October  2019, 13(5): 1067-1094. doi: 10.3934/ipi.2019048

A convergent numerical method for a multi-frequency inverse source problem in inhomogenous media

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

* Corresponding author: Loc H. Nguyen

Received  January 2019 Revised  April 2019 Published  July 2019

Fund Project: The work of Nguyen and Klibanov was supported by US Army Research Laboratory and US Army Research Office grant W911NF-19-1-0044. In addition, the effort of Nguyen and Li was supported by research funds FRG 111172 provided by The University of North Carolina at Charlotte

A new numerical method to solve an inverse source problem for the Helmholtz equation in inhomogenous media is proposed. This method reduces the original inverse problem to a boundary value problem for a coupled system of elliptic PDEs, in which the unknown source function is not involved. The Dirichlet boundary condition is given on the entire boundary of the domain of interest and the Neumann boundary condition is given on a part of this boundary. To solve this problem, the quasi-reversibility method is applied. Uniqueness and existence of the minimizer are proven. A new Carleman estimate is established. Next, the convergence of those minimizers to the exact solution is proven using that Carleman estimate. Results of numerical tests are presented.

Citation: Loc H. Nguyen, Qitong Li, Michael V. Klibanov. A convergent numerical method for a multi-frequency inverse source problem in inhomogenous media. Inverse Problems & Imaging, 2019, 13 (5) : 1067-1094. doi: 10.3934/ipi.2019048
References:
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R. Albanese and P. Monk, The inverse source problem for Maxwell's equations, Inverse Problems, 22 (2006), 1023-1035. doi: 10.1088/0266-5611/22/3/018. Google Scholar

[2]

H. AmmariG. Bao and J. Flemming, An inverse source problem for Maxwell's equations in magnetoencephalography, SIAM J. Appl. Math., 62 (2002), 1369-1382. doi: 10.1137/S0036139900373927. Google Scholar

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G. BaoJ. Lin and F. Triki, A multi-frequency inverse source problem, Journal of Differential Equations, 249 (2010), 3443-3465. doi: 10.1016/j.jde.2010.08.013. Google Scholar

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G. BaoJ. Lin and F. Triki, An inverse source problem with multiple frequency data, C. R. Math., 349 (2011), 855-859. doi: 10.1016/j.crma.2011.07.009. Google Scholar

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G. BaoJ. Lin and F. Triki, Numerical solution of the inverse source problem for the Helmholtz equation with multiple frequency data, Contemp. Math, 548 (2011), 45-60. doi: 10.1090/conm/548/10835. Google Scholar

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E. BécacheL. BourgeoisL. Franceschini and J. Dardé, Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: The 1d case, Inverse Problems & Imaging, 9 (2015), 971-1002. doi: 10.3934/ipi.2015.9.971. Google Scholar

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L. Beilina and M. V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York, 2012. doi: 10.1007/978-1-4419-7805-9. Google Scholar

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

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L. Bourgeois, Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace's equation, Inverse Problems, 22 (2006), 413-430. doi: 10.1088/0266-5611/22/2/002. Google Scholar

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J. ChengV. Isakov and S. Lu, Increasing stability in the inverse source problem with many frequencies, Journal of Differential Equations, 260 (2016), 4786-4804. doi: 10.1016/j.jde.2015.11.030. Google Scholar

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C. Clason and M. V. Klibanov, The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium, SIAM J. Sci. Comput., 30 (2007), 1-23. doi: 10.1137/06066970X. Google Scholar

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V. Isakov and S. Lu, Inverse source problems without (pseudo) convexity assumptions, Inverse Probl. Imaging, 12 (2018), 955-970. doi: 10.3934/ipi.2018040. Google Scholar

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M. V. Klibanov, Convexification of restricted Dirichlet to Neumann map, J. Inverse and Ill-Posed Problems, 25 (2017), 669-685. doi: 10.1515/jiip-2017-0067. Google Scholar

