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A Projection Method Based on Discrete Normalized Dynamical System for Computing C-eigenpairs

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Abstract

The largest C-eigenvalue of piezoelectric tensors determines the highest piezoelectric coupling constant, which reflects the coupling between the elastic and dielectric properties of crystal. Here, a projection method based on discrete normalized dynamical system (PDND) is established for computing the largest C-eigenvalue. Theoretical analysis of the convergence for PDND algorithm is given. In numerical experiments, the longitudinal piezoelectric modulus and the unit uniaxial direction that the extreme piezoelectric effect along took place of different piezoelectric materials are given to display the physical meaning of the C-eigenvalues and eigenvectors. Furthermore, the largest C-eigenvalue and all the corresponding eigenvectors can be obtained, which is the advantage of the proposed method.

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References

  1. Benson, A.R., Gleich, D.F.: Computing tensor Z-eigenvectors with dynamical systems. SIAM J. Matrix Anal. Appl. 40(4), 1311–1324 (2019)

    Article  MathSciNet  Google Scholar 

  2. Che, H., Chen, H., Wang, Y.: C-eigenvalue inclusion theorems for piezoelectric-type tensors. Appl. Math. Lett. 89, 41–49 (2019)

    Article  MathSciNet  Google Scholar 

  3. Che, H., Chen, H., Wang, Y.: On the M-eigenvalue estimation of fourth order partially symmetric tensors. J. Ind. Manag. Optim. 16(1), 309–324 (2020)

    Article  MathSciNet  Google Scholar 

  4. Chen, L., Han, L., Zhou, L.: Computing tensor eigenvalues via homotopy methods. SIAM J. Matrix Anal. Appl. 37(1), 290–319 (2016)

    Article  MathSciNet  Google Scholar 

  5. Chen, Y., Jákli, A., Qi, L.: The C-eigenvalue of third order tensors and its application in crystals. J. Ind. Manag. Optim. 19(1), 265–281 (2023)

    Article  MathSciNet  Google Scholar 

  6. Cui, L., Hu, Q., Chen, Y., Song, Y.: A Rayleigh quotient-gradient neural network method for computing \({\cal{Z} }\)-eigenpairs of general tensors. Numer. Linear Algebra Appl. 29(3), e2420 (2022)

    Article  MathSciNet  Google Scholar 

  7. Curie, J., Curie, P.: Développement par compression de l’électricité polaire dans les cristaux hémièdres à faces inclinées. Bull. Minér. 3(4), 90–93 (1880)

    Google Scholar 

  8. Curie, J., Curie, P.: Contractions et dilatations produites par des tensions électriques dans les cristaux hémièdres à faces inclinées. C. R. 93, 1137–1140 (1881). (in French)

    Google Scholar 

  9. de Jong, M., Chen, W., Geerlings, H., Asta, M., Persson, K.A.: A database to enable discovery and design of piezoelectric materials. Sci. Data 2(1), 1–13 (2015)

    Google Scholar 

  10. De Lathauwer, L., De Moor, B., Vandewalle, J.: A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21, 1253–1278 (2000)

    Article  MathSciNet  Google Scholar 

  11. Gaeta, G., Virga, E.G.: Octupolar order in three dimensions. Eur. Phys. J. E 39(11), 113 (2016)

    Article  Google Scholar 

  12. Gonze, X.: Adiabatic density-functional perturbation theory. Phys. Rev. A 52(2), 1096 (1995)

    Article  Google Scholar 

  13. Guo, D., Yan, L., Nie, Z.: Design, analysis, and representation of novel five-step DTZD algorithm for time-varying nonlinear optimization. IEEE Trans. Neural Netw. Learn. Syst. 29(9), 4248–4260 (2017)

    Article  Google Scholar 

  14. Guo, C., Lin, W., Liu, C.: A modified Newton iteration for finding nonnegative Z-eigenpairs of a nonnegative tensor. Numer. Algorithms 80, 595–616 (2019)

