# American Institute of Mathematical Sciences

February  2022, 16(1): 1-18. doi: 10.3934/ipi.2021038

## A stable non-iterative reconstruction algorithm for the acoustic inverse boundary value problem

 Department of Computational Mathematics Science and Engineering, Michigan State University, East Lansing, MI 48824, USA

* Corresponding author: Yang Yang

Received  January 2021 Revised  March 2021 Published  February 2022 Early access  April 2021

We present a non-iterative algorithm to reconstruct the isotropic acoustic wave speed from the measurement of the Neumann-to-Dirichlet map. The algorithm is designed based on the boundary control method and involves only computations that are stable. We prove the convergence of the algorithm and present its numerical implementation. The effectiveness of the algorithm is validated on both constant speed and variable speed, with full and partial boundary measurement as well as different levels of noise.

Citation: Tianyu Yang, Yang Yang. A stable non-iterative reconstruction algorithm for the acoustic inverse boundary value problem. Inverse Problems & Imaging, 2022, 16 (1) : 1-18. doi: 10.3934/ipi.2021038
##### References:
 [1] I. B. Aïcha, Stability estimate for a hyperbolic inverse problem with time-dependent coefficient, Inverse Problems, 31 (2015), 125010, 21 pp. doi: 10.1088/0266-5611/31/12/125010.  Google Scholar [2] G. Bao and H. Zhang, Sensitivity analysis of an inverse problem for the wave equation with caustics, J. Amer. Math. Soc., 27 (2014), 953-981.  doi: 10.1090/S0894-0347-2014-00787-6.  Google Scholar [3] M. I. Belishev, An approach to multidimensional inverse problems for the wave equation, Soviet Math. Dokl., 36 (1988), 481-484.   Google Scholar [4] M. I. Belishev, Recent progress in the boundary control method, Inverse Problems, 23 (2007), R1–R67. doi: 10.1088/0266-5611/23/5/R01.  Google Scholar [5] M. I. Belishev and V. Y. Gotlib, Dynamical variant of the BC-method: Theory and numerical testing, J. Inverse Ill-Posed Probl., 7 (1999), 221-240.  doi: 10.1515/jiip.1999.7.3.221.  Google Scholar [6] M. I. Belishev, I. B. Ivanov, I. V. Kubyshkin and V. S. Semenov, Numerical testing in determination of sound speed from a part of boundary by the BC-method, J. Inverse Ill-Posed Probl., 24 (2016), 159-180.  doi: 10.1515/jiip-2015-0052.  Google Scholar [7] M. I. Belishev and Y. V. Kuryiev, To the reconstruction of a Riemannian manifold via its spectral data (BC–method), Comm. Partial Differential Equations, 17 (1992), 767-804.  doi: 10.1080/03605309208820863.  Google Scholar [8] M. Bellassoued and I. B. Aïcha, Stable determination outside a cloaking region of two time-dependent coefficients in an hyperbolic equation from Dirichlet to Neumann map, J. Math. Anal. Appl., 449 (2017), 46-76.  doi: 10.1016/j.jmaa.2016.11.082.  Google Scholar [9] M. Bellassoued and D. Dos Santos Ferreira, Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map, Inverse Probl. Imaging, 5 (2011), 745-773.  doi: 10.3934/ipi.2011.5.745.  Google Scholar [10] K. Bingham, Y. Kurylev, M. Lassas and S. Siltanen, Iterative time-reversal control for inverse problems, Inverse Probl. Imaging, 2 (2008), 63-81.  doi: 10.3934/ipi.2008.2.63.  Google Scholar [11] A. S. Blagoveščenskiǐ, The inverse problem in the theory of seismic wave propagation, in Spectral Theory and Wave Processes, Springer, 1 (1966), 68–81.  Google Scholar [12] E. Blåsten, F. Zouari, M. Louati and M. S. Ghidaoui, Blockage detection in networks: The area reconstruction method, Math. Eng., 1 (2019), 849–880. arXiv: 1909.05497. doi: 10.3934/mine.2019.4.849.  Google Scholar [13] A. Bogomolny, Fundamental solutions method for elliptic boundary value problems, SIAM J. Numer. Anal., 22 (1985), 644-669.  doi: 10.1137/0722040.  