Krylov Subspace Methods for Linear Systems with Tensor Product Structure

By Daniel Kressner, Christine Tobler. 2010

192 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Electrical and Electronic Engineering, Numerical Methods & Scientific Computing

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “This applies, in particular, to discretizations of the multi-dimensional Poisson equation [22,101,170,188].”
Low-rank tensor methods for linear systems and eigenvalue problems (2012, methodology): “Alternatively, the tensor Krylov subspace method from [47] could be used for directly approximating the action of A−1 L .”
Preconditioned Low-Rank Methods for High-Dimensional Elliptic PDE Eigenvalue Problems (2011, methodology): “Alternatively, the tensor Krylov subspace method from [18] could be used for directly approximating the action of A 1 L .”
Algorithm 941 (2014, methodology): “A more detailed description of truncation to HTD without initial orthogonalization can be found in Kressner and Tobler [2012, Algorithm 7] and is implemented in truncate nonorthog(x, opts) in the toolbox. The result of this algorithm satisfies practically the same error bound as in (17); see also Le”
Numerical solution of the Hartree-Fock equation by multilevel tensor-structured methods (2011, unknown): “6.1 Brief summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.2 Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140”
Low-Rank Tensor Krylov Subspace Methods for Parametrized Linear Systems (2011, methodology): “An extension of such Krylov subspace methods to several parameters is described in [31] under the condition that the coefficients A1, .”
F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig a Projection Method to Solve Linear Systems in Tensor Format a Projection Method to Solve Linear Systems in Tensor Format (unknown): “Kressner and Tobler [15] have introduced a so-called tensor Krylov subspace method.”
ALTERNATING MINIMAL ENERGY METHODS FOR LINEAR SYSTEMS IN HIGHER DIMENSIONS∗ (2014, methodology): “, 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].”
The geometry of algorithms using hierarchical tensors (2013, background): “Besides their longstanding tradition in the computational physics community (see [20] [23] for recent overviews), the (Q)TT and HT formats have more recently been used, amongst others, in [24] [25] [26] [27] [28] [29] [30] [31] for solving a variety of high-dimensional problems.”
An Error Analysis of Galerkin Projection Methods for Linear Systems with Tensor Product Structure (2013, methodology): “For A being Hermitian positive definite, error bounds in the energy norm, ‖x− x̃‖A have been given in [21] for the case of standard Krylov subspaces, that is, zμ,j ≡ ∞.”
Towards an ADI iteration for Tensor Structured Equations (2014, methodology): “The solution of tensor structured equations using a Krylov subspace method was recently investigated by Tobler and Kressner in [19].”
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].”
A preconditioned low‐rank CG method for parameter‐dependent Lyapunov matrix equations (2014, methodology): “For this purpose, we combine our recently developed low-rank techniques for solving parameter-dependent linear systems [28] with low-rank techniques for Lyapunov equations.”
TT-GMRES: on solution to a linear system in the structured tensor format (2012, unknown): “It is worth to mention the previous papers devoted to Krylov methods with tensor computations: [14] (a FOM-like method in the case of a Laplace-like matrix), [15] (Richardson, PCG and BiCGStab meth-ods with application to parametric and stochastic PDEs), and, the closest to our work, a projection…”
Low-rank solvers for unsteady Stokes–Brinkman optimal control problem with random data (2016, unknown): “For problems of some special forms, such as the Lyapunov equations, one can employ ADI [53] or tensor product Krylov methods [29].”
Low-Rank Explicit QTT Representation of the Laplace Operator and Its Inverse (2012, background): “molecular dynamics, [18] on electronic structure calculations, and also the numerical experiments in [27, 9].”
Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey (2013, methodology): “In this view it is clear that their approach has to be seen as close relative of the Krylov subspace methods applied to more generally tensor structured equations as for example treated by Kressner and Tobler [34] that have shown in [8] to be effective also in the Lyapunov-plus-positive case discuss”