Constructive Representation of Functions in Low-Rank Tensor Formats
Tóm tắt
Từ khóa
Tài liệu tham khảo
Beylkin, G., Mohlenkamp, M.J.: Numerical operator calculus in higher dimensions. Proc. Natl. Acad. Sci. USA 99, 10246–10251 (2002)
Beylkin, G., Mohlenkamp, M.J.: Algorithms for numerical analysis in high dimensions. SIAM J. Sci. Comput. 26, 2133–2159 (2005)
Bieri, M., Schwab, C.: Sparse high order FEM for elliptic sPDEs. Comput. Methods Appl. Mech. Eng. 198, 1149–1170 (2009)
Buhmann, M.: Multivariate cardinal interpolation with radial-basis functions. Constr. Approx. 6, 225–255 (1990)
Bungartz, H.-J., Griebel, M., Röschke, D., Zenger, C.: Pointwise convergence of the combination technique for Laplace’s equation. East-West J. Numer. Math. 2, 21–45 (1994)
Caroll, J.D., Chang, J.J.: Analysis of individual differences in multidimensional scaling via n-way generalization of Eckart–Young decomposition. Psychometrika 35, 283–319 (1970)
de Lathauwer, L., de Moor, B., Vandewalle, J.: A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21, 1253–1278 (2000)
de Silva, V., Lim, L.-H.: Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl. 30, 1084–1127 (2008)
Espig, M., Grasedyck, L., Hackbusch, W.: Black box low tensor rank approximation using fibre-crosses. Constr. Approx. 30, 557–597 (2009)
Falco, A., Hackbusch, W.: On minimal subspaces in tensor representation. Preprint 70, MPI MIS, Leipzig (2010)
Grasedyck, L.: Polynomial approximation in hierarchical Tucker format by vector-tensorization. DFG-SPP1324 Preprint 43, Philipps-Univ., Marburg (2010)
Hackbusch, W., Khoromskij, B.N.: Low-rank Kronecker-product approximation to multi-dimensional nonlocal operators. I. Separable approximation of multi-variate functions. Computing 76, 177–202 (2006)
Hackbusch, W., Kühn, S.: A new scheme for the tensor representation. J. Fourier Anal. Appl. 15, 706–722 (2009)
Harshman, R.A.: Foundations of the PARAFAC procedure: models and conditions for an explanatory multimodal factor analysis. UCLA Working Papers in Phonetics, 16, pp. 1–84 (1970)
Khoromskaia, V.: Numerical solution of the Hartree–Fock equation by multilevel tensor-structured methods. PhD thesis, TU, Berlin (2010)
Khoromskij, B.N.: Tensor-structured preconditioners and approximate inverse of elliptic operators in ℝ d . Constr. Approx. 599–620 (2009)
Khoromskij, B.N.: Fast and accurate tensor approximation of multivariate convolution with linear scaling in dimension. J. Comput. Appl. Math. 234, 3122–3139 (2010)
Khoromskij, B.N.: $\mathcal{O}(d \log n)$ —quantics approximation of N−d tensors in high-dimensional numerical modeling. Constr. Approx. 34, 257–280 (2011)
Khoromskij, B.N., Khoromskaia, V.: Multigrid accelerated tensor approximation of function related multidimensional arrays. SIAM J. Sci. Comput. 31, 3002–3026 (2009)
Khoromskij, B.N., Khoromskaia, V., Chinnamsetty, S.R., Flad, H.-J.: Tensor decomposition in electronic structure calculations on 3D Cartesian grids. J. Comput. Phys. 228, 5749–5762 (2009)
Oseledets, I.V.: Lower bounds for separable approximations of the Hilbert kernel. Mat. Sb. 198, 137–144 (2007)
Oseledets, I.V.: Approximation of matrices with logarithmic number of parameters. Dokl. Math. 428, 23–24 (2009)
Oseledets, I.V.: Compact matrix form of the d-dimensional tensor decomposition. Preprint 2009-01, INM RAS, Moscow (2009)
Oseledets, I.V.: Approximation of 2 d ×2 d matrices using tensor decomposition. SIAM J. Matrix Anal. Appl. 31, 2130–2145 (2010)
Oseledets, I.V., Tyrtyshnikov, E.E.: Breaking the curse of dimensionality, or how to use SVD in many dimensions. SIAM J. Sci. Comput. 31, 3744–3759 (2009)
Oseledets, I.V., Tyrtyshnikov, E.E.: TT-cross approximation for multidimensional arrays. Linear Algebra Appl. 432, 70–88 (2010)
Östlund, S., Rommer, S.: Thermodynamic limit of density matrix renormalization. Phys. Rev. Lett. 75, 3537–3540 (1995)
Sloan, I., Wozniakowski, H.: When are quasi-Monte Carlo algorithms efficient for high dimensional integrals. J. Complex. 14, 1–33 (1998)
Tucker, L.R.: Some mathematical notes on three-mode factor analysis. Psychometrika 31, 279–311 (1966)
Vidal, G.: Efficient classical simulation of slightly entangled quantum computations. Phys. Rev. Lett. 91, 147902 (2003)