    Article  MathSciNet  Google Scholar 

  15. Han, L.: An unconstrained optimization approach for finding real eigenvalues of even order symmetric tensors. Numer. Algebra Control Optim. 3, 583–599 (2013)

    Article  MathSciNet  Google Scholar 

  16. Hao, L., Cui, F., Dai, H.: A sequential subspace projection method for extreme Z-eigenvalues of supersymmetric tensors. Numer. Linear Algebra Appl. 22(2), 283–298 (2015)

    Article  MathSciNet  Google Scholar 

  17. Haussüuhl, S.: Physical Properties of Crystals: An Introduction. Wiley, Weinheim (2007)

    Book  Google Scholar 

  18. He, J., Liu, Y., Xu, G.: An S-type inclusion set for C-eigenvalues of a piezoelectric-type tensor. Appl. Math. Lett. 121, 107448 (2021)

    Article  MathSciNet  Google Scholar 

  19. Hu, S., Huang, Z., Qi, L.: Finding the extreme Z-eigenvalues of tensors via a sequential semidefinite programming method. Numer. Linear Algebra Appl. 20(6), 972–984 (2013)

    Article  MathSciNet  Google Scholar 

  20. Jerphagnon, J.: Invariants of the third-rank Cartesian tensor: optical nonlinear susceptibilities. Phys. Rev. B 2(4), 1091 (1970)

    Article  Google Scholar 

  21. Kholkin, A.L., Pertsev, N.A., Goltsev, A.V.: Piezolelectricity and crystal symmetry. In: Safari, A., Akdog̃an, E.K. (eds.) Piezoelectric and Acoustic Materials, pp. 17–38. Springer, New York (2008)

    Google Scholar 

  22. Kolda, T.G., Mayo, J.R.: Shifted power method for computing tensor eigenpairs. SIAM J. Matrix Anal. Appl. 32(4), 1095–1124 (2011)

    Article  MathSciNet  Google Scholar 

  23. Kolda, T.G., Mayo, J.R.: An adaptive shifted power method for computing generalized tensor eigenpairs. SIAM J. Matrix Anal. Appl. 35(4), 1563–1581 (2014)

    Article  MathSciNet  Google Scholar 

  24. Kulagin, I.A., Ganeev, R.A., Tugushev, R., Ryasnyansky, A.I., Usmanov, T.: Components of the third-order nonlinear susceptibility tensors in KDP, DKDP and LiNbO3 nonlinear optical crystals. Quantum Electron. 34(7), 657 (2004)

    Article  Google Scholar 

  25. Kuo, Y., Lin, W., Liu, C.: Continuation methods for computing Z-/H-eigenpairs of nonnegative tensors. J. Comput. Appl. Math. 340, 71–88 (2018)

    Article  MathSciNet  Google Scholar 

  26. Li, C., Liu, Y., Li, Y.: C-eigenvalues intervals for piezoelectric-type tensors. Appl. Math. Comput. 358, 244–250 (2019)

    MathSciNet  Google Scholar 

  27. Liang, C., Yang, Y.: Shifted eigenvalue decomposition method for computing C-eigenvalues of a piezoelectric-type tensor. Comput. Appl. Math. 40(7), 1–22 (2021)

    Article  MathSciNet  Google Scholar 

  28. Lim, L.: Singular values and eigenvalues of tensors: a variational approach. In: CAMSAP05: Proceeding of IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing 1, pp. 129–132 (2005)

  29. Lippmann, G.: Principe de la conservation de l’électricité, ou second principe de la théorie des phénomènes électriques. Ann. Chim. Phys. 10(1), 381–394 (1881)

    Google Scholar 

  30. Lovett, D.: Tensor Properties of Crystals, 2nd edn. Institute of Physics Publishing, Bristol (1989)

    Google Scholar 

  31. Nye, J.F.: Physical Properties of Crystals: Their Representation by Tensors and Matrices. Oxford University Press, Oxford (1985)

