Tensor Computation

 

A.  Eigenvalues of Tensors:

 

[1]            L. Qi,"Eigenvalues of a real supersymmetric tensor",  Journal of Symbolic Computation  40 (2005) 1302-1324.

 

[2]            L. Qi,"Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines", Journal of Symbolic Computation 41 (2006) 1309-1327.

 

[3]            L. Qi, "Eigenvalues and invariants of tensors", Journal of Mathematical Analysis & Applications 325 (2007) 1363-1377.

 

[4]            G. Ni, L. Qi, F. Wang and Y. Wang, "The degree of the E-characteristic polynomial of an even order tensor",   Journal of Mathematical Analysis & Applications  329 (2007) 1218-1229.

 

[5]            Y. Wang and L. Qi, "On the successive supersymmetric rank-1 decomposition of higher order supersymmetric tensors", Numerical Linear Algebra with Applications 14 (2007) 503-519.

 

[6]           L. Qi, W. Sun and Y. Wang, "Numerical multilinear algebra and its applications", Frontiers of Mathematics in China  2 (2007) 501-526.

 

[7]          Q. Ni, L. Qi and F. Wang, "An eigenvalue method for the positive definiteness identification problem", IEEE Transactions on Automatic Control 53 (2008) 1096-1107.

 

[8]           L. Qi, F. Wang and Y. Wang, "Z-eigenvalue methods for a global polynomial optimization problem", Mathematical Programming 118 (2009) 301-316.

 

[9]           Y.Wang, L. Qi and X. Zhang, "A Practical method for computing the largest M-eigenvalue of a fourth-order partially symmetric tensor", Numerical Linear Algebra and Applications 16 (2009) 589-601.

 

[10]         D. Han and L. Qi, "A successive approximation method for quantum separability", February 2009.

 

[11]         K.C. Chang, L. Qi and G. Zhou, "Singular values of a real rectangular tensor", Journal of Mathematical Analysis & Applications 370 (2010) 284-294.

 

[12]         G. Li, L. Qi and G. Yu, "Semismoothness of the maximum eigenvalue function of a symmetric tensor and its application'', Linear Algebra and Its Applications 438 (2013) 813-833.

 

B.  Survey

 

[13]     L. Qi, W. Sun and Y. Wang, "Numerical multilinear algebra and its applications'', Frontiers of Mathematics in China  2 (2007) 501-526.

 

[14]     L. Qi, "The spectral theory of tensors (rough version)", arXiv:1201.3424v1 [math.SP] 17 Jan 2012.

 

C.  Applications in Biomedical Engineering

 

[15]        L. Qi, Y. Wang and E.X. Wu, "D-eigenvalues of diffusion kurtosis tensors", Journal of Computational and Applied Mathematics 221 (2008) 150-157.

 

[16]        E.S. Hui, M.M. Cheung, L. Qi and E.X. Wu, "Towards better MR characterization of neural tissues using directional diffusion kurtosis analysis", Neuroimage 42 (2008) 122-134.

 

[17]        L. Qi, D. Han and E.X. Wu, "Principal invariants and inherent parameters of diffusion kurtosis tensors", Journal of Mathematical Analysis and Applications 349 (2009) 165-180.

 

[18]        D. Han, L. Qi and E.X. Wu, "Extreme diffusion values for non-Gaussian diffusions", Optimization Methods and Software 23 (2008) 703-716.

 

[19]         E.S. Hui, M.M. Cheung, L. Qi and E.X. Wu, "Advanced MR diffusion characterization of neural tissue using directional diffusion kurtosis analysis",  Conf. Proc. IEEE Eng. Med. Biol. Soc. 2008 (2008) 3941-3944.

 

[20]        M.M. Cheung, E.S Hui, K.C Chan, J.A Helpern, L. Qi and E.X Wu: "Does diffusion kurtosis imaging lead to better neural tissue characterization? A rat brain maturation study", Neuroimage 45 (2009) 386-392.

 

[21]       X. Zhang, C. Ling, L. Qi and E.X. Wu, "The measure of diffusion skewness and kurtosis in magnetic resonance imaging", Pacific Journal of Optimization 6 (2010)391-404.

 

[22]        L. Qi, G. Yu and E.X. Wu, "Higher order positive semi-definite diffusion tensor imaging", SIAM Journal on Imaging Sciences 3 (2010) 416-433.

 

[23]        S.L. Hu, Z.H. Huang, H.Y. Ni and L. Qi, "Positive definiteness of diffusion kurtosis imaging'', to appear in: Inverse Problems and Imaging.

 

[24]        L. Qi, G. Yu and Y. Xu, "Nonnegative diffusion orientation distribution function'', Journal of Mathematical Imaging and Vision 45 (2013) 103-113.

 

D.  Applications in Solid Mechanics

 

[25]        L. Qi, H.H. Dai and D. Han. "Conditions for strong ellipticity and M-eigenvalues", Frontiers of Mathematics in China  4 (2009) 349-364.

 

[26]        D. Han, H.H. Dai and L. Qi, "Conditions for strong ellipticity of anisotropic elastic materials", Journal of Elasticity 97 (2009) 1-13.

 

E.  Nonnegative Tensors

 

[27]         M. Ng, L. Qi and G. Zhou, "Finding the largest eigenvalue of a non-negative tensor", SIAM Journal on Matrix Analysis and Applications 31 (2009) 1090-1099.