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M. V. Klibanov, A. E. Kolesov, A. Sullivan and L. Nguyen, A new version of the convexification method for a 1-D coefficient inverse problem with experimental data, Inverse Problems, 34 (2018), 115014, 29 pp, https://doi.org/10.1088/1361-6420/aadbc6. doi: 10.1088/1361-6420/aadbc6. Google Scholar

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M. V. Klibanov and L. H. Nguyen, PDE-based numerical method for a limited angle X-ray tomography, Inverse Problems, 35 (2019), 045009, 32 pp, https://doi.org/10.1088/1361-6420/ab0133, see also arXiv: 1809.06012. doi: 10.1088/1361-6420/ab0133. Google Scholar

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[35]

M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, Inverse and Ill-Posed Problems Series, VSP, Utrecht, 2004. doi: 10.1515/9783110915549. Google Scholar

[36]

M. V. Klibanov and N. T. Thành, Recovering of dielectric constants of explosives via a globally strictly convex cost functional, SIAM J. Appl. Math., 75 (2015), 518-537. doi: 10.1137/140981198. Google Scholar

[37]

R. Lattès and J. L. Lions, The Method of Quasireversibility: Applications to Partial Differential Equations, Elsevier, New York, 1969. Google Scholar

[38]

J. LiH. Liu and H. Sun, On a gesture-computing technique using eletromagnetic waves, Inverse Probl. Imaging, 12 (2018), 677-696. doi: 10.3934/ipi.2018029. Google Scholar

[39]

H. Liu and G. Uhlmann, Determining both sound speed and internal source in thermo- and photo-acoustic tomography, Inverse Problems, 31 (2015), 105005, 10pp. doi: 10.1088/0266-5611/31/10/105005. Google Scholar

[40]

L. H. Nguyen, An inverse space-dependent source problem for hyperbolic equations and the Lipschitz-like convergence of the quasi-reversibility method, Inverse Problems, 35 (2019), 035007, 28pp. doi: 10.1088/1361-6420/aafe8f. Google Scholar

[41]

A. N. Tikhonov, A. Goncharsky, V. V. Stepanov and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Kluwer Academic Publishers Group, Dordrecht, 1995. doi: 10.1007/978-94-015-8480-7. Google Scholar

[42]

G. WangF. MaY. Guo and J. Li, Solving the multi-frequency electromagnetic inverse source problem by the Fourier method, J. Differential Equations, 265 (2018), 417-443. doi: 10.1016/j.jde.2018.02.036. Google Scholar

[43]

X. Wang, Y. Guo, J. Li and H. Liu, Mathematical design of a novel input/instruction device using a moving acoustic emitter, Inverse Problems, 33 (2017), 105009, 19pp. doi: 10.1088/1361-6420/aa873f. Google Scholar

[44]

X. Wang, Y. Guo, D. Zhang and H. Liu, Fourier method for recovering acoustic sources from multi-frequency far-field data, Inverse Problems, 33 (2017), 035001, 18pp. doi: 10.1088/1361-6420/aa573c. Google Scholar

[45]

X. WangM. SongY. GuoH. Li and H. Liu, Fourier method for identifying electromagnetic sources with multi-frequency far-field data, Journal of Computational and Applied Mathematics, 358 (2019), 279-292. doi: 10.1016/j.cam.2019.03.013. Google Scholar

[46]

X. Xiang and H. Sun, Sparse reconstructions of acoustic source for inverse scattering problems in measure space, Inverse Problems, 2019. doi: 10.1088/1361-6420/ab28cb. Google Scholar

[47]

D. Zhang and Y. Guo, Fourier method for solving the multi-frequency inverse source problem for the Helmholtz equation, Inverse Problems, 31 (2015), 035007, 30pp. doi: 10.1088/0266-5611/31/3/035007. Google Scholar

[48]

