{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,11,22]],"date-time":"2023-11-22T15:23:46Z","timestamp":1700666626918},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"3","license":[{"start":{"date-parts":[[2011,1,1]],"date-time":"2011-01-01T00:00:00Z","timestamp":1293840000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Comput. Methods Appl. Math."],"published-print":{"date-parts":[[2011]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p> In the present paper, we propose and analyse a class of tensor methods for the efficient \n\t\t\tnumerical computation of the dynamics and spectrum of high-dimensional Hamiltonians. We focus \n\t\t\ton the complex-time evolution problems. We apply the quantized-TT (QTT) matrix product states \n\t\t\ttype tensor approximation that allows to represent N-d tensors generated by the grid \n\t\t\trepresentation of d-dimensional functions and operators with log-volume complexity, O(d log N), \n\t\t\twhere N is the univariate discretization parameter in space. Making use of the truncated \n\t\t\tCayley transform method allows us to recursively separate the time and space variables and \n\t\t\tthen introduce the efficient QTT representation of both the temporal and the spatial parts of \n\t\t\tthe solution to the high-dimensional evolution equation. We prove the exponential convergence \n\t\t\tof the m-term time-space separation scheme and describe the efficient tensor-structured \n\t\t\tpreconditioners for the arising system with multidimensional Hamiltonians. For the class of \n\t\t\t\"analytic\" and low QTT-rank input data, our method allows to compute the solution at a fixed \n\t\t\tpoint in time t=T&gt;0 with an asymptotic complexity of order O(d log N ln^q (1\/\u03b5)), where \u03b5&gt;0 \n\t\t\tis the error bound and q is a fixed small number. The time-and-space separation method via \n\t\t\tthe QTT-Cayley-transform enables us to construct a global m-term separable (x,t)-representation \n\t\t\tof the solution on a very fine time-space grid with complexity of order O(dm^4 log N_t log N), \n\t\t\twhere N_t is the number of sampling points in time. The latter allows efficient energy \n\t\t\tspectrum calculations by FFT (or QTT-FFT) of the autocorrelation function computed on a \n\t\t\tsufficiently long time interval [0,T]. Moreover, we show that the spectrum of the Hamiltonian \n\t\t\tcan also be represented by the poles of the t-Laplace transform of a solution. In particular, \n\t\t\tthe approach can be an option to compute the dynamics and the spectrum in the time-dependent \n\t\t\tmolecular Schr\u00f6dinger equation.<\/jats:p>","DOI":"10.2478\/cmam-2011-0015","type":"journal-article","created":{"date-parts":[[2013,4,15]],"date-time":"2013-04-15T16:12:56Z","timestamp":1366042376000},"page":"273-290","source":"Crossref","is-referenced-by-count":12,"title":["Quantized-TT-Cayley Transform for Computing the Dynamics and the Spectrum of High-Dimensional Hamiltonians"],"prefix":"10.2478","volume":"11","author":[{"given":"Ivan","family":"Gavrilyuk","sequence":"first","affiliation":[{"name":"1Staatliche Studienakademie Th\u00fcringen, Berufsakademie Eisenach, University of Cooperative Education, Am Wartenberg 2, D-99817 Eisenach, Germany."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Boris","family":"Khoromskij","sequence":"additional","affiliation":[{"name":"2Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22-26, D-04103 Leipzig, Germany."}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/11\/3\/article-p273.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.2478\/cmam-2011-0015\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,4,22]],"date-time":"2021-04-22T14:21:55Z","timestamp":1619101315000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/11\/3\/article-p273.xml"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011]]},"references-count":0,"journal-issue":{"issue":"3"},"URL":"https:\/\/doi.org\/10.2478\/cmam-2011-0015","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"value":"1609-9389","type":"electronic"},{"value":"1609-4840","type":"print"}],"subject":[],"published":{"date-parts":[[2011]]}}}