On the approximation of high-dimensional differential equations in the hierarchical Tucker format

By Andreas Arnold, Tobias Jahnke. 2013

40 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Mechanics

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In Computational Mathematics

In Computational Mechanics

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.”
A projector-splitting integrator for dynamical low-rank approximation (2013, background): “Very recently, in [1,9, 15] the dynamical low-rank approximation approach was extended to tensors in the tensor train (TT) format studied in [14] and the hierarchical Tucker (HT) format studied in [3].”
Tensor product methods and entanglement optimization for ab initio quantum chemistry (2014, methodology): “,νd) ∈ Vd N for the eigenvector, with the help of ι, we obtain the discrete eigenvalue problem from (13) in the N -electron subspace of Λ 〈Φ(μ1,.”
Geometric Structures in Tensor Representations (Final Release) (2013, unknown): “Recently, similar results have been obtained for the TT format [20] and the HT format [33] (see also [3]).”
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].”
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).”
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.”
The geometry of algorithms using hierarchical tensors (2013, background): “3] and [48] have appeared that also extend the dynamical low-rank algorithm from [37, 38] to HT tensors using the same tangent space and choice of gauge as ours.”