D. Zhang, Y. Guo, J. Li and H. Liu, Retrieval of acoustic sources from multi-frequency phaseless data, Inverse Problems, 34 (2018), 094001, 21pp. doi: 10.1088/1361-6420/aaccda. Google Scholar

show all references

References:
[1]

R. Albanese and P. Monk, The inverse source problem for Maxwell's equations, Inverse Problems, 22 (2006), 1023-1035. doi: 10.1088/0266-5611/22/3/018. Google Scholar

[2]

H. AmmariG. Bao and J. Flemming, An inverse source problem for Maxwell's equations in magnetoencephalography, SIAM J. Appl. Math., 62 (2002), 1369-1382. doi: 10.1137/S0036139900373927. Google Scholar

[3]

G. BaoJ. Lin and F. Triki, A multi-frequency inverse source problem, Journal of Differential Equations, 249 (2010), 3443-3465. doi: 10.1016/j.jde.2010.08.013. Google Scholar

[4]

G. BaoJ. Lin and F. Triki, An inverse source problem with multiple frequency data, C. R. Math., 349 (2011), 855-859. doi: 10.1016/j.crma.2011.07.009. Google Scholar

[5]

G. BaoJ. Lin and F. Triki, Numerical solution of the inverse source problem for the Helmholtz equation with multiple frequency data, Contemp. Math, 548 (2011), 45-60. doi: 10.1090/conm/548/10835. Google Scholar

[6]

E. BécacheL. BourgeoisL. Franceschini and J. Dardé, Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: The 1d case, Inverse Problems & Imaging, 9 (2015), 971-1002. doi: 10.3934/ipi.2015.9.971. Google Scholar

[7]

L. Beilina and M. V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York, 2012. doi: 10.1007/978-1-4419-7805-9. Google Scholar

[8]

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

[9]

L. Bourgeois, Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace's equation, Inverse Problems, 22 (2006), 413-430. doi: 10.1088/0266-5611/22/2/002. Google Scholar

[10]

L. Bourgeois and J. Dardé, A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data, Inverse Problems, 26 (2010), 095016, 21pp. doi: 10.1088/0266-5611/26/9/095016. Google Scholar

[11]

L. BourgeoisD. Ponomarev and J. Dardé, An inverse obstacle problem for the wave equation in a finite time domain, Inverse Probl. Imaging, 13 (2019), 377-400. doi: 10.3934/ipi.2019019. Google Scholar

[12]

A. L. Bukhgeim and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Soviet Math. Doklady, 17 (1981), 244-247. Google Scholar

[13]

X. Cao and H. Liu, Determining a fractional Helmholtz system with unknown source and medium parameter determining a fractional Helmholtz system with unknown source and medium parameter, preprint, arXiv: 1803.09538v1.Google Scholar

[14]

J. ChengV. Isakov and S. Lu, Increasing stability in the inverse source problem with many frequencies, Journal of Differential Equations, 260 (2016), 4786-4804. doi: 10.1016/j.jde.2015.11.030. Google Scholar

[15]

C. Clason and M. V. Klibanov, The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium, SIAM J. Sci. Comput., 30 (2007), 1-23. doi: 10.1137/06066970X. Google Scholar

[16]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. Applied Mathematical Sciences, 3rd edition, Springer, New York, 2013. doi: 10.1007/978-1-4614-4942-3. Google Scholar

[17]

J. Dardé, Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems, Inverse Problems and Imaging, 10 (2016), 379-407. doi: 10.3934/ipi.2016005. Google Scholar

[18]

G. Dassios and F. Kariotou, Magnetoencephalography in ellipsoidal geometry, J. Math. Physics, 44 (2003), 220-241. doi: 10.1063/1.1522135. Google Scholar

[19]

A. El Badia and T. Ha-Duong, An inverse source problem in potential analysis, Inverse Problems, 16 (2000), 651-663. doi: 10.1088/0266-5611/16/3/308. Google Scholar

[20]

