October  2020, 14(5): 913-938. doi: 10.3934/ipi.2020042

Convexification for a 1D hyperbolic coefficient inverse problem with single measurement data

Department of Mathematics and Statistics, University of North Carolina Charlotte, Charlotte, NC 28223, USA

* Corresponding author: Michael Klibanov (mklibanv@uncc.edu)

Received  February 2020 Revised  May 2020 Published  July 2020

Fund Project: The work was supported by US Army Research Laboratory and US Army Research Office grant W911NF-19-1-0044

A version of the convexification numerical method for a Coefficient Inverse Problem for a 1D hyperbolic PDE is presented. The data for this problem are generated by a single measurement event. This method converges globally. The most important element of the construction is the presence of the Carleman Weight Function in a weighted Tikhonov-like functional. This functional is strictly convex on a certain bounded set in a Hilbert space, and the diameter of this set is an arbitrary positive number. The global convergence of the gradient projection method is established. Computational results demonstrate a good performance of the numerical method for noisy data.

Citation: Alexey Smirnov, Michael Klibanov, Loc Nguyen. Convexification for a 1D hyperbolic coefficient inverse problem with single measurement data. Inverse Problems & Imaging, 2020, 14 (5) : 913-938. doi: 10.3934/ipi.2020042
References:
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A. B. BakushinskiiM. V. Klibanov and N. A. Koshev, Carleman weight functions for a globally convergent numerical method for ill-posed Cauchy problems for some quasilinear PDEs, Nonlinear Analysis: Real World Applications, 34 (2017), 201-224.  doi: 10.1016/j.nonrwa.2016.08.008.  Google Scholar

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Y. Ahajjam, O. Aghzout, and J. M. Catalá-Civera, F. Peñaranda-Foix and A. Driouach, A compact UWB sub-nanosecond pulse generator for microwave radar sensor with ringing miniaturization, in 2016 5th International Conference on Multimedia Computing and Systems (ICMCS), IEEE, 2016,497–501. Google Scholar

[3]

L. Baudouin, M. de Buhan, S. Ervedoza and A. Osses, Carleman-based reconstruction algorithm for the waves, preprint, available at https: //hal.archives-ouvertes.fr/hal-02458787, 2020. Google Scholar

[4]

L. BaudouinM. de Buhan and S. Ervedoza, Convergent algorithm based on Carleman estimates for the recovery of a potential in the wave equation, SIAM Journal on Numerical Analysis, 55 (2017), 1578-1613.  doi: 10.1137/16M1088776.  Google Scholar

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L. Beilina and M. V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer Science & Business Media, 2012. Google Scholar

[6]

L. Beilina and M. V. Klibanov, Globally strongly convex cost functional for a coefficient inverse problem, Nonlinear Analysis: Real World Applications, 22 (2015), 272-288.  doi: 10.1016/j.nonrwa.2014.09.015.  Google Scholar

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M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer, 2017. doi: 10.1007/978-4-431-56600-7.  Google Scholar

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M. Boulakia, M. de Buhan and E. Schwindt, Numerical reconstruction based on Carleman estimates of a source term in a reaction-diffusion equation, 2019. Google Scholar

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A. L. Buchgeim, Uniqueness in the large of a class of multidimensional inverse problems, Sov. Math. Dokl., 24 (1981), 244-247.   Google Scholar

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D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, Springer Nature, 93 (2019). doi: 10.1007/978-3-030-30351-8.  Google Scholar

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B. Engquist and A. Majda, Absorbing boundary conditions for numerical simulation of waves, Proceedings of the National Academy of Sciences, 74 (1977), 1765-1766.  doi: 10.1073/pnas.74.5.1765.  Google Scholar

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S. I. Kabanikhin, A. D. Satybaev and M. A. Shishlenin, Direct methods of solving multidimensional inverse hyperbolic problems, Walter de Gruyter, 48 (2013).  Google Scholar