Google Scholar [14] R. Bosi, Y. Kurylev and M. Lassas, Reconstruction and stability in Gel'fand's inverse interior spectral problem, arXiv preprint, arXiv: 1702.07937. Google Scholar [15] M. V. de Hoop, P. Kepley and L. Oksanen, An exact redatuming procedure for the inverse boundary value problem for the wave equation, SIAM J. Appl. Math., 78 (2018), 171-192.  doi: 10.1137/16M1106729.  Google Scholar [16] M. V. de Hoop, P. Kepley and L. Oksanen, Recovery of a smooth metric via wave field and coordinate transformation reconstruction, SIAM J. Appl. Math., 78 (2018), 1931-1953.  doi: 10.1137/17M1151481.  Google Scholar [17] G. Eskin, A new approach to hyperbolic inverse problems, Inverse Problems, 22 (2006), 815-831.  doi: 10.1088/0266-5611/22/3/005.  Google Scholar [18] G. Eskin, Inverse hyperbolic problems with time-dependent coefficients, Comm. Partial Differential Equations, 32 (2007), 1737-1758.  doi: 10.1080/03605300701382340.  Google Scholar [19] G. Eskin, Inverse problems for the Schrödinger equations with time-dependent electromagnetic potentials and the Aharonov–Bohm effect, J. Math. Phys., 49 (2008), 022105, 18 pp. doi: 10.1063/1.2841329.  Google Scholar [20] G. Eskin, Inverse problems for general second order hyperbolic equations with time-dependent coefficients, Bull. Math. Sci., 7 (2017), 247-307.  doi: 10.1007/s13373-017-0100-2.  Google Scholar [21] G. Fairweather and A. Karageorghis, The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9 (1998), 69-95.  doi: 10.1023/A:1018981221740.  Google Scholar [22] A. Feizmohammadi and Y. Kian, Recovery of non-smooth coefficients appearing in anisotropic wave equations, SIAM J. Math. Anal., 51 (2019), 4953–4976. arXiv: 1903.08118. doi: 10.1137/19M1251394.  Google Scholar [23] G. Hu and Y. Kian, Determination of singular time-dependent coefficients for wave equations from full and partial data, Inverse Probl. Imaging 12 (2018), 745–772. arXiv: 1706.07212. doi: 10.3934/ipi.2018032.  Google Scholar [24] V. Isakov and Z. Sun, Stability estimates for hyperbolic inverse problems with local boundary data, Inverse Problems, 8 (1992), 193-206.  doi: 10.1088/0266-5611/8/2/003.  Google Scholar [25] Y. Kian, Recovery of time-dependent damping coefficients and potentials appearing in wave equations from partial data, SIAM J. Math. Anal., 48 (2016), 4021-4046.  doi: 10.1137/16M1076708.  Google Scholar [26] Y. Kian and L. Oksanen, Recovery of time-dependent coefficient on Riemannian manifold for hyperbolic equations, Int. Math. Res. Not. IMRN, 2019 (2019), 5087-5126.  doi: 10.1093/imrn/rnx263.  Google Scholar [27] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, vol. 120, Springer Science & Business Media, 2011. doi: 10.1007/978-1-4419-8474-6.  Google Scholar [28] J. Korpela, M. Lassas and L. Oksanen, Discrete regularization and convergence of the inverse problem for $1+1$ dimensional wave equation, Inverse Probl. Imaging, 13 (2019), 575-596.  doi: 10.3934/ipi.2019027.  Google Scholar [29] V. D. Kupradze, On the approximate solution of problems in mathematical physics, (Russian) Uspehi Mat., 22 (1967), 59-107.   Google Scholar [30] Y. V. Kurylev and M. Lassas, Hyperbolic inverse problem with data on a part of the boundary, in UAB-GIT International Conference on Differential Equations and Mathematical Physics, American Mathematical Society, 16 (2000), 259–272.  Google Scholar [31] Y. Kurylev, L. Oksanen and G. P. Paternain, Inverse problems for the connection laplacian, J. Differential Geom., 110 (2018), 457-494.  doi: 10.4310/jdg/1542423627.  Google Scholar [32] I. Lasiecka and R. Triggiani, Regularity theory of hyperbolic equations with non-homogeneous Neumann boundary conditions. II. General boundary data, J. Differential Equations, 94 (1991), 112-164.  doi: 10.1016/0022-0396(91)90106-J.  Google Scholar [33] M. Lassas and L. Oksanen, Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets, Duke Math. J., 163 (2014), 1071-1103.  doi: 10.1215/00127094-2649534.  