M. N. Entekhabi and V. Isakov, On increasing stability in the two dimensional inverse source scattering problem with many frequencies, Inverse Problems, 34 (2018), 055005, 14pp. doi: 10.1088/1361-6420/aab465. Google Scholar

[21]

S. He and V. G. Romanov, Identification of dipole sources in a bounded domain for Maxwell's equations, Wave Motion, 28 (1998), 25-44. doi: 10.1016/S0165-2125(97)00063-2. Google Scholar

[22]

V. Isakov and S. Lu, Increasing stability in the inverse source problem with attenuation and many frequencies, SIAM J. Appl. Math., 78 (2018), 1-18. doi: 10.1137/17M1112704. Google Scholar

[23]

V. Isakov and S. Lu, Inverse source problems without (pseudo) convexity assumptions, Inverse Probl. Imaging, 12 (2018), 955-970. doi: 10.3934/ipi.2018040. Google Scholar

[24]

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

[25]

S. I. Kabanikhin, A. D. Satybaev and M. A. Shishlenin, Direct Methods of Solving Inverse Hyperbolic Problems, VSP, Utrecht, 2005. Google Scholar

[26]

S. I. Kabanikhin and M. A. Shishlenin, Numerical algorithm for two-dimensional inverse acoustic problem based on Gel'fand–Levitan–Krein equation, Journal of Inverse and Ill-posed Problems, 18 (2011), 979-995. doi: 10.1515/JIIP.2011.016. Google Scholar

[27]

B. Kaltenbacher and W. Rundell, Regularization of a backwards parabolic equation by fractional operators, Inverse Probl. Imaging, 13 (2019), 401-430. doi: 10.3934/ipi.2019020. Google Scholar

[28]

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

[29]

M. V. Klibanov, Carleman estimates for the regularization of ill-posed Cauchy problems, Applied Numerical Mathematics, 94 (2015), 46-74. doi: 10.1016/j.apnum.2015.02.003. Google Scholar

[30]

M. V. Klibanov, Convexification of restricted Dirichlet to Neumann map, J. Inverse and Ill-Posed Problems, 25 (2017), 669-685. doi: 10.1515/jiip-2017-0067. Google Scholar

[31]

M. V. Klibanov, A. E. Kolesov, A. Sullivan and L. Nguyen, A new version of the convexification method for a 1-D coefficient inverse problem with experimental data, Inverse Problems, 34 (2018), 115014, 29 pp, https://doi.org/10.1088/1361-6420/aadbc6. doi: 10.1088/1361-6420/aadbc6. Google Scholar

[32]

M. V. Klibanov, J. Li and W. Zhang, Convexification for the inversion of a time dependent wave front in a heterogeneous medium, Inverse Problems, 35 (2019), 035005, 33pp. doi: 10.1088/1361-6420/aafecd. Google Scholar

[33]

M. V. Klibanov and L. H. Nguyen, PDE-based numerical method for a limited angle X-ray tomography, Inverse Problems, 35 (2019), 045009, 32 pp, https://doi.org/10.1088/1361-6420/ab0133, see also arXiv: 1809.06012. doi: 10.1088/1361-6420/ab0133. Google Scholar

[34]

M. V. Klibanov and F. Santosa, A computational quasi-reversibility method for Cauchy problems for Laplace's equation, SIAM J. Appl. Math., 51 (1991), 1653-1675. doi: 10.1137/0151085. Google Scholar

[35]

M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, Inverse and Ill-Posed Problems Series, VSP, Utrecht, 2004. doi: 10.1515/9783110915549. Google Scholar

[36]

M. V. Klibanov and N. T. Thành, Recovering of dielectric constants of explosives via a globally strictly convex cost functional, SIAM J. Appl. Math., 75 (2015), 518-537. doi: 10.1137/140981198. Google Scholar

[37]

R. Lattès and J. L. Lions, The Method of Quasireversibility: Applications to Partial Differential Equations, Elsevier, New York, 1969. Google Scholar