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S. I. KabanikhinK. K. SabelfeldN. S. Novikov and M. A. Shishlenin, Numerical solution of the multidimensional Gelfand–Levitan equation, Journal of Inverse and Ill-Posed Problems, 23 (2015), 439-450.  doi: 10.1515/jiip-2014-0018.  Google Scholar

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S. I. KabanikhinN. S. NovikovI. V. Oseledets and M. A. Shishlenin, Fast Toeplitz linear system inversion for solving two-dimensional acoustic inverse problem, Journal of Inverse and Ill-Posed Problems, 23 (2015), 687-700.  doi: 10.1515/jiip-2015-0083.  Google Scholar

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V. A. KhoaM. V. Klibanov and L. H. Nguyen, Convexification for a three-dimensional inverse scattering problem with the moving point source, SIAM Journal on Imaging Sciences, 13 (2020), 871-904.  doi: 10.1137/19M1303101.  Google Scholar

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M. V. Klibanov, Inverse problems in the large and Carleman bounds, Differential Equations, 20 (1984), 755-760.   Google Scholar

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[20]

J. S. Lee and C. Nguyen, Novel low-cost ultra-wideband, ultra-short-pulse transmitter with MESFET impulse-shaping circuitry for reduced distortion and improved pulse repetition rate, IEEE Microwave and Wireless Components Letters, 11 (2001), 208-210.   Google Scholar

[21]

V. A. Khoa, G. W. Bidney, M. V. Klibanov, L. H. Nguyen, L. H. Nguyen, A. J. Sullivan and V. N. Astratov, Convexification and experimental data for a 3D inverse scattering problem with the moving point source, preprint, arXiv: 2003.11513 doi: 10.1137/19M1303101.  Google Scholar

[22]

I. M. Gelfand and B. M. Levitan, On determining a differential equation from its spectral function, Amer. Math. Soc. Transl., 2 (1955), 253-304.  doi: 10.1090/trans2/001/11.  Google Scholar

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A. L. KarchevskyM. V. KlibanovL. NguyenN. Pantong and A. Sullivan, The Krein method and the globally convergent method for experimental data, Appl. Numer. Math., 74 (2013), 111-127.  doi: 10.1016/j.apnum.2013.09.003.  Google Scholar

[25]

M. V. Klibanov and A. A. Timonov, Carleman estimates for coefficient inverse problems and numerical applications, Walter de Gruyter, 46 (2012). doi: 10.1515/9783110915549.  Google Scholar

[26]

M. V. KlibanovA. V. KuzhugetS. I. Kabanikhin and D. V. Nechaev, A new version of the quasi-reversibility method for the thermoacoustic tomography and a coefficient inverse problem, Applicable Analysis, 87 (2008), 1227-1254.  doi: 10.1080/00036810802001297.  Google Scholar

[27]

M. V. Klibanov, Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, Journal of Inverse and Ill-Posed Problems, 21 (2013), 477-560.  doi: 10.1515/jip-2012-0072.  Google Scholar

[28]

M. V. Klibanov and V. G. Kamburg, Globally strictly convex cost functional for an inverse parabolic problem, Mathematical Methods in the Applied Sciences, 39 (2016), 930-940.  doi: 10.1002/mma.3531.  Google Scholar

[29]

M. V. Klibanov, L. H. Nguyen, A. Sullivan and L. Nguyen, A globally convergent numerical method for a 1-d inverse medium problem with experimental data, preprint, arXiv: 1602.09092. doi: 10.3934/ipi.2016032.  Google Scholar

[30]

M. V. Klibanov, A. E. Kolesov, A. Sullivan and L. Nguyen, A new version of the convexification method for a 1D coefficient inverse problem with experimental data, Inverse Problems, 34 (2018), 115014. doi: 10.1088/1361-6420/aadbc6.  Google Scholar

[31]

L. Baudouin, M. de Buhan, S. Ervedoza and A. Osses, Carleman-based reconstruction algorithm for the waves, 2020. Google Scholar

