{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,15]],"date-time":"2026-03-15T13:46:01Z","timestamp":1773582361973,"version":"3.50.1"},"reference-count":60,"publisher":"Walter de Gruyter GmbH","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,7,1]]},"abstract":"Abstract<\/jats:title>\n This paper introduces and analyzes the new grid-based tensor approach\nto approximate solutions of the elliptic eigenvalue problem\nfor the 3D lattice-structured systems.\nWe consider the linearized Hartree\u2013Fock equation over a spatial\n\n \n \n \n \n L<\/m:mi>\n 1<\/m:mn>\n <\/m:msub>\n \u00d7<\/m:mo>\n \n L<\/m:mi>\n 2<\/m:mn>\n <\/m:msub>\n \u00d7<\/m:mo>\n \n L<\/m:mi>\n 3<\/m:mn>\n <\/m:msub>\n <\/m:mrow>\n <\/m:math>\n {L_{1}\\times L_{2}\\times L_{3}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> lattice for both periodic and non-periodic problem setting,\ndiscretized in the localized Gaussian-type orbitals basis.\nIn the periodic case, the Galerkin system matrix obeys a\nthree-level block-circulant structure that allows the FFT-based diagonalization, while\nfor the finite extended systems in a box (Dirichlet boundary conditions)\nwe arrive at the perturbed block-Toeplitz representation providing fast\nmatrix-vector multiplication and low storage size.\nThe proposed grid-based tensor techniques manifest the twofold benefits:\n(a) the entries of the Fock matrix are computed by 1D operations using\nlow-rank tensors represented on a 3D grid, (b) in the periodic case the low-rank tensor\nstructure in the diagonal blocks of the Fock matrix in the Fourier space\nreduces the conventional 3D FFT to the product of 1D FFTs.\nLattice type systems in a box with Dirichlet boundary conditions\nare treated numerically by our previous tensor solver for single\nmolecules, which makes possible calculations on rather\nlarge \n \n \n \n \n L<\/m:mi>\n 1<\/m:mn>\n <\/m:msub>\n \u00d7<\/m:mo>\n \n L<\/m:mi>\n 2<\/m:mn>\n <\/m:msub>\n \u00d7<\/m:mo>\n \n L<\/m:mi>\n 3<\/m:mn>\n <\/m:msub>\n <\/m:mrow>\n <\/m:math>\n {L_{1}\\times L_{2}\\times L_{3}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> lattices due to reduced numerical cost for 3D problems.\nThe numerical simulations for both box-type and periodic \n \n \n \n L<\/m:mi>\n \u00d7<\/m:mo>\n 1<\/m:mn>\n \u00d7<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n <\/m:math>\n {L\\times 1\\times 1}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\nlattice chain in a 3D rectangular \u201ctube\u201d with L<\/jats:italic> up to several hundred\nconfirm the theoretical complexity bounds for the block-structured\neigenvalue solvers in the limit of large L<\/jats:italic>.<\/jats:p>","DOI":"10.1515\/cmam-2017-0004","type":"journal-article","created":{"date-parts":[[2017,6,2]],"date-time":"2017-06-02T13:21:55Z","timestamp":1496409715000},"page":"431-455","source":"Crossref","is-referenced-by-count":3,"title":["Block Circulant and Toeplitz Structures in the Linearized Hartree\u2013Fock Equation on Finite Lattices: Tensor Approach"],"prefix":"10.1515","volume":"17","author":[{"given":"Venera","family":"Khoromskaia","sequence":"first","affiliation":[{"name":"Max-Planck-Institute for Mathematics in the Sciences , Inselstr. 22-26, 04103 Leipzig ; and Max Planck Institute for Dynamics of Complex Systems, Magdeburg , Germany"}]},{"given":"Boris N.","family":"Khoromskij","sequence":"additional","affiliation":[{"name":"Max-Planck-Institute for Mathematics in the Sciences , Inselstr. 22-26, 04103 Leipzig ; and Max Planck Institute for Dynamics of Complex Systems, Magdeburg , Germany"}]}],"member":"374","published-online":{"date-parts":[[2017,5,31]]},"reference":[{"key":"2023033114491608191_j_cmam-2017-0004_ref_001_w2aab3b7d384b1b6b1ab2b1b1Aa","doi-asserted-by":"crossref","unstructured":"P. Benner, S. Dolgov, V. Khoromskaia and B. N. Khoromskij,\nFast iterative solution of the Bethe-Salpeter eigenvalue problem using low-rank and QTT tensor approximation,\nJ. Comput. Phys. 334 (2017), 221\u2013239.","DOI":"10.1016\/j.jcp.2016.12.047"},{"key":"2023033114491608191_j_cmam-2017-0004_ref_002_w2aab3b7d384b1b6b1ab2b1b2Aa","unstructured":"P. Benner, H. Fa\u00dfbender and C. Yang,\nSome remarks on the complex J{{J}}-symmetric eigenproblem,\npreprint (2015), http:\/\/www2.mpi-magdeburg.mpg.de\/preprints\/2015\/12\/."},{"key":"2023033114491608191_j_cmam-2017-0004_ref_003_w2aab3b7d384b1b6b1ab2b1b3Aa","doi-asserted-by":"crossref","unstructured":"P. Benner, V. Khoromskaia and B. N. Khoromskij,\nA reduced basis approach for calculation of the Bethe\u2013Salpeter excitation energies using low-rank tensor factorizations,\nMol. 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