[38]

J. LiH. Liu and H. Sun, On a gesture-computing technique using eletromagnetic waves, Inverse Probl. Imaging, 12 (2018), 677-696. doi: 10.3934/ipi.2018029. Google Scholar

[39]

H. Liu and G. Uhlmann, Determining both sound speed and internal source in thermo- and photo-acoustic tomography, Inverse Problems, 31 (2015), 105005, 10pp. doi: 10.1088/0266-5611/31/10/105005. Google Scholar

[40]

L. H. Nguyen, An inverse space-dependent source problem for hyperbolic equations and the Lipschitz-like convergence of the quasi-reversibility method, Inverse Problems, 35 (2019), 035007, 28pp. doi: 10.1088/1361-6420/aafe8f. Google Scholar

[41]

A. N. Tikhonov, A. Goncharsky, V. V. Stepanov and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Kluwer Academic Publishers Group, Dordrecht, 1995. doi: 10.1007/978-94-015-8480-7. Google Scholar

[42]

G. WangF. MaY. Guo and J. Li, Solving the multi-frequency electromagnetic inverse source problem by the Fourier method, J. Differential Equations, 265 (2018), 417-443. doi: 10.1016/j.jde.2018.02.036. Google Scholar

[43]

X. Wang, Y. Guo, J. Li and H. Liu, Mathematical design of a novel input/instruction device using a moving acoustic emitter, Inverse Problems, 33 (2017), 105009, 19pp. doi: 10.1088/1361-6420/aa873f. Google Scholar

[44]

X. Wang, Y. Guo, D. Zhang and H. Liu, Fourier method for recovering acoustic sources from multi-frequency far-field data, Inverse Problems, 33 (2017), 035001, 18pp. doi: 10.1088/1361-6420/aa573c. Google Scholar

[45]

X. WangM. SongY. GuoH. Li and H. Liu, Fourier method for identifying electromagnetic sources with multi-frequency far-field data, Journal of Computational and Applied Mathematics, 358 (2019), 279-292. doi: 10.1016/j.cam.2019.03.013. Google Scholar

[46]

X. Xiang and H. Sun, Sparse reconstructions of acoustic source for inverse scattering problems in measure space, Inverse Problems, 2019. doi: 10.1088/1361-6420/ab28cb. Google Scholar

[47]

D. Zhang and Y. Guo, Fourier method for solving the multi-frequency inverse source problem for the Helmholtz equation, Inverse Problems, 31 (2015), 035007, 30pp. doi: 10.1088/0266-5611/31/3/035007. Google Scholar

[48]

D. Zhang, Y. Guo, J. Li and H. Liu, Retrieval of acoustic sources from multi-frequency phaseless data, Inverse Problems, 34 (2018), 094001, 21pp. doi: 10.1088/1361-6420/aaccda. Google Scholar

