Existence and Computation of Low Kronecker-Rank Approximations for Large Linear Systems of Tensor Product Structure

By Lars Grasedyck. 2004

174 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Numerical Analysis

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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): “Suitable coefficients αj , ωj and error bounds are described in [26, 30].”
Algorithm 941 (2014, methodology): “It is described in Grasedyck [2010b, Algorithm 2] and Kressner and Tobler [2012, Algorithm 5], and implemented in the function x = htensor.truncate ltr(x, opts) of the toolbox. The computational complexity of leaves-to-root truncation is O(dnd+1), which is a significant reduction compared to the roo”
An Error Analysis of Galerkin Projection Methods for Linear Systems with Tensor Product Structure (2013, methodology): “Such linear systems arise, for example, from the discretization of a separable d-dimensional PDE with tensorized finite elements [14, 21].”
Adaptive wavelet methods for solving operator equations: An overview (2009, result): “As demonstrated in [ Gra04 , BM02, HK07], a class of functions that can be accurately approximated in high space dimensions are the solutions of boundary value problems with righthand sides that can be well approximated by a small number of separable functions.”
Further results on generalized inverses of tensors via the Einstein product (2016, methodology): “This appears in the finite element method,[15] finite difference or spectral methods,[16,17] and plays an important role in discretization of a linear partial differential equation in high dimension.”
Towards an ADI iteration for Tensor Structured Equations (2014, methodology): “Equations like (1) have been investigate in [12,19] and more recently in [10].”
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): “In [8] we have shown that u has an integral representation which can be approximated by vectors of low rank.”
Concepts of Data-Sparse Tensor-Product Approximation in Many-Particle Modelling (2008, background): “Numerical examples for still higher dimensions d ≤ 1024 are presented in [45].”
Krylov Subspace Methods for Linear Systems with Tensor Product Structure (2010, background): “Grasedyck [9] has combined an integral representation of the solution x to (1.3) with quadrature based on sinc interpolation [25] to show that x can be well approximated by vectors of low tensor rank and to develop a numerical algorithm that scales linearly with d.”
Tensor-Train Decomposition (2011, background): “Multilinear algebra, tensor analysis, and the theory of tensor approximations play increasingly important roles in computational mathematics and numerical analysis [8, 7, 9, 5, 14]; see also the review [26].”
Black Box Low Tensor-Rank Approximation Using Fiber-Crosses (2009, methodology): “(2) Initial low rank approximation: In [9] a method for the fast solution of special high-dimensional partial differential equations (PDE) is presented, which requires the right-hand side of the PDE to be in low rank format.”
Efficient time-stepping scheme for dynamics on TT-manifolds (2012, unknown): “It can be used to approximate high-dimensional operators and their inverses (see, for example, [2, 3, 11, 14, 15, 12]) using certain analytical expansions.”
Preconditioned Low-Rank Methods for High-Dimensional Elliptic PDE Eigenvalue Problems (2011, methodology): “As a preconditioner in LOBPCG, we use the results from [7] to construct an approximate inverse of the AL having the form A 1 L » 8”
Moore–Penrose inverse of tensors via Einstein product (2016, methodology): “It comes from the finite element,[11] finite difference or spectral method [12,13] and plays an important role in discretization of a linear partial differential equation in high dimension.”
Low-Rank Tensor Krylov Subspace Methods for Parametrized Linear Systems (2011, methodology): “The inverse of M̂ can be approximated by an exponential sum [20], M̂ = ∞”