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数学进展 - 2016, Vol. 45(6): 955-960
研究论文
一个应变梯度有限元的新基函数
New Basis Function of a Strain Gradient Finite Element

李鸿亮1, 明平兵2,3
LI Hongliang1, *, MING Pingbing2,3, **

1. 中国工程物理研究院电子工程研究所, 绵阳, 四川, 621900;
2. 中国科学院数学与系统科学研究院计算数学与科学工程计算研究所, 科学与工程计算国家重点实验室, 北京, 100190;
3. 中国科学院大学数学科学学院, 北京, 100049
1. Institute of Electronic Engineering, China Academy of Engineering Physics, Mianyang, Sichuan, 621900, P. R. China;
2. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, CAS, Beijing, 100190, P. R. China;
3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, P. R. China

收稿日期: 2016-01-16
出版日期: 2016-11-10
2016, Vol. 45(6): 955-960
DOI: 10.11845/sxjz.2016012b


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摘要 本文获得了魏悦广在[ Eur. J. Mech. A Solids}, 2006, 25: 897-913]中提出的一种简单有效的应变梯度有限元的一组基函数. 它与已有基函数等价, 但形式更加简单.
Abstract:We derived an equivalent but simpler basis function of Wei's finite element in~[ Eur. J. Mech. A Solids},2006, 25: 897-913], which is a simple and efficient finite element method in the strain gradient theory.
PACS:  O241.82  
  O175.25  
[1] Acar, E., Dunlavy, D.M. and Kolda, T.G., A scalable optimization approach for fitting canonical tensor decompositions, J. Chemom., 2011, 25(2): 67-86.
[2] Bauckhage, C., Robust tensor classifiers for color object recognition, In: Image Analysis and Recognition, Lecture Notes in Comput. Sci., Vol. 4633, Berlin: Springer-Verlag, 2007, 352-363.
[3] Berman, A. and Plemmons, R.J., Nonnegative Matrices in the Mathematical Sciences, Classics Appl. Math., Vol. 9, Philadelphia, PA: SIAM, 1994.
[4] Bloy, L. and Verma, R., On computing the underlying fiber directions from the diffusion orientation distribution function, In: Medical Image Computing and Computer-Assisted Intervention---MICCAI 2008, Lecture Notes in Comput. Sci., Vol. 5241, Berlin: Springer-Verlag, 2008, 1-8.
[5] Bourennane, S., Fossati, C. and Cailly, A., Improvement of classification for hyperspectral images based on tensor modeling, IEEE Geosci. Remote Sens. Lett., 2010, 7(4): 801-805.
[6] Caiafa, C.F. and Cichocki, A., Multidimensional compressed sensing and their applications, Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery, 2013, 3(6): 355-380.
[7] Cao, X.C., Wei, X.X., Han, Y.H. and Lin, D.D., Robust face clustering via tensor decomposition, IEEE Trans. Cybern., 2015, 45(11): 2546-2557.
[8] Carroll, J.D. and Chang, J.J., Analysis of individual differences in multidimensional scaling via an $N$-way generalization of ``Eckart-Young'' decomposition, Psychometrika}, 1970, 35(3): 283-319.
[9] Chang, K.C., Pearson, K. and Zhang, T., {Perron-Frobenius} theorem for nonnegative tensors, Commun. Math. Sci., 2008, 6(2): 507-520.
[10] Chang, K.C., Pearson, K. and Zhang, T., On eigenvalue problems of real symmetric tensors, J. Math. Anal. Appl.,2009, 350(1): 416-422.
[11] Chang, K.C., Pearson, K. and Zhang, T., Primitivity, the convergence of the {NQZ} method, and the largest eigenvalue for nonnegative tensors, SIAM J. Matrix Anal. Appl., 2011, 32(3): 806-819.
[12] Chang, K.C., Pearson, K. and Zhang, T., Some variational principles for {Z}-eigenvalues of nonnegative tensors, Linear Algebra Appl., 2013, 438(11): 4166-4182.
[13] Chang, K.C., Qi, L.Q. and Zhang, T., A survey on the spectral theory of nonnegative tensors, Numer. Linear Algebra Appl., 2013,20(6): 891-912.
[14] Chang, K.C. and Zhang, T., On the uniqueness and nonuniqueness of the {Z}-eigenvector for transition probability, J. Math. Anal. Appl., 2013, 408(2): 525-540.
[15] Chen, Y.N., Han, D.R. and Qi, L.Q., New {ALS} methods with extrapolating search directions and optimal step size for complex-valued tensor decompositions,IEEE Trans. Signal Process., 2011, 59(12): 5888-5898.
