Low-Rank Explicit QTT Representation of the Laplace Operator and Its Inverse

By Vladimir A. Kazeev, Boris N. Khoromskij. 2012

146 citations recorded by the source corpus.

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

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, positive): “For the following matrices, QTT representations have been discussed: Toeplitz matrices [136,216], (inverse) Laplace operators [139,205], linear diffusion operators [62,138].”
Low-rank tensor structure of linear diffusion operators in the TT and QTT formats☆ (2013, methodology): “Following [7], we use the following notation for TT cores and operations with them.”
Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions (2013, unknown): “Note that the matrix and the right–hand side have exact low–rank representations, see [21, 31].”
Tensor-product approach to global time-space-parametric discretization of chemical master equation (2012, background): “Moreover, the QTT-format allows simple constructive representations of basic operators (Laplacian, gradient and divergence operators) [48] on uniform tensor grids.”
Direct Solution of the Chemical Master Equation Using Quantized Tensor Trains (2014, methodology): “4 we revisit the notions of core matrices and strong Kronecker product according to the papers [3–5].”
ALTERNATING MINIMAL ENERGY METHODS FOR LINEAR SYSTEMS IN HIGHER DIMENSIONS∗ (2014, background): “Both A and b have moderate QTT ranks [7, 30].”
Fast Solution of Parabolic Problems in the Tensor Train/Quantized Tensor Train Format with Initial Application to the Fokker-Planck Equation (2012, background): “Moreover, the QTT-format allows simple constructive representations of basic operators (Laplacian, gradient, and divergence operators) [21] on uniform tensor grids.”
Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 1 Low-Rank Tensor Decompositions (2016, unknown): “Not only QTT ranks are typically small (usually, below 20) but they are also almost independent4 of the data size (even for I 250), thus providing a logarithmic (sub-linear) reduction of storage requirements from OpIN q to OpNR2 logqpIqq which is referred to as super-compression (Khoromskij, 2011a; ”
Low-rank tensor methods for linear systems and eigenvalue problems (2012, methodology): “Similarly as in the TT format [37, 62], the discretization AL of the Laplace operator can be represented by a tensor in HTD, with all hierarchical ranks equal to 2, see Example 3.”