Google Scholar [34] S. Liu and L. Oksanen, A Lipschitz stable reconstruction formula for the inverse problem for the wave equation, Trans. Amer. Math. Soc., 368 (2016), 319-335.  doi: 10.1090/tran/6332.  Google Scholar [35] R. Mathon and R. L. Johnston, The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM J. Numer. Anal., 14 (1977), 638-650.  doi: 10.1137/0714043.  Google Scholar [36] T. P. Matthews and M. A. Anastasio, Joint reconstruction of the initial pressure and speed of sound distributions from combined photoacoustic and ultrasound tomography measurements, Inverse Problems, 33 (2017), 124002, 24 pp. doi: 10.1088/1361-6420/aa9384.  Google Scholar [37] C. Montalto, Stable determination of a simple metric, a covector field and a potential from the hyperbolic Dirichlet-to-Neumann map, Comm. Partial Differential Equations, 39 (2014), 120-145.  doi: 10.1080/03605302.2013.843429.  Google Scholar [38] A. I. Nachman, Reconstructions from boundary measurements, Ann. of Math., 128 (1988), 531-576.  doi: 10.2307/1971435.  Google Scholar [39] L. Oksanen, Solving an inverse obstacle problem for the wave equation by using the boundary control method, Inverse Problems, 29 (2013), 035004, 12 pp. doi: 10.1088/0266-5611/29/3/035004.  Google Scholar [40] L. N. Pestov, On reconstruction of the speed of sound from a part of boundary, J. Inverse Ill-Posed Probl., 7 (1999), 481-486.  doi: 10.1515/jiip.1999.7.5.481.  Google Scholar [41] L. Pestov, V. Bolgova and O. Kazarina, Numerical recovering of a density by the BC-method, Inverse Probl. Imaging, 4 (2010), 703-712.  doi: 10.3934/ipi.2010.4.703.  Google Scholar [42] A. G. Ramm and Ra kesh, Property $C$ and an inverse problem for a hyperbolic equation, J. Math. Anal. Appl., 156 (1991), 209-219.  doi: 10.1016/0022-247X(91)90391-C.  Google Scholar [43] R. Salazar, Determination of time-dependent coefficients for a hyperbolic inverse problem, Inverse Problems, 29 (2013), 095015, 17 pp. doi: 10.1088/0266-5611/29/9/095015.  Google Scholar [44] P. D. Stefanov, Uniqueness of the multi-dimensional inverse scattering problem for time dependent potentials, Math. Z., 201 (1989), 541-559.  doi: 10.1007/BF01215158.  Google Scholar [45] P. Stefanov and G. Uhlmann, Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media, J. Funct. Anal., 154 (1998), 330-358.  doi: 10.1006/jfan.1997.3188.  Google Scholar [46] P. Stefanov and G. Uhlmann, Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map, Int. Math. Res. Not., 2005 (2005), 1047-1061.  doi: 10.1155/IMRN.2005.1047.  Google Scholar [47] P. Stefanov and Y. Yang, The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds, Anal. PDE, 11 (2018), 1381-1414.  doi: 10.2140/apde.2018.11.1381.  Google Scholar [48] 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.  Google Scholar [49] D. Tataru, On the regularity of boundary traces for the wave equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 185-206.   Google Scholar

show all references

##### References:
 [1] I. B. Aïcha, Stability estimate for a hyperbolic inverse problem with time-dependent coefficient, Inverse Problems, 31 (2015), 125010, 21 pp. doi: 10.1088/0266-5611/31/12/125010.  Google Scholar [2] G. Bao and H. Zhang, Sensitivity analysis of an inverse problem for the wave equation with caustics, J. Amer. Math. Soc., 27 (2014), 953-981.  doi: 10.1090/S0894-0347-2014-00787-6.  Google Scholar [3] M. I. Belishev, An approach to multidimensional inverse problems for the wave equation, Soviet Math. Dokl., 36 (1988), 481-484.   Google Scholar [4] M. I. Belishev, Recent progress in the boundary control method, Inverse Problems, 23 (2007), R1–R67. doi: 10.1088/0266-5611/23/5/R01.  Google Scholar [5] M. I. Belishev and V. Y. Gotlib, Dynamical variant of the BC-method: Theory and numerical testing, J. Inverse Ill-Posed Probl., 7 (1999), 221-240.  doi: 10.1515/jiip.1999.7.3.221.  