    Google Scholar 

  32. Ouyang, J., Ramesh, R., Roytburd, A.L.: Intrinsic effective piezoelectric coefficient e 31, f for ferroelectric thin films. Appl. Phys. Lett. 86(15), 152901 (2005)

    Article  Google Scholar 

  33. Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 6, 1302–1324 (2005)

    Article  MathSciNet  Google Scholar 

  34. Sang, C.: A new Brauer-type Z-eigenvalue inclusion set for tensors. Numer. Algorithms 80(3), 781–794 (2019)

    Article  MathSciNet  Google Scholar 

  35. Wang, W., Chen, H., Wang, Y., Zhou, G.: A proximal alternating minimization algorithm for the largest C-eigenvalue of piezoelectric-type tensors. J. Glob. Optim. 1–18 (2022)

  36. Wang, G., Zhou, G., Caccetta, L.: Z-eigenvalue inclusion theorems for tensors. Discrete Contin. Dyn. Syst.-Ser. B 22, 187–197 (2017)

    MathSciNet  Google Scholar 

  37. Wang, X., Che, M., Wei, Y.: Neural network approach for solving nonsingular multi-linear tensor systems. J. Comput. Appl. Math. 368, 112569 (2020)

    Article  MathSciNet  Google Scholar 

  38. Warner, A.W., Onoe, M., Coquin, G.A.: Determination of elastic and piezoelectric constants for crystals in class (3m). J. Acoust. Soc. Am. 42(6), 1223–1231 (1967)

    Article  Google Scholar 

  39. Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Clarendon, Oxford (1965)

    Google Scholar 

  40. Yang, Y., Liang, C.: Computing the largest C-eigenvalue of a tensor using convex relaxation. J. Optim. Theory Appl. 192(2), 648–677 (2022)

    Article  MathSciNet  Google Scholar 

  41. Zhao, J., Luo, J.: Properties and calculation for C-eigenvalues of a piezoelectric-type tensor. J. Ind. Manag. Optim. (2021). https://doi.org/10.3934/jimo.2021162

    Article  Google Scholar 

  42. Zhao, R., Zheng, B., Liang, M., Xu, Y.: A locally and cubically convergent algorithm for computing \({\cal{Z} }\)-eigenpairs of symmetric tensors. Numer. Linear Algebra Appl. 27(3), e2284 (2021)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are grateful to the handling editor and anonymous referees for useful comments and suggestions that contributed to improving the quality of the manuscript.

Funding

This research is supported in part by National Natural Science Foundations of China (No.12171087), Shanghai Municipal Science and Technology Commission under Grant 23WZ2501400 (China), Foundation of Henan Educational Committee (No.21A110013), Foundation of Henan Normal University (No.2021PL03), Xinxiang aviation industry (Group) Co., Ltd (No. 5201019160013).

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Authors and Affiliations

  1. School of Mathematics and Information Sciences, Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, Henan Normal University, Xinxiang, 453007, Henan, People’s Republic of China

    Lu-Bin Cui

  2. School of Mathematics and Information Sciences, Henan Normal University, Xinxiang, 453007, Henan, People’s Republic of China

    Jia-Le Yao

  3. School of Mathematics Sciences, East China Normal University, Shanghai, 200241, People’s Republic of China

    Jia-Le Yao

  4. School of Computer Science and Technology, Dongguan University of Technology, Dongguan, 523808, People’s Republic of China

    Jin-Yun Yuan

  5. Departamento de Matemática, Universidade Federal do Paraná, Centro Politécnico, Curitiba, 81531-980, Brazil

    Jin-Yun Yuan

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  1. Lu-Bin Cui
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Correspondence to Jin-Yun Yuan.

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Communicated by Liqun Qi.

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Cui, LB., Yao, JL. & Yuan, JY. A Projection Method Based on Discrete Normalized Dynamical System for Computing C-eigenpairs. J Optim Theory Appl 200, 768–793 (2024). https://doi.org/10.1007/s10957-023-02341-x

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