Low-Rank Tensor Krylov Subspace Methods for Parametrized Linear Systems

By Daniel Kressner, Christine Tobler. 2011

237 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Statistical and Nonlinear Physics

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “…on Hartree-Fock or DFT models [23,24,31,44,45,47,87,141–146,150,156–159]; • the solution of stochastic and parametric PDEs [68,69,76,77,161,165,171,189,202]; • approximation of Green’s functions of high-dimensional PDEs [30,113]; • the solution of the Boltzmann equation [129, 149], and the…”
Low-rank solvers for unsteady Stokes–Brinkman optimal control problem with random data (2016, unknown): “These properties allow to adopt classical iterative methods such as MINRES or GMRES in an inexact fashion, keeping all Krylov vectors in the TT format and performing the TT-SVD re-compression (or TT truncation) [30, 4, 1, 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.”
Multilevel preconditioning and low‐rank tensor iteration for space–time simultaneous discretizations of parabolic PDEs (2015, background): “It has been observed in [21] that a good preconditioner is essential for such low-rank variants of classical iterative solvers, in order to keep the ranks of the iterates moderate without significant loss of accuracy.”
Low-rank methods for high-dimensional approximation and model order reduction (2015, unknown): “ ui+1 ≈ M(ui), where the iterates {ui}i≥1 are in low-rank format. The resulting algorithm can be analyzed as a perturbed version of the initial algorithm (see e.g. 53 [47]). The reader is referred to [54, 52, 3] for a detailed introduction to these techniques in a general algebraic setting, and to [”
Computational Methods for Linear Matrix Equations (2016, background): “Another approach for solving multilinear systems in Kronecker form was analyzed in [165], for which a tensor-based form for the approximate solution is considered.”
Approximation rates for the hierarchical tensor format in periodic Sobolev spaces (2014, background): “A direct elementary proof of the stated inequality can be found in [27].”
Preconditioned Low-Rank Methods for High-Dimensional Elliptic PDE Eigenvalue Problems (2011, methodology): “Using the low-rank CG method described in [19], the tensor C is already in HTD, which we truncate further up to a relative error eps 10 (4).”
Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives (2017, methodology): “…et al., 2014], TT format [Dolgov, 2013, Khoromskij and Oseledets, 2010], HT format [Bachmayr and Dahmen, 2015, Bachmayr and Schneider, 2016, Kressner and Tobler, 2011a], and a subspace projection method combined with HT format [Ballani and Grasedyck, 2013]; • A subspace projection method…”
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].”
Low-Rank Explicit QTT Representation of the Laplace Operator and Its Inverse (2012, methodology): “Note that the straightforward discretizations of the Laplace operator we consider turn out to be very efficient in tensor-structured solvers based on the TT, QTT, and hierarchical tensor [15, 13] formats and yield convincing experimental results when constructed on fine meshes; see the papers [24, 8”
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].”
Geometric Methods on Low-Rank Matrix and Tensor Manifolds (2020, background): “, ω), the tensor X can be shown [62] to be well approximated by low TT rank and it satisfies a very large linear system L(X) = B.”
Algorithm 941 (2014, background): “As explained in [25], this tensor can be interpreted as the solution of a (huge) symmetric positive definite linear system ApX q B.”
Low-rank tensor methods for linear systems and eigenvalue problems (2012, background): “The contents of this chapter are an updated version of [48].”