Dynamical Approximation by Hierarchical Tucker and Tensor-Train Tensors

By Christian Lubich, Thorsten Rohwedder, Reinhold Schneider, Bart Vandereycken. 2013

188 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Nuclear and High Energy Physics

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “Initially proposed for low-rank matrix manifolds in [166], dynamical low-rank methods have been extended to low-rank tensors in Tucker [167, 199], TT/MPS [118, 123, 163, 186], and HT [9, 186, 255] decomposition.”
Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions (2013, unknown): “Several theoretical issues were addressed in [43], and the numerical splitting w.”
A projector-splitting integrator for dynamical low-rank approximation (2013, methodology): “roach was proposed and analyzed in [5], and ﬁrst numerical applications were given in [12]. The approach and its error analysis were extended to tensors in the Tucker format in [6]. Very recently, in [1, 8, 14] the dynamical low-rank approximation approach was extended to tensors in the tensor train”
Multilevel preconditioning and low‐rank tensor iteration for space–time simultaneous discretizations of parabolic PDEs (2015, background): “In [28, 29, 30, 31], other low rank tensor approaches for the solution of evolution equations are proposed.”
On the approximation of high-dimensional differential equations in the hierarchical Tucker format (2013, methodology): “The same result has also been obtained with other techniques in [12] and [14].”
On Local Convergence of Alternating Schemes for Optimization of Convex Problems in the Tensor Train Format (2013, background): “differential equations [19, 31, 21, 43], and although ALS might be refined by using locally quadratically convergent methods near the sought optimum, the ALS approach is in fact at the moment our basic method of choice for the treatment of optimization problems in the sense that even with its striki”
Algorithm 941 (2014, methodology): “Candidates include dynamical low-rank methods for HTD [Arnold and Jahnke 2012; Lubich et al. 2013] and low-rank tensor cross-approximation for HTD [Ballani 2012; Ballani et al. 2012].”
Geometric Methods on Low-Rank Matrix and Tensor Manifolds (2020, methodology): “Most of the analysis was also extended to manifolds of fixed TT rank (as well as to Tucker and hierarchical) tensors in [60, 73] and to Hilbert spaces [83].”
Low-Rank Tensor Learning with Discriminant Analysis for Action Classification and Image Recovery (2014, background): “…representation learning approaches have been proposed (Chen and Saad 2009; Koch and Lubich 2010; Haegeman et al. 2011; Holtz, Rohwedder, and Schneider 2012; Khoromskij, Oseledets, and Schneider 2012; Arnold and Jahnke 2012; Lubich et al. 2013; Uschmajew and Vandereycken 2013; Mu et al. 2013).”
ALTERNATING MINIMAL ENERGY METHODS FOR LINEAR SYSTEMS IN HIGHER DIMENSIONS∗ (2014, background): “Several theoretical issues were addressed in [43], and the numerical splitting w.”
The geometry of algorithms using hierarchical tensors (2013, background): “For a theoretical and numerical investigation of such an approximation, we refer to [48, 70].”