), Springer New
York, 2011, pp. 817-865. |
[26] |
L. Kunyansky, B. 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. |
[27] |
L. V. Nguyen and L. A. Kunyansky,
A dissipative time reversal technique for photoacoustic tomography in a cavity, SIAM J. Imaging Sci., 9 (2016), 748-769.
doi: 10.1137/15M1049683. |
[28] |
J. Qian, P. Stefanov, G. Uhlmann and H. Zhao,
An efficient Neumann series-based algorithm for thermoacoustic and photoacoustic tomography with variable sound speed, SIAM J. Imaging Sci., 4 (2011), 850-883.
doi: 10.1137/100817280. |
[29] |
M. Reed and B. Simon,
Methods of Modern Mathematical Physics. III, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979, Scattering theory. |
[30] |
______,
Methods of Modern Mathematical Physics. I, second ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980, Functional analysis. |
[31] |
P. Stefanov and G. Uhlmann,
Linearizing non-linear inverse problems and an application to inverse backscattering, J. Funct. Anal., 256 (2009), 2842-2866.
doi: 10.1016/j.jfa.2008.10.017. |
[32] |
______, Thermoacoustic tomography with variable sound speed,
Inverse Problems, 25 (2009), 075011, 16pp. |
[33] |
______, Thermoacoustic tomography arising in brain imaging,
Inverse Problems, 27 (2011), 045004, 26pp. |
[34] |
______, Multi-wave methods via ultrasound, Inside Out, MSRI Publications, 60 (2012), 271-324. . |
[35] |
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. |
[36] |
D. Tataru,
Unique continuation for solutions to PDE's; between Hörmander's theorem and Holmgren's theorem, Comm. Partial Differential Equations, 20 (1995), 855-884.
doi: 10.1080/03605309508821117. |
[37] |
______, On the regularity of boundary traces for the wave equation, Ann. Scuola Norm. Sup.
Pisa Cl. Sci., (4), 26 (1998), 185-206 |
[38] |
G. M. Vainikko,
Error estimates of the successive approximation method for ill-posed problems, Automat. Remote Control, (1980), 84-92.
|
[39] |
G. M. Vainikko and A. Yu. Veretennikov,
Iteration Procedures in Ill-Posed Problems (in russian)
"Nauka", Moscow, 1986. |
[40] |
M. Xu and L. V. Wang, Photoacoustic imaging in biomedicine Review of Scientific Instruments, 77 (2006), 041101.
doi: 10.1063/1.2195024. |
[41] |
Y. Xu and L. V. Wang,
Rhesus monkey brain imaging through intact skull with thermoacoustic tomography, IEEE Trans. Ultrason., Ferroelectr., Freq. Control, 53 (2006), 542-548.
|
[42] |
X. Yang and L. V. Wang, Monkey brain cortex imaging by photoacoustic tomography Proc. SPIE, 6856 (2008), 762396.
doi: 10.1117/12.762396. |
[43] |
E. Zuazua,
Numerics for the control of partial differential equations, Reference Work Entry: Encyclopedia of Applied and Computational Mathematics, (2015), 1076-1080.
doi: 10.1007/978-3-540-70529-1_362. |















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