Linear algebra for tensor problems

By Ivan V. Oseledets, Dmitry V. Savostyanov, E. E. Tyrtyshnikov. 2009

57 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Hardware and Architecture

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mathematics

1. Boris N. Khoromskij
2. Ivan V. Oseledets
3. Dmitry V. Savostyanov
4. Sergey V. Dolgov
5. E. E. Tyrtyshnikov
6. Venera Khoromskaia
7. Christine Tobler
8. V. Dolgov

In Computational Theory and Mathematics

1. Ivan V. Oseledets
2. Boris N. Khoromskij
3. Dmitry V. Savostyanov
4. Venera Khoromskaia
5. E. E. Tyrtyshnikov
6. Thomas Mach
7. V. Dolgov
8. Sergey V. Dolgov

In Hardware and Architecture

1. Ivan V. Oseledets
2. Dmitry V. Savostyanov
3. Venera Khoromskaia
4. V. Dolgov
5. E. E. Tyrtyshnikov
6. Thomas Schulte-Herbrüggen
7. Thomas K. Huckle
8. Konrad Waldherr

Citations in the corpus

• A literature survey of low‐rank tensor approximation techniques (2013, neutral): “Other techniques include the use of approximate inverses for high-dimensional Laplace operators [151, 165, 172, 210], low-rank manipulation of the PDE coefficients [60], low-rank tensor approximation of multilevel preconditioners [8], and low-rank tensor diagonal preconditioners for wavelet…”
• Fast truncation of mode ranks for bilinear tensor operations (2010, unknown): “The residual between two Tucker formats is computed as proposed in [10].”
• Cross approximation in tensor electron density computations (2010, methodology): “• Develop algorithms for approximate multiplication in the Tucker formats (the work that was already started in [6, 27, 33]) and apply them to compute pointwise functions (for example, cubic root) of a vector in the Tucker format by the Newton method.”
• Tensor-Train Decomposition (2011, methodology): “⊗ Ud(αd), where Uk(αk) is an n × n matrix, and the TT-decomposition is applied to the core G; see [31], where a detailed description for the three-dimensional case is given.”
• 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].”
• Computation of the Hartree-Fock Exchange by the Tensor-Structured Methods (2010, background): “A comprehensive description of the multi-linear tensor-product operations for the multidimensional tensors is presented in the survey [18], see also [14, 13, 22] for details on function related tensors.”
• Computing Inner Eigenvalues of Matrices in Tensor Train Matrix Format (2013, background): “The concept of tensor trains, TT for short, as described in [17] permits the computation of a data-sparse approximation of tensors.”
• Computations in quantum tensor networks (2012, background): “The canonical decomposition (candecomp), which is also known as Parallel Factorization( parafac ), has the form Hence, the parafac decomposes the given tensor into a sum of rank one tensors (see [54, 55]), which is illustrated by Figure 10(b).”
• Numerical solution of the Hartree-Fock equation by multilevel tensor-structured methods (2011, unknown): “6.1 Brief summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.2 Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140”
• Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions (2013, unknown): “For ALS there is also a lack of convergence estimates both in the classical papers [16, 4], and in the recent ones, where ALS was applied to the Tucker model [5, 33], tensor trains [29], hierarchical Tucker format [25] and high–dimensional interpolation [34].”