Google Scholar [6] M. I. Belishev, I. B. Ivanov, I. V. Kubyshkin and V. S. Semenov, Numerical testing in determination of sound speed from a part of boundary by the BC-method, J. Inverse Ill-Posed Probl., 24 (2016), 159-180.  doi: 10.1515/jiip-2015-0052.  Google Scholar [7] M. I. Belishev and Y. V. Kuryiev, To the reconstruction of a Riemannian manifold via its spectral data (BC–method), Comm. Partial Differential Equations, 17 (1992), 767-804.  doi: 10.1080/03605309208820863.  Google Scholar [8] M. Bellassoued and I. B. Aïcha, Stable determination outside a cloaking region of two time-dependent coefficients in an hyperbolic equation from Dirichlet to Neumann map, J. Math. Anal. Appl., 449 (2017), 46-76.  doi: 10.1016/j.jmaa.2016.11.082.  Google Scholar [9] M. Bellassoued and D. Dos Santos Ferreira, Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map, Inverse Probl. Imaging, 5 (2011), 745-773.  doi: 10.3934/ipi.2011.5.745.  Google Scholar [10] K. Bingham, Y. Kurylev, M. Lassas and S. Siltanen, Iterative time-reversal control for inverse problems, Inverse Probl. Imaging, 2 (2008), 63-81.  doi: 10.3934/ipi.2008.2.63.  Google Scholar [11] A. S. Blagoveščenskiǐ, The inverse problem in the theory of seismic wave propagation, in Spectral Theory and Wave Processes, Springer, 1 (1966), 68–81.  Google Scholar [12] E. Blåsten, F. Zouari, M. Louati and M. S. Ghidaoui, Blockage detection in networks: The area reconstruction method, Math. Eng., 1 (2019), 849–880. arXiv: 1909.05497. doi: 10.3934/mine.2019.4.849.  Google Scholar [13] A. Bogomolny, Fundamental solutions method for elliptic boundary value problems, SIAM J. Numer. Anal., 22 (1985), 644-669.  doi: 10.1137/0722040.  Google Scholar [14] R. Bosi, Y. Kurylev and M. Lassas, Reconstruction and stability in Gel'fand's inverse interior spectral problem, arXiv preprint, arXiv: 1702.07937. Google Scholar [15] M. V. de Hoop, P. Kepley and L. Oksanen, An exact redatuming procedure for the inverse boundary value problem for the wave equation, SIAM J. Appl. Math., 78 (2018), 171-192.  doi: 10.1137/16M1106729.  Google Scholar [16] M. V. de Hoop, P. Kepley and L. Oksanen, Recovery of a smooth metric via wave field and coordinate transformation reconstruction, SIAM J. Appl. Math., 78 (2018), 1931-1953.  doi: 10.1137/17M1151481.  Google Scholar [17] G. Eskin, A new approach to hyperbolic inverse problems, Inverse Problems, 22 (2006), 815-831.  doi: 10.1088/0266-5611/22/3/005.  Google Scholar [18] G. Eskin, Inverse hyperbolic problems with time-dependent coefficients, Comm. Partial Differential Equations, 32 (2007), 1737-1758.  doi: 10.1080/03605300701382340.  Google Scholar [19] G. Eskin, Inverse problems for the Schrödinger equations with time-dependent electromagnetic potentials and the Aharonov–Bohm effect, J. Math. Phys., 49 (2008), 022105, 18 pp. doi: 10.1063/1.2841329.  Google Scholar [20] G. Eskin, Inverse problems for general second order hyperbolic equations with time-dependent coefficients, Bull. Math. Sci., 7 (2017), 247-307.  doi: 10.1007/s13373-017-0100-2.  Google Scholar [21] G. Fairweather and A. Karageorghis, The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9 (1998), 69-95.  doi: 10.1023/A:1018981221740.  Google Scholar [22] A. Feizmohammadi and Y. Kian, Recovery of non-smooth coefficients appearing in anisotropic wave equations, SIAM J. Math. Anal., 51 (2019), 4953–4976. arXiv: 1903.08118. doi: 10.1137/19M1251394.  Google Scholar [23] G. Hu and Y. Kian, Determination of singular time-dependent coefficients for wave equations from full and partial data, Inverse Probl. Imaging 12 (2018), 745–772. arXiv: 1706.07212. doi: 10.3934/ipi.2018032.  Google Scholar [24] V. Isakov and Z. Sun, Stability estimates for hyperbolic inverse problems with local boundary data, Inverse Problems, 8 (1992), 193-206.  doi: 10.1088/0266-5611/8/2/003.  