[32]

M. V. Klibanov, J. Li and W. Zhang, Convexification of electrical impedance tomography with restricted Dirichlet-to-Neumann map data, Inverse Problems, (2019). doi: 10.1088/1361-6420/aafecd.  Google Scholar

[33]

M. V. KlibanovJ. Li and W. Zhang, Convexification for the inversion of a time dependent wave front in a heterogeneous medium, SIAM J. Appl. Math., 79 (2019), 1722-1747.  doi: 10.1137/18M1236034.  Google Scholar

[34]

M. V. Klibanov and A. E. Kolesov, Convexification of a 3-D coefficient inverse scattering problem, Computers & Mathematics with Applications, 77 (2019), 1681-1702.  doi: 10.1016/j.camwa.2018.03.016.  Google Scholar

[35]

M. V. KlibanovA. E. Kolesov and D.-L. Nguyen, Convexification method for an inverse scattering problem and its performance for experimental backscatter data for buried targets, SIAM J. Imag. Sci., 12 (2019), 576-603.  doi: 10.1137/18M1191658.  Google Scholar

[36]

A. Einstein, Onthe electrodynamics of moving bodies, Annalen der Physik, 322 (1905), 891-921.   Google Scholar

[37]

M. V. Klibanov, J. Li and W. Zhang, Convexification for an inverse parabolic problem, preprint, arXiv: 2001.01880. Google Scholar

[38]

J. Korpela, M. Lassas and L. Oksanen, Regularization strategy for an inverse problem for a 1+ 1 dimensional wave equation, Inverse Problems, 32 (2016), 065001. doi: 10.1088/0266-5611/32/6/065001.  Google Scholar

[39]

J. Korpela, M. Lassas and L. Oksanen, Discrete regularization and convergence of the inverse problem for 1+ 1 dimensional wave equation, preprint, arXiv: 1803.10541. doi: 10.3934/ipi.2019027.  Google Scholar

[40]

A. V. Kuzhuget, et al., Blind backscattering experimental data collected in the field and an approximately globally convergent inverse algorithm, Inverse Problems, 28 (2012), 095007. Google Scholar

[41]

T. T. Le and L. H. Nguyen, A convergent numerical method to recover the initial condition of nonlinear parabolic equations from lateral Cauchy data, preprint, arXiv: 1910.05584 Google Scholar

[42]

B. T. Polyak, Introduction to optimization. Optimization software, Inc., Publications Division, New York, 1 (1987).  Google Scholar

[43]

V. G. Romanov, Inverse Problems of Mathematical Physics, Walter de Gruyter GmbH & Co KG, 2018. Google Scholar

[44]

J. A. ScalesM. L. Smith and T. L. Fischer, Global optimization methods for multimodal inverse problems, J. Comput. Phys., 103 (1992), 258-268.   Google Scholar

[45]

A. N. Tikhonov, A. Goncharsky, V. Stepanov and A. G. Yagola, Numerical methods for the solution of ill-posed problems, Springer Science & Business Media, 328 (2013). doi: 10.1007/978-94-015-8480-7.  Google Scholar

show all references

References:
[1]

A. B. BakushinskiiM. V. Klibanov and N. A. Koshev, Carleman weight functions for a globally convergent numerical method for ill-posed Cauchy problems for some quasilinear PDEs, Nonlinear Analysis: Real World Applications, 34 (2017), 201-224.  doi: 10.1016/j.nonrwa.2016.08.008.  Google Scholar

[2]

Y. Ahajjam, O. Aghzout, and J. M. Catalá-Civera, F. Peñaranda-Foix and A. Driouach, A compact UWB sub-nanosecond pulse generator for microwave radar sensor with ringing miniaturization, in 2016 5th International Conference on Multimedia Computing and Systems (ICMCS), IEEE, 2016,497–501. Google Scholar

[3]