[16] Costantini, R., Sbaiz, L. and S{\"u}sstrunk, S., Higher order {SVD} analysis for dynamic texture synthesis, IEEE Trans. Image Process., 2008,17(1): 42-52.
[17] Cox, D.A., Little, J. and O'shea, D., Using Algebraic Geometry, 2nd Edition,Grad. Texts in Math., Vol. 185, New York: Springer-Verlag, 2005.
[18] Cui, C.F., Dai, Y.H. and Nie, J.W., All real eigenvalues of symmetric tensors,SIAM J. Matrix Anal. Appl., 2014, 35(4): 1582-1601.
[19] Ding, W.Y. and Wei, Y.M., Generalized tensor eigenvalue problems, SIAM J. Matrix Anal. Appl., 2015, 36(3): 1073-1099.
[20] Eld{\'e}n, L. and Savas, B., A Newton-Grassmann method for computing the best multilinear rank-$(r_1, r_2, r_3)$ approximation of a tensor, SIAM J. Matrix Anal. Appl., 2009, 31(2): 248-271.
[21] Espig, M. and Hackbusch, W., A regularized {Newton} method for the efficient approximation of tensors represented in the canonical tensor format,Numer. Math., 2012, 122(3): 489-525.
[22] Fanaee-T, H. and Gama, J., Tensor-based anomaly detection: An interdisciplinary survey, Knowledge-based Systems}, 2016, 98: 130-147.
[23] Feng, B.W., Lu, W., Sun, W., Huang, J.W. and Shi, Y.Q., Robust image watermarking based on Tucker decomposition and adaptive-lattice quantization index modulation, Signal Process. Image Commun., 2016, 41: 1-14.
[24] Friedland, S., Gaubert, S. and Hanc, L., {Perron-Frobenius} theorem for nonnegative multilinear forms and extensions, Linear Algebra Appl.,2013, 438(2): 738-749.
[25] Furukawa, R., Kawasaki, H., Ikeuchi, K. and Sakauchi, M., Appearance based object modeling using texture database: Acquisition compression and rendering, In: Proceedings of the 13th Eurographics Workshop on Rendering Techniques, Italy, 2002, 257-266.
[26] Geng, X.R., Ji, L.Y., Zhao, Y.C. and Wang, F.X., A small target detection method for the hyperspectral image based on higher order singular value decomposition (HOSVD), IEEE Geosci. Remote Sens. Lett., 2013, 10(6): 1305-1308.
[27] Grasedyck, L., Hierarchical singular value decomposition of tensors, SIAM J. Matrix Anal. Appl., 2010, 31(4): 2029-2054.
[28] Guo, X., Huang, X., Zhang, L.P. and Zhang, L.F., Hyperspectral image noise reduction based on rank-1 tensor decomposition, ISPRS J. Photogramm. Remote Sens., 2013, 83: 50-63.
[29] Hackbusch, W. and K{\"u}hn, S., A new scheme for the tensor representation,J. Fourier Anal. Appl., 2009, 15(5): 706-722.
[30] Han, L.X., An unconstrained optimization approach for finding real eigenvalues of even order symmetric tensors, Numer. Algebra Control Optim., 2013, 3(3): 583-599.
[31]Hao, C.L., Cui, C.F. and Dai, Y.H., A sequential subspace projection method for extreme {Z}-eigenvalues of supersymmetric tensors, Numer. Linear Algebra Appl., 2015, 22(2): 283-298.
[32]Harshman, R.A., Foundations of the {PARAFAC} procedure: Model and conditions for an ``explanatory'' multi-mode factor analysis, UCLA Working Papers in Phonetics}, 1970, 16: 1-84.
[33]Hillar, C.J. and Lim, L.H., Most tensor problems are {NP}-hard, J. ACM}, 2013, 60(6): Article 45, 39 pages.
[34] Hitchcock, F.L., The expression of a tensor or a polyadic as a sum of products,J. Math. and Phys., 1927, 6: 164-189.
[35] Hitchcock, F.L., Multiple invariants and generalized rank of a $p$-way matrix or tensor, J. Math. and Phys., 1928, 7: 39-79.
[36] Hu, H.F., Multiview gait recognition based on patch distribution features and uncorrelated multilinear sparse local discriminant canonical correlation analysis, IEEE Trans.Circuits Syst.Video Technol., 2014, 24(4): 617-630.
[37] Hu, S.L., Huang, Z.H., Ni, H. and Qi, L.Q., Positive definiteness of diffusion kurtosis imaging, Inverse Probl. Imaging}, 2012, 6: 57-75.