Figure 1.  The comparison of the true function $ v(\cdot , k = 1.5) = \sum_{m = 1}^{\infty }v_{m}(\mathbf{x})\Psi _{m}(k) $ and the test function $ \sum_{m = 1}^{10}v_{m}(\cdot )\Psi _{m}(k) $ in Test 5, see Section 4. In this test, we consider the case $ n = 2 $ and $ \Omega = (-2, 2)^{2} $. On $ \Omega , $ we arrange a uniform grid of $ 121\times 121 $ points in $ \Omega $. Those points are numbered from $ 1 $ to $ 121^{2} $. In (a) and (b), we respectively show the real and imaginary parts of the two functions at 300 points numbered from 7170 to 7470. It is evident that reconstructing the first 10 terms of the Fourier coefficients of $ v(\mathbf{x }, k) $ is sufficient to solve our inverse source problems
Figure 2.  Test 1. The true and reconstructed source functions and the true and reconstructed functions $ v(\mathbf{x}, k) = u(\mathbf{x}, k)/g(k) $ when $ k = 1.5. $ The reconstructed positive value of the source function is 2.76 (relative error 10.5%). The reconstructed negative value of the source function is -2.17 (relative error 8.5%). (A) The function $ f_{\rm true} $; (B) The real part of the function $ v_{\rm true}(\cdot, k = 1.5) $; (C) The imaginary part of the function $ v_{\rm true}(\cdot, k = 1.5) $; (D) The function $ f_{\rm comp} $; (E) The real part of the function $ v_{\rm comp}(\cdot, k = 1.5) $; (F) The imaginary part of the function $ v_{\rm comp}(\cdot, k = 1.5) $
Figure 3.  Test 2. The true and reconstructed source functions and the true and reconstructed functions $ v(\mathbf{x}, k) = u(\mathbf{x}, k)/g(k) $ when $ k = 1.5. $ The reconstructed positive value of the source function is 1.11 (relative error 11.1%). The reconstructed negative value of the source function is -1.11 (relative error 11.1%). A) The function $ f_{\rm true} $; (B) The real part of the function $ v_{\rm true}(\cdot, k = 1.5) $; (C) The imaginary part of the function $ v_{\rm true}(\cdot, k = 1.5) $; (D) The function $ f_{\rm comp} $; (E) The real part of the function $ v_{\rm comp}(\cdot, k = 1.5) $; (F) The imaginary part of the function $ v_{\rm comp}(\cdot, k = 1.5) $
Figure 4.  Test 3. The true and reconstructed source functions and the true and reconstructed functions $ v(\mathbf{x}, k) = u(\mathbf{x}, k)/g(k) $ when $ k = 1.5. $ The reconstructed positive value of the source function is 1.09 (relative error 9.0%). The reconstructed negative value of the source function is -0.89 (relative error 11.0%). A) The function $ f_{\rm true} $; (B) The real part of the function $ v_{\rm true}(\cdot, k = 1.5) $; (C) The imaginary part of the function $ v_{\rm true}(\cdot, k = 1.5) $; (D) The function $ f_{\rm comp} $; (E) The real part of the function $ v_{\rm comp}(\cdot, k = 1.5) $; (F) The imaginary part of the function $ v_{\rm comp}(\cdot, k = 1.5) $
Figure 5.  Test 4. The true and reconstructed source functions and the true and reconstructed functions $ v(\mathbf{x}, k) = u(\mathbf{x}, k)/g(k) $ when $ k = 1.5. $ The reconstructed positive value of the source function is 1.12 (relative error 12.0%). The reconstructed negative value of the source function is -1.94 (relative error 3.0%). A) The function $ f_{\rm true} $; (B) The real part of the function $ v_{\rm true}(\cdot, k = 1.5) $; (C) The imaginary part of the function $ v_{\rm true}(\cdot, k = 1.5) $; (D) The function $ f_{\rm comp} $; (E) The real part of the function $ v_{\rm comp}(\cdot, k = 1.5) $; (F) The imaginary part of the function $ v_{\rm comp}(\cdot, k = 1.5) $
Figure 6.  Test 5. The true and reconstructed source functions and the true and reconstructed functions $ v(\mathbf{x}, k) = u(\mathbf{x}, k)/g(k) $ when $ k = 1.5. $ The true and reconstructed maximal positive value of the source function are 8.10 and 7.36 (relative error 9.1%) respectively. The true and reconstructed minimal negative value of the source function are -6.55 and -5.48 (relative error 16.0%) respectively. A) The function $ f_{\rm true} $; (B) The real part of the function $ v_{\rm true}(\cdot, k = 1.5) $; (C) The imaginary part of the function $ v_{\rm true}(\cdot, k = 1.5) $; (D) The function $ f_{\rm comp} $; (E) The real part of the function $ v_{\rm comp}(\cdot, k = 1.5) $; (F) The imaginary part of the function $ v_{\rm comp}(\cdot, k = 1.5) $
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