Google Scholar [25] Y. Kian, Recovery of time-dependent damping coefficients and potentials appearing in wave equations from partial data, SIAM J. Math. Anal., 48 (2016), 4021-4046.  doi: 10.1137/16M1076708.  Google Scholar [26] Y. Kian and L. Oksanen, Recovery of time-dependent coefficient on Riemannian manifold for hyperbolic equations, Int. Math. Res. Not. IMRN, 2019 (2019), 5087-5126.  doi: 10.1093/imrn/rnx263.  Google Scholar [27] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, vol. 120, Springer Science & Business Media, 2011. doi: 10.1007/978-1-4419-8474-6.  Google Scholar [28] J. Korpela, M. Lassas and L. Oksanen, Discrete regularization and convergence of the inverse problem for $1+1$ dimensional wave equation, Inverse Probl. Imaging, 13 (2019), 575-596.  doi: 10.3934/ipi.2019027.  Google Scholar [29] V. D. Kupradze, On the approximate solution of problems in mathematical physics, (Russian) Uspehi Mat., 22 (1967), 59-107.   Google Scholar [30] Y. V. Kurylev and M. Lassas, Hyperbolic inverse problem with data on a part of the boundary, in UAB-GIT International Conference on Differential Equations and Mathematical Physics, American Mathematical Society, 16 (2000), 259–272.  Google Scholar [31] Y. Kurylev, L. Oksanen and G. P. Paternain, Inverse problems for the connection laplacian, J. Differential Geom., 110 (2018), 457-494.  doi: 10.4310/jdg/1542423627.  Google Scholar [32] I. Lasiecka and R. Triggiani, Regularity theory of hyperbolic equations with non-homogeneous Neumann boundary conditions. II. General boundary data, J. Differential Equations, 94 (1991), 112-164.  doi: 10.1016/0022-0396(91)90106-J.  Google Scholar [33] M. Lassas and L. Oksanen, Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets, Duke Math. J., 163 (2014), 1071-1103.  doi: 10.1215/00127094-2649534.  Google Scholar [34] S. Liu and L. Oksanen, A Lipschitz stable reconstruction formula for the inverse problem for the wave equation, Trans. Amer. Math. Soc., 368 (2016), 319-335.  doi: 10.1090/tran/6332.  Google Scholar [35] R. Mathon and R. L. Johnston, The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM J. Numer. Anal., 14 (1977), 638-650.  doi: 10.1137/0714043.  Google Scholar [36] T. P. Matthews and M. A. Anastasio, Joint reconstruction of the initial pressure and speed of sound distributions from combined photoacoustic and ultrasound tomography measurements, Inverse Problems, 33 (2017), 124002, 24 pp. doi: 10.1088/1361-6420/aa9384.  Google Scholar [37] C. Montalto, Stable determination of a simple metric, a covector field and a potential from the hyperbolic Dirichlet-to-Neumann map, Comm. Partial Differential Equations, 39 (2014), 120-145.  doi: 10.1080/03605302.2013.843429.  Google Scholar [38] A. I. Nachman, Reconstructions from boundary measurements, Ann. of Math., 128 (1988), 531-576.  doi: 10.2307/1971435.  Google Scholar [39] L. Oksanen, Solving an inverse obstacle problem for the wave equation by using the boundary control method, Inverse Problems, 29 (2013), 035004, 12 pp. doi: 10.1088/0266-5611/29/3/035004.  Google Scholar [40] L. N. Pestov, On reconstruction of the speed of sound from a part of boundary, J. Inverse Ill-Posed Probl., 7 (1999), 481-486.  doi: 10.1515/jiip.1999.7.5.481.  Google Scholar [41] L. Pestov, V. Bolgova and O. Kazarina, Numerical recovering of a density by the BC-method, Inverse Probl. Imaging, 4 (2010), 703-712.  doi: 10.3934/ipi.2010.4.703.  Google Scholar [42] A. G. Ramm and Ra kesh, Property $C$ and an inverse problem for a hyperbolic equation, J. Math. Anal. Appl., 156 (1991), 209-219.  doi: 10.1016/0022-247X(91)90391-C.  Google Scholar [43] R. Salazar, Determination of time-dependent coefficients for a hyperbolic inverse problem, Inverse Problems, 29 (2013), 095015, 17 pp. doi: 10.1088/0266-5611/29/9/095015.  Google Scholar [44] P. D. Stefanov, Uniqueness of the multi-dimensional inverse scattering problem for time dependent potentials, Math. Z., 201 (1989), 541-559.  