[38] Hu, S.L., Huang, Z.H. and Qi, L.Q., Finding the extreme {Z}-eigenvalues of tensors via a sequential semidefinite programming method, Numer. Linear Algebra Appl., 2013, 20(6): 972-984.
[39] Ishteva, M., Absil, P.A., Van~Huffel, S. and De~Lathauwer, L., Best low multilinear rank approximation of higher-order tensors, based on the
{Riemannian} trust-region scheme, SIAM J. Matrix Anal. Appl., 2011, 32(1): 115-135.
[40] Jensen, J.H., Helpern, J.A., Ramani, A., Lu, H.Z. and Kaczynski, K., Diffusional kurtosis imaging: the quantification of non-Gaussian water diffusion by means of magnetic resonance imaging, Magn.Reson. Med., 2005, 53(6): 1432-1440.
[41] Kapteyn, A., Neudecker, H. and Wansbeek, T., An approach ton-mode components analysis, Psychometrika}, 1986, 51(2): 269-275.
[42] Karami, A., Yazdi, M. and Asli, A.Z., Noise reduction of hyperspectral images using kernel non-negative tucker decomposition, IEEE J. Sel. Top. Sign. Proces., 2011, 5(3): 487-493.
[43] Kazeev, V.A. and Tyrtyshnikov, E.E., Structure of the {Hessian} matrix and an economical implementation of {Newton's} method in the problem of canonical approximation of tensors, Comput. Math. Math. Phys., 2010, 50(6): 927-945.
[44] Kiers, H.A.L., Towards a standardized notation and terminology in multiway analysis, J. Chemom., 2000, 14(3): 105-122.
[45] Kolda, T.G. and Bader, B.W., Tensor decompositions and applications, SIAM Rev., 2009, 51(3): 455-500.
[46] Kolda, T.G. and Mayo, J.R., Shifted power method for computing tensor eigenpairs, SIAM J. Matrix Anal. Appl., 2011, 32(4): 1095-1124.
[47] Kolda, T.G. and Mayo, J.R., An adaptive shifted power method for computing generalized tensor eigenpairs, SIAM J. Matrix Anal. Appl., 2014, 35(4): 1563--1581.
[48] Kroonenberg, P.M. and De~Leeuw, J., Principal component analysis of three-mode data by means of alternating least squares algorithms, Psychometrika}, 1980, 45(1): 69-97.
[49] Kruskal, J.B., Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics,Linear Algebra Appl., 1977, 18(2): 95-138.
[50] Lathauwer, L.D., Moor, B.D. and Vandewalle, J., A multilinear singular value decomposition, SIAM J. Matrix Anal. Appl., 2000, 21(4): 1253-1278.
[51] Lathauwer, L.D., Moor, B.D. and Vandewalle, J., On the best rank-$1$ and rank-{$(R_1,R_2,\cdots,R_N)$} approximation of higher-order tensors,SIAM J. Matrix Anal. Appl., 2000, 21(4): 1324-1342.
[52] Letexier, D. and Bourennane, S., Noise removal from hyperspectral images by multidimensional filtering, IEEE Trans. Geosci. Remote Sens., 2008, 46(7): 2061-2069.
[53] Letexier, D., Bourennane, S. and Blanc-Talon, J., Nonorthogonal tensor matricization for hyperspectral image filtering, IEEE Geosci. Remote Sens. Lett., 2008, 5(1): 3-7.
[54] Lim, L.H., Singular values and eigenvalues of tensors: A variational approach, In: Proceedings of the 1st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, 2005, 129-132.
[55] Lim, L.H. and Comon, P., Multiarray signal processing: Tensor decomposition meets compressed sensing, C. R. M\'{e}c., 2010, 338(6): 311-320.
[56] Li{\c{t}}{\u{a}}, L. and Pelican, E., A low-rank tensor-based algorithm for face recognition, Appl. Math. Modell., 2015, 39(3): 1266-1274.
[57] Liu, S. and Ruan, Q.Q., Orthogonal tensor neighborhood preserving embedding for facial expression recognition, Pattern Recognit., 2011, 44(7): 1497-1513.
[58] Liu, X.F., Bourennane, S. and Fossati, C., Denoising of hyperspectral images using the {PARAFAC} model and statistical performance analysis, IEEE Trans. Geosci. Remote Sens., 2012, 50(10): 3717-3724.
[59] Liu, X.F., Bourennane, S. and Fossati, C., Nonwhite noise reduction in hyperspectral images, IEEE Geosci. Remote Sens. Lett., 2012, 9(3): 368-372.