L. Baudouin, M. de Buhan, S. Ervedoza and A. Osses, Carleman-based reconstruction algorithm for the waves, preprint, available at https: //hal.archives-ouvertes.fr/hal-02458787, 2020. Google Scholar

[4]

L. BaudouinM. de Buhan and S. Ervedoza, Convergent algorithm based on Carleman estimates for the recovery of a potential in the wave equation, SIAM Journal on Numerical Analysis, 55 (2017), 1578-1613.  doi: 10.1137/16M1088776.  Google Scholar

[5]

L. Beilina and M. V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer Science & Business Media, 2012. Google Scholar

[6]

L. Beilina and M. V. Klibanov, Globally strongly convex cost functional for a coefficient inverse problem, Nonlinear Analysis: Real World Applications, 22 (2015), 272-288.  doi: 10.1016/j.nonrwa.2014.09.015.  Google Scholar

[7]

M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer, 2017. doi: 10.1007/978-4-431-56600-7.  Google Scholar

[8]

M. Boulakia, M. de Buhan and E. Schwindt, Numerical reconstruction based on Carleman estimates of a source term in a reaction-diffusion equation, 2019. Google Scholar

[9]

A. L. Buchgeim, Uniqueness in the large of a class of multidimensional inverse problems, Sov. Math. Dokl., 24 (1981), 244-247.   Google Scholar

[10]

D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, Springer Nature, 93 (2019). doi: 10.1007/978-3-030-30351-8.  Google Scholar

[11]

B. Engquist and A. Majda, Absorbing boundary conditions for numerical simulation of waves, Proceedings of the National Academy of Sciences, 74 (1977), 1765-1766.  doi: 10.1073/pnas.74.5.1765.  Google Scholar

[12]

S. I. Kabanikhin, A. D. Satybaev and M. A. Shishlenin, Direct methods of solving multidimensional inverse hyperbolic problems, Walter de Gruyter, 48 (2013).  Google Scholar

[13]

S. I. KabanikhinK. K. SabelfeldN. S. Novikov and M. A. Shishlenin, Numerical solution of the multidimensional Gelfand–Levitan equation, Journal of Inverse and Ill-Posed Problems, 23 (2015), 439-450.  doi: 10.1515/jiip-2014-0018.  Google Scholar

[14]

S. I. KabanikhinN. S. NovikovI. V. Oseledets and M. A. Shishlenin, Fast Toeplitz linear system inversion for solving two-dimensional acoustic inverse problem, Journal of Inverse and Ill-Posed Problems, 23 (2015), 687-700.  doi: 10.1515/jiip-2015-0083.  Google Scholar

[15]

V. A. KhoaM. V. Klibanov and L. H. Nguyen, Convexification for a three-dimensional inverse scattering problem with the moving point source, SIAM Journal on Imaging Sciences, 13 (2020), 871-904.  doi: 10.1137/19M1303101.  Google Scholar

[16]

M. V. Klibanov, Inverse problems in the large and Carleman bounds, Differential Equations, 20 (1984), 755-760.   Google Scholar

[17]

M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), 575. doi: 10.1088/0266-5611/8/4/009.  Google Scholar

[18]

M. V. Klibanov and O. V. Ioussoupova, Uniform strict convexity of a cost functional for three-dimensional inverse scattering problem, SIAM Journal on Mathematical Analysis, 26 (1995), 147-179.  doi: 10.1137/S0036141093244039.  Google Scholar

[19]

M. V. Klibanov, Global convexity in a three-dimensional inverse acoustic problem, SIAM Journal on Mathematical Analysis, 28 (1997), 1371-1388.  doi: 10.1137/S0036141096297364.  Google Scholar

[20]

J. S. Lee and C. Nguyen, Novel low-cost ultra-wideband, ultra-short-pulse transmitter with MESFET impulse-shaping circuitry for reduced distortion and improved pulse repetition rate, IEEE Microwave and Wireless Components Letters, 11 (2001), 208-210.   Google Scholar