doi: 10.1007/BF01215158.  Google Scholar [45] P. Stefanov and G. Uhlmann, Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media, J. Funct. Anal., 154 (1998), 330-358.  doi: 10.1006/jfan.1997.3188.  Google Scholar [46] P. Stefanov and G. Uhlmann, Stable determination of generic simple metrics from the hyperbolic Dirichlet-to-Neumann map, Int. Math. Res. Not., 2005 (2005), 1047-1061.  doi: 10.1155/IMRN.2005.1047.  Google Scholar [47] P. Stefanov and Y. Yang, The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds, Anal. PDE, 11 (2018), 1381-1414.  doi: 10.2140/apde.2018.11.1381.  Google Scholar [48] 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.  Google Scholar [49] D. Tataru, On the regularity of boundary traces for the wave equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 185-206.   Google Scholar
]">Figure 1.  Data acquisition scheme in USCT [36]
The structure of $[\Lambda_c]$ with $I = 15$ and $L = 63$. nz is the number of nonzero elements in the matrix
The structure of $[J],[P_T],[R],[K]$ with $I = 15$ and $L = 63$. nz is the number of nonzero elements in the matrix
The singular values of $[K]$ with $I = 15,L = 63$. $[K]$ has 60 zero singular values
Reconstructions of the constant speed $c = 1$. Top row: reconstructed $c$. Bottom row: error between the reconstruction and the ground truth. First column: $0\%$ noise; the relative $L^2$-error is $0.4769\%$. Second column: $5\%$ noise; the relative $L^2$-error is $0.4873\%$. Third column: $50\%$ noise; the relative $L^2$-error is $0.5454\%$. Grid: $283\times51\times51$, $I = 50,L = 282$
Left: the variable speed $c^{-2} = \sum\limits_{i = 1}^6\frac{i}{10}\phi^{(i)}$. Middle: the variable speed $c(x,y) = 1+0.08\sin{\pi x}+0.06\cos{\pi y}$. Right: orthogonal projection of $c(x,y) = 1+0.08\sin{\pi x}+0.06\cos{\pi y}$ on $S_6$
Reconstructions of the variable speed $c^{-2} = \sum\limits_{i = 1}^6\frac{i}{10}\phi^{(i)}$. Top row: reconstructed $c$. Bottom row: error between the reconstruction and the ground truth. First column: First $2$ harmonic functions; the relative $L^2$-error is $15.6987\%$. Second column: First $4$ harmonic functions; the relative $L^2$-error is $0.7939\%$. Third column: All $6$ harmonic functions; the relative $L^2$-error is $0.7907\%$. Grid: $323\times51\times51$, $I = 50,L = 163$
Reconstructions of the variable speed $c(x,y) = 1+0.08\sin{\pi x}+0.06\cos{\pi y}$. Top row: reconstructed $c$. Bottom row: error between the reconstruction and the orthogonal projection of the ground truth. First column: First $2$ harmonic functions; the relative $L^2$-error is $12.3535\%$. Second column: First $4$ harmonic functions; the relative $L^2$-error is $0.4139\%$. Third column: All $6$ harmonic functions; the relative $L^2$-error is $0.3104\%$. Grid: $323\times51\times51$, $I = 50,L = 322$
Top row: reconstructed $c$. Bottom row: error between the reconstruction and the ground truth. First column: no data on $y = -1$; the relative $L^2$-error is $0.4954\%$. Second column: no data on $y = -1$ and $x = 1$; the relative $L^2$-error is $0.6583\%$. Third column: no data on $y = \pm 1$ and $x = 1$; the relative $L^2$-error is $1.2518\%$. Grid: $283\times51\times51$, $I = 50,L = 282$
The ground truth speed c and its projection
Left: reconstructed $c$. Right: error between the reconstruction and the orthogonal projection of the ground truth. The relative $L^2$ error is $1.7289\%$. Grid: $283\times51\times51$, $I = 50,L = 282$
Impact of random and constant noises on $[K]$
 Random Noise Relative Frobenius Error 10% 3.74% 20% 7.51% 50% 18.80% Constant Noise Relative Frobenius Error 0.01 119.01% 0.02 238.01% 0.05 595.03%
 Random Noise Relative Frobenius Error 10% 3.74% 20% 7.51% 50% 18.80% Constant Noise Relative Frobenius Error 0.01 119.01% 0.02 238.01% 0.05 595.03%
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