[60] Ng, M., Qi, L.Q. and Zhou, G.L., Finding the largest eigenvalue of a nonnegative tensor, SIAM J. Matrix Anal. Appl., 2009, 31(3): 1090-1099.
[61] Oseledets, I.V., Tensor-train decomposition, SIAM J. Sci. Comput., 2011, 33(5): 2295-2317.
[62] Oseledets, I.V. and Tyrtyshnikov, E.E., Breaking the curse of dimensionality, or how to use {SVD} in many dimensions, SIAM J. Sci. Comput., 2009, 31(5): 3744-3759.
[63] Park, S.W. and Savvides, M., Individual kernel tensor-subspaces for robust face recognition: A computationally efficient tensor framework without requiring mode factorization, IEEE Trans. Syst., Man}, Cybern. B, Cybern., 2007, 37(5): 1156-1166.
[64] Phan, A.H., Tichavsky, P. and Cichocki, A., Low complexity damped {Gauss-Newton} algorithms for {CANDECOMP/PARAFAC}, SIAM J. Matrix Anal. Appl., 2013, 34(1): 126-147.
[65] Qi, L.Q., Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 2005, 40: 1302-1324.
[66] Qi, L.Q., Eigenvalues and invariants of tensors, J. Math. Anal. Appl., 2007, 325: 1363-1377.
[67] Qi, L.Q., Han, D.R. and Wu, E.X., Principal invariants and inherent parameters of diffusion kurtosis tensors, J. Math. Anal. Appl., 2009, 349(1): 165-180.
[68] Qi, L.Q., Sun, W.Y. and Wang, Y.J., Numerical multilinear algebra and its applications, Front. Math. China}, 2007, 2(4): 501-526.
[69] Qi, L.Q., Wang, F. and Wang, Y.J., {Z}-eigenvalue methods for a global polynomial optimization problem, Math. Program., 2009, 118(2): 301-316.
[70] Qi, L.Q., Wang, Y.J. and Wu, E.X., {D}-eigenvalues of diffusion kurtosis tensors,
J. Comput. Appl. Math., 2008, 221: 150-157.
[71] Qi, L.Q., Yu, G.H. and Wu, E.X., Higher order positive semidefinite diffusion tensor imaging, SIAM J. Imag. Sci., 2010, 3(3): 416-433.
[72] Rajwade, A., Rangarajan, A. and Banerjee, A., Image denoising using the higher order singular value decomposition, IEEE Trans. Pattern Anal. Mach. Intell., 2013, 35(4): 849-862.
[73] Savas, B. and Eld{\'e}n, L., Handwritten digit classification using higher order singular value decomposition, Pattern Recognit., 2007, 40(3): 993-1003.
[74] Savas, B. and Lim, L.H., {Quasi-Newton} methods on Grassmannians and multilinear approximations of tensors, SIAM J. Sci. Comput., 2010, 32(6): 3352-3393.
[75] Shashua, A. and Levin, A., Linear image coding for regression and classification using the tensor-rank principle, In: Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Vol. 1, Los Alamitos, CA: IEEE Computer Soc., 2001, I-42.
[76] Sidiropoulos, N.D. and Bro, R., On the uniqueness of multilinear decomposition of $N$-way arrays, J. Chemom., 2000, 14(3): 229-239.
[77] Song, B., Hassan, M.M., Tian, Y., Hossain, M.S. and Alamri, A., Remote display solution for video surveillance in multimedia cloud, Multimedia Tools
Appl., 2015, 1-22, doi: 10.1007/s11042-015-2816-x, 22 pages.
[78] Sterck, H.D., A nonlinear {GMRES} optimization algorithm for canonical tensor decomposition, SIAM J. Sci. Comput., 2012, 34(3): A1351-A1379.
[79] Sun, C., Junejo, I.N., Tappen, M. and Foroosh, H., Exploring sparseness and self-similarity for action recognition, IEEE Trans. Image Process}, 2015, 24(8): 2488-2501.
[80] Takallou, H.M. and Kasaei, S., Head pose estimation and face recognition using a non-linear tensor-based model, IET Comput. Vision}, 2014, 8(1): 54-65.
[81] Tao, D.C., Li, X.L., Wu, X.D. and Maybank, S.J., General tensor discriminant analysis and Gabor features for gait recognition, IEEE Trans. Pattern Anal. Mach. Intell., 2007, 29(10): 1700-1715.
[82] Tian, C.N., Fan, G.L., Gao, X.B. and Tian, Q., Multiview face recognition: from Tensorface to V-TensorFace and K-TensorFace, IEEE Trans. Syst., Man}, Cybern. B, Cybern., 2012, 42(2): 320-333.