[21]

V. A. Khoa, G. W. Bidney, M. V. Klibanov, L. H. Nguyen, L. H. Nguyen, A. J. Sullivan and V. N. Astratov, Convexification and experimental data for a 3D inverse scattering problem with the moving point source, preprint, arXiv: 2003.11513 doi: 10.1137/19M1303101.  Google Scholar

[22]

I. M. Gelfand and B. M. Levitan, On determining a differential equation from its spectral function, Amer. Math. Soc. Transl., 2 (1955), 253-304.  doi: 10.1090/trans2/001/11.  Google Scholar

[23]

M. G. Krein, On a method of effective solution of an inverse boundary problem, Dokl. Akad. Nauk SSSR, 94 (1954), 987-990.   Google Scholar

[24]

A. L. KarchevskyM. V. KlibanovL. NguyenN. Pantong and A. Sullivan, The Krein method and the globally convergent method for experimental data, Appl. Numer. Math., 74 (2013), 111-127.  doi: 10.1016/j.apnum.2013.09.003.  Google Scholar

[25]

M. V. Klibanov and A. A. Timonov, Carleman estimates for coefficient inverse problems and numerical applications, Walter de Gruyter, 46 (2012). doi: 10.1515/9783110915549.  Google Scholar

[26]

M. V. KlibanovA. V. KuzhugetS. I. Kabanikhin and D. V. Nechaev, A new version of the quasi-reversibility method for the thermoacoustic tomography and a coefficient inverse problem, Applicable Analysis, 87 (2008), 1227-1254.  doi: 10.1080/00036810802001297.  Google Scholar

[27]

M. V. Klibanov, Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, Journal of Inverse and Ill-Posed Problems, 21 (2013), 477-560.  doi: 10.1515/jip-2012-0072.  Google Scholar

[28]

M. V. Klibanov and V. G. Kamburg, Globally strictly convex cost functional for an inverse parabolic problem, Mathematical Methods in the Applied Sciences, 39 (2016), 930-940.  doi: 10.1002/mma.3531.  Google Scholar

[29]

M. V. Klibanov, L. H. Nguyen, A. Sullivan and L. Nguyen, A globally convergent numerical method for a 1-d inverse medium problem with experimental data, preprint, arXiv: 1602.09092. doi: 10.3934/ipi.2016032.  Google Scholar

[30]

M. V. Klibanov, A. E. Kolesov, A. Sullivan and L. Nguyen, A new version of the convexification method for a 1D coefficient inverse problem with experimental data, Inverse Problems, 34 (2018), 115014. doi: 10.1088/1361-6420/aadbc6.  Google Scholar

[31]

L. Baudouin, M. de Buhan, S. Ervedoza and A. Osses, Carleman-based reconstruction algorithm for the waves, 2020. Google Scholar

[32]

M. V. Klibanov, J. Li and W. Zhang, Convexification of electrical impedance tomography with restricted Dirichlet-to-Neumann map data, Inverse Problems, (2019). doi: 10.1088/1361-6420/aafecd.  Google Scholar

[33]

M. V. KlibanovJ. Li and W. Zhang, Convexification for the inversion of a time dependent wave front in a heterogeneous medium, SIAM J. Appl. Math., 79 (2019), 1722-1747.  doi: 10.1137/18M1236034.  Google Scholar

[34]

M. V. Klibanov and A. E. Kolesov, Convexification of a 3-D coefficient inverse scattering problem, Computers & Mathematics with Applications, 77 (2019), 1681-1702.  doi: 10.1016/j.camwa.2018.03.016.  Google Scholar

[35]

M. V. KlibanovA. E. Kolesov and D.-L. Nguyen, Convexification method for an inverse scattering problem and its performance for experimental backscatter data for buried targets, SIAM J. Imag. Sci., 12 (2019), 576-603.  doi: 10.1137/18M1191658.  Google Scholar

[36]

