QTT approximation of elliptic solution operators in higher dimensions

By Boris N. Khoromskij, Ivan V. Oseledets. 2011

53 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Mechanics of Materials, Numerical Methods & Scientific Computing

How researchers see this paper

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

In Computational Mechanics

1. Sergey V. Dolgov
2. Boris N. Khoromskij
3. Christine Tobler
4. Dmitry V. Savostyanov
5. Ivan V. Oseledets
6. Vladimir A. Kazeev
7. Christoph Schwab
8. Daniel Kressner

In Computational Theory and Mathematics

1. Christine Tobler
2. Sergey V. Dolgov
3. Dmitry V. Savostyanov
4. Boris N. Khoromskij
5. Vladimir A. Kazeev
6. Ivan V. Oseledets
7. Daniel Kressner
8. Christoph Schwab

In Mechanics of Materials

1. Ivan V. Oseledets
2. Vladimir A. Kazeev
3. Christine Tobler
4. Boris N. Khoromskij
5. Dmitry V. Savostyanov
6. Sergey V. Dolgov
7. Daniel Kressner
8. Christoph Schwab

In Numerical Methods & Scientific Computing

1. Christine Tobler
2. Dmitry V. Savostyanov
3. Daniel Kressner
4. Sergey V. Dolgov
5. Boris N. Khoromskij
6. Ivan V. Oseledets
7. Vladimir A. Kazeev
8. Christoph Schwab

Citations in the corpus

• A literature survey of low‐rank tensor approximation techniques (2013, neutral): “QTT has been applied to the solution of PDEs and eigenvalue problems [144, 154, 162], evaluation of boundary integrals in BEM [164], convolution [107] and the FFT [63, 108, 230].”
• 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”
• DMRG Approach to Fast Linear Algebra in the TT-Format (2011, methodology): “A simple approach is the add-andcompress algorithm proposed in [31].”
• Algorithm 941 (2014, background): “This includes the tensor train decomposition [Khoromskij and Oseledets 2011], the closely related but somewhat more general HTD (hierarchical Tucker decomposition) [Grasedyck 2010b; Hackbusch and Kühn 2009], and even more general tensor networks [Espig et al. 2011].”
• Low-rank tensor structure of linear diffusion operators in the TT and QTT formats☆ (2013, background): “: ; The computational efficiency of the TT, QTT and Hierarchical Tensor [19, 20] representations has been demonstrated in many papers, including [21, 22, 23] on elliptic PDEs and eigenvalue problems, [24, 25, 26] on parabolic PDEs, [27] on nonlinear EVPs, [28] on multi-parametric problems, [29] on…”
• Superfast Fourier Transform Using QTT Approximation (2012, methodology): “There are more examples of high-dimensional problems which were efficiently solved using the QTT approximation [4, 36, 37] as well as H -Tucker format [15].”
• Tensor-Train Decomposition (2011, unknown): “Ideas presented in this paper, have already been applied to the solution of different problems: stochastic partial differential equations [25], high-dimensional elliptic equations [24], and elliptic equations with variable coefficients [11].”
• Low-rank tensor methods for linear systems and eigenvalue problems (2012, background): “This includes the tensor train decomposition [40] and the closely related but more general HTD (hierarchical tensor decomposition) [28, 32].”