[83] Torkamani-Azar, F., Imani, H. and Fathollahian, H., Video quality measurement based on {3-D.} Singular Value Decomposition, J. Visual Commun. Image Represent., 2015, 27: 1-6.
[84] Tucker, L.R., Some mathematical notes on three-mode factor analysis,
Psychometrika}, 1966, 31: 279-311.
[85] Vannieuwenhoven, N., Vandebril, R. and Meerbergen, K., A new truncation strategy for the higher-order singular value decomposition, SIAM J. Sci. Comput., 2012, 34(2): A1027-A1052.
[86] Vasilescu, M.A.O. and Terzopoulos, D., Multilinear analysis of image ensembles: Tensorfaces,In: Computer Vision---ECCV 2002, Lecture Notes in Comput. Sci., Vol. 2350, Berlin: Springer-Verlag, 2002, 447-460.[87] Vo, T., Tran, D. and Ma, W.L., Tensor decomposition and application in image
classification with histogram of oriented gradients, Neurocomputing}, 2015, 165: 38-45.
[88] Wang, C., He, X.F., Bu, J.J., Chen, Z.G., Chen, C.C. and Guan, Z.Y., Image representation using {Laplacian} regularized nonnegative tensor factorization, Pattern Recognit., 2011, 44(11): 2516-2526.
[89] Wong, W.K., Lai, Z.H., Xu, Y., Wen, J.J. and Ho, C.P., Joint tensor feature analysis for visual object recognition, IEEE Trans. Cybern., 2015, 45(11): 2425-2436.
[90] Yang, Q.Z. and Yang, Y.N., Further results for {Perron-Frobenius} theorem for nonnegative tensors {II}, SIAM J. Matrix Anal. Appl., 2011, 32(4): 1236-1250.
[91] Yang, Y.N. and Yang, Q.Z., Further results for {Perron-Frobenius} theorem for nonnegative tensors, SIAM J. Matrix Anal. Appl., 2010, 31(5): 2517-2530.
[92] Zhang, F., Zhou, B.Y. and Peng, L.Z., Dynamic texture analysis using eigenvectors of gradient skewness tensors, In: Proceedings of the 2nd International Conference on Computer Science and Service System, Washington, DC: IEEE Computer Soc., 2012, 2297-2302.
[93] Zhang, F., Zhou, B.Y. and Peng, L.Z., Gradient skewness tensors and local illumination detection for images, J. Comput. Appl. Math., 2013, 237(1): 663-671.[94] Zhang, L.F., Zhang, L.P., Tao, D.C. and Du, B., A sparse and discriminative tensor to vector projection for human gait feature representation, Signal Process., 2015, 106: 245-252.
[95] Zhang, L.F., Zhang, L.P., Tao, D.C., Huang, X. and Du, B., Compression of hyperspectral remote sensing images by tensor approach,
Neurocomputing}, 2015, 147: 358-363.
[96] Zhang, L.P. and Qi, L.Q., Linear convergence of an algorithm for computing the largest eigenvalue of a nonnegative tensor, Numer. Linear Algebra Appl., 2012, 19(5): 830-841.
[97] Zhang, Q., Wang, Y.B., Levine, M.D., Yuan, X.Q. and Wang, L., Multisensor video fusion based on higher order singular value decomposition, Information Fusion}, 2015, 24: 54-71.
[98] Zhao, H.T. and Sun, S.Y., Sparse tensor embedding based multispectral face recognition, Neurocomputing}, 2014, 133: 427-436.
[99] Zhao, Q.B., Caiafa, C.F., Mandic, D.P., Chao, Z.C., Nagasaka, Y., Fujii, N., Zhang, L.Q. and Cichocki, A., Higher order partial least squares (HOPLS): A generalized multilinear regression method, IEEE Trans. Pattern Anal. Mach. Intell., 2013, 35(7): 1660-1673.
[100] Zhou, B.Y., Zhang, F. and Peng, L.Z., Higher-order {SVD} analysis for crowd density estimation, Comput. Vision Image Understanding}, 2012, 116(9): 1014-1021.
[101] Zhou, B.Y., Zhang, F. and Peng, L.Z., Compact representation for dynamic texture video coding using tensor method, IEEE Trans. Circuits Syst. Video Technol., 2013, 23(2): 280-288.
[102] Zhou, B.Y., Zhang, F. and Peng, L.Z., Background modeling for dynamic scenes using tensor decomposition,In: Proceedings of the 6th International Conference on Electronics Information and Emergency Communication,Piscataway, NJ: IEEE Press, 2016, 206-210.
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