A. Einstein, Onthe electrodynamics of moving bodies, Annalen der Physik, 322 (1905), 891-921.   Google Scholar

[37]

M. V. Klibanov, J. Li and W. Zhang, Convexification for an inverse parabolic problem, preprint, arXiv: 2001.01880. Google Scholar

[38]

J. Korpela, M. Lassas and L. Oksanen, Regularization strategy for an inverse problem for a 1+ 1 dimensional wave equation, Inverse Problems, 32 (2016), 065001. doi: 10.1088/0266-5611/32/6/065001.  Google Scholar

[39]

J. Korpela, M. Lassas and L. Oksanen, Discrete regularization and convergence of the inverse problem for 1+ 1 dimensional wave equation, preprint, arXiv: 1803.10541. doi: 10.3934/ipi.2019027.  Google Scholar

[40]

A. V. Kuzhuget, et al., Blind backscattering experimental data collected in the field and an approximately globally convergent inverse algorithm, Inverse Problems, 28 (2012), 095007. Google Scholar

[41]

T. T. Le and L. H. Nguyen, A convergent numerical method to recover the initial condition of nonlinear parabolic equations from lateral Cauchy data, preprint, arXiv: 1910.05584 Google Scholar

[42]

B. T. Polyak, Introduction to optimization. Optimization software, Inc., Publications Division, New York, 1 (1987).  Google Scholar

[43]

V. G. Romanov, Inverse Problems of Mathematical Physics, Walter de Gruyter GmbH & Co KG, 2018. Google Scholar

[44]

J. A. ScalesM. L. Smith and T. L. Fischer, Global optimization methods for multimodal inverse problems, J. Comput. Phys., 103 (1992), 258-268.   Google Scholar

[45]

A. N. Tikhonov, A. Goncharsky, V. Stepanov and A. G. Yagola, Numerical methods for the solution of ill-posed problems, Springer Science & Business Media, 328 (2013). doi: 10.1007/978-94-015-8480-7.  Google Scholar

Figure 1.  The rectangle $ D(x,t) = \left\{ (\xi ,\tau ):\left\vert \xi \right\vert <\tau <t-\left\vert x-\xi \right\vert \right\} $ and the triangle $ Tr $
Figure 3.  Numerical reconstructions (the black marked dots) of functions $ a(x) $ (the solid lines). Noise level $ \xi = 0.1 $.
Figure 2.  The comparison of noiseless and noisy data. Figure 2(A) shows the norm of the functional (6.2) for each iteration of the gradient descent for the test function depicted on Figure 3(A). Figure 2(D) corresponds to our test for $ \lambda $ = 0; see the text.
Figure 4.  Limiting testing of different values of the parameter $ \lambda $ for the test function of Test 1, see comments in the text. The data are noiseless.
Figure 5.  Testing of different values of the parameter $ \alpha , $ see comments in the text. Solid line is the correct function of Test 1. The data are noiseless.
Table 6.2.  Summary of numerical results. Here $ \left\Vert \cdot\right\Vert _{\infty } $ denotes the $ L_{\infty } $
Test $ n^{\ast} $ Error $ \Vert J^h_{\lambda,\beta,\mu}(w_0) \Vert_{\infty} $ $ \Vert J^h_{\lambda,\beta,\mu}(w_{n^{\ast}}) \Vert_{\infty} $
1 30 0.1628 2570 2.7465
2 33 0.2907 34.42 0.22
3 51 0.0804 3.12 0.0007
4 41 0.3222 0.82 0.0003
Test $ n^{\ast} $ Error $ \Vert J^h_{\lambda,\beta,\mu}(w_0) \Vert_{\infty} $ $ \Vert J^h_{\lambda,\beta,\mu}(w_{n^{\ast}}) \Vert_{\infty} $
1 30 0.1628 2570 2.7465
2 33 0.2907 34.42 0.22
3 51 0.0804 3.12 0.0007
4 41 0.3222 0.82 0.0003
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