 

[28]        G. Zhou, L. Caccetta and L. Qi, "Convergence of an algorithm for the largest singular value of a nonnegative rectangular tensor'', Linear Algebra and Its Applications 438 (2013) 959-968

 

[29]      L. Zhang and L. Qi, "Linear convergence of an algorithm for computing the largest eigenvalue of a nonnegative tensor'', Numerical Linear Algebra with Applications 19 (2012) 830-841.

 

[30]       L. Zhang, L. Qi and Y. Xu, "Linear convergence of the LZI algorithm for weakly positive tensors'', Journal of Computational Mathematics 30 (2012) 24-33.

 

[31]       G. Zhou, L. Qi and S.Y. Wu, "Efficient algorithms for computing the largest eigenvalue of a nonnegative tensor'', Frontiers of Mathematics in China 8 (2013) 155-168.

 

[32]       L. Zhang, L. Qi, Z. Luo and Y. Xu, "The dominant eigenvalue of an essentially nonnegative tensor'', arXiv: 1110.6261v1 [math.NA] 28 Oct 2011, Revised in January 2013.    Matlab Code of the Algorithm in this paper.

 

[33]       S. Hu, Z.H. Huang and L. Qi, "Finding the spectral radius of a nonnegative tensor", arXiv: 1111.2138v1 [math.NA] 9 Nov 2011.

 

[34]       L. Qi and Q. Yang, "Eigenvalues of nonnegative tensors", January 9, 2012.

 

F.  Spectral Hypergraph Theory

 

[35]    S. Hu and L. Qi, "Algebraic connectivity of an even uniform hypergraph", Journal of Combinatorial Optimization 24 (2012) 564-579.

 

G.  Further Theory on Eigenvalues of Tensors

 

[36]        L. Qi, "The Best Rank-One Approximation Ratio of a Tensor Space'', SIAM Journal on Matrix Analysis and Applications 32 (2011) 430-442.

 

[37]        X. Zhang, C. Ling and L. Qi, "The best rank-1 approximation of a symmetric tensor and related spherical optimization problems'', SIAM Journal on Matrix Analysis and Applications 33 (2012) 806-821.

 

[38]        Y. Song and L. Qi, "The existence and uniqueness of eigenvalue for monotone homogeneous mapping pairs'', Nonlinear Analysis 75 (2012) 5283-5293.

 

[39]         A.M. Li, L. Qi and B. Zhang, "E-characteristic polynomials of tensors'', Communications in Mathematical Sciences 11 (2013) 33-53.

 

[40]         S. Hu and L. Qi, "E-characteristic polynomial of a tensor of dimension two'', Applied Mathematics Letters 26 (2013) 225-231.

 

[41]     S. Hu, Z. Huang, C. Ling and L. Qi, "On Determinants and Eigenvalue Theory of Tensors'', Journal of Symbolic Computation 50 (2013) 508-531.

 

[42]     Y. Song and L. Qi, "Positive eigenvalue-eigenvector of nonlinear positive mappings'', to appear in: Frontiers of Mathematics in China.

 

[43]     C. Ling and L. Qi, "On lk-singular values and spectral radius of rectangular tensors'', Frontiers of Mathematics in China 8 (2013) 63-84.

 

H.  Computational Polynomial Optimization

 

[44]        C. Ling, J. Nie, L. Qi and Y. Ye, "Bi-quadratic optimization over unit spheres and semidefinite programming relaxations", SIAM Journal on Optimization 20 (2009) 1286-1310.

 

[45]       X. Zhang, L. Qi and Y. Ye, "The cubic spherical optimization problems'', Mathematics of Computation 81 (2012) 1513-1525.

 

[46]       X. Zhang, C. Ling and L. Qi, "Semidefinite relaxation bounds for bi-quadratic optimization problems with quadratic constraints'', Journal of Global Optimization  49 (2011) 293-311.

 

[47]        C. Ling, X. Zhang and L. Qi, "Semidefinite relaxation approximation for multivariate bi-quadratic optimization with quadratic constraints", Numerical Linear Algebra with Applications 19 (2012) 113-131.

 

[48]         I.M. Bomze, C. Ling, L. Qi and X. Zhang, "Standard bi-quadratic optimization problems and unconstrained polynomial reformulations'', Journal of Global Optimization 52 (2012) 663-687.

 

[W1]      Workshop of Computational Polynomial Optimization and Multilinear Algebra

 

 

I.  Space Tensor Conic Programming

 

[49]         L. Qi and Y. Ye, "Space tensor conic programming", June 2009.

 

J.  Quantum Physics

 

[50]     L. Qi, "The minimum Hartree value for the quantum entanglement problem'', arXiv:1202.2983v1 [quant-ph] 14 Feb 2012.

 

K.  Tensor Decomposition

 

[51]        Y. Chen, D. Han and L. Qi, "New ALS methods with extrapolating search directions and optimal step size for complex-valued tensor decompositions'', IEEE Transactions on Signal Processing 59 (2011) 5888-5898.

 

L.  Articles and Photo

 

[P1]      Article 1 –– 四十二年后再登东嶽泰山.

 

[P2]      Photo –– 一覽众山小..

 

[P3]   Article 2 –– 张量研究的两点心得.

 

[P4]   Article 3 –– 开拓应用数学和计算数学的新疆土.

 

M.  Special Issues: Call-of-Papers

 

[S1]      NLA Special Issue: The Spectral Theory of Tensors and Its Applications