Use of tensor formats in elliptic eigenvalue problems

By Wolfgang Hackbusch, Boris N. Khoromskij, Stefan A. Sauter, E. E. Tyrtyshnikov. 2012

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “…• the (shift-and-invert) power method combined with CP [26,27]; • the (shift-and-invert) power and Lanczos methods combined with CP and Tucker [11,114]; • a restarted Lanczos method combined with TT [126]; • conjugate gradient type-methods for symmetric eigenvalue problems combined with…”
QTT approximation of elliptic solution operators in higher dimensions (2011, methodology): “The numerical algorithms are based on the idea of truncated preconditioned iteration [15, 19, 21] via the TT-approximation method [30, 36], combined with the quantics reshaping of arising multidimensional arrays [21, 32, 33].”
Adaptive low rank wavelet methods and applications to two-electron Schrödinger equations (2012, background): “, in [101, 5, 96, 140], and for eigenvalue problems in [92, 6, 74, 102, 95, 140].”
Numerical Solution of the Hartree - Fock Equation in Multilevel Tensor-Structured Format (2011, background): “1 Introduction In the recent years the concept of tensor-structured numerical methods has opened new perspectives for solving the basic equations of mathematical physics in R, d ≥ 3, in particular, the many-particle Schrödinger equation [1, 11, 2, 27], and multidimensional elliptic spectral problems”
Quantized-TT-Cayley transform to compute dynamics and spectrum of high-dimensional Hamiltonians (2011, methodology): “Max-Planck-Institut für Mathematik in den Naturwissenschaften Leipzig Quantized-TT-Cayley transform to compute dynamics and spectrum of high-dimensional Hamiltonians (revised version: September 2011) by Ivan P. Gavrilyuk, and Boris N. Khoromskij Preprint no.”
Cross approximation in tensor electron density computations (2010, methodology): “Finally, we suggest to solve spectral problems in the Tucker format (preliminary experiments are provided in [32]) and construct solver for Hartree–Fock/Kohn–Sham model with the Tucker representation of electron density.”
Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces (2011, unknown): “A first family of numerical methods based on classical constructions of tensor approximations [18,24,35] have been recently investigated for the solution of highdimensional partial differential equations [3,19,21,22,28].”
Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions (2013, unknown): “, algorithms for the solution of linear systems [51, 21, 47, 38, 34, 31, 39, 2, 8, 4, 3], partial eigenproblems [41, 40, 24, 20], and Fourier transform [11].”
Preconditioned Low-Rank Methods for High-Dimensional Elliptic PDE Eigenvalue Problems (2011, background): “A number of recent, more mathematically oriented papers illustrate how these ideas can be adapted and transferred to other applications [3, 10, 12, 13, 14, 27].”
Krylov Subspace Methods for Linear Systems with Tensor Product Structure (2010, methodology): “For example, they could be used in a preconditioned inverse iteration for high-dimensional PDE eigenvalue problems [13].”
Low-rank tensor methods for linear systems and eigenvalue problems (2012, background): “A number of recent, more mathematically oriented papers illustrate how these ideas can be adapted and transferred to other applications [18, 31, 35, 34, 36, 63].”
Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances (2012, background): “The preconditioner B = BM can be chosen as inverse of the shifted Laplacian (see [37, 54] for more details).”