O(dlog N)-Quantics Approximation of N-d Tensors in High-Dimensional Numerical Modeling

By Boris N. Khoromskij. 2011

346 citations recorded by the source corpus.

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

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “The exact and approximate ranks of X for various functions f have been discussed for the TT and HT decompositions in [105, 109, 155].”
ALTERNATING MINIMAL ENERGY METHODS FOR LINEAR SYSTEMS IN HIGHER DIMENSIONS∗ (2014, methodology): “Additional cost reduction is achieved using the quantized TT (QTT) format [32], for which we increase the dimensionality of x and b by the reshape and apply the TT format (2.2) with dimension D = d log 2 n +log 2 L and mode sizes 2.”
Polynomial Approximation in Hierarchical Tucker Format by Vector – Tensorization (2010, background): “Financial support from the DFG under grant GRA2179/2-1 gratefully acknowledged.”
Mathematik in den Naturwissenschaften Leipzig Tensor-Structured Galerkin Approximation of parametric and stochastic Elliptic PDEs (2010, methodology): “[22]) it might be possible to achieve the log-scaling of the numerical algorithm in both the volume size and the approximation error, O(M log n log ε).”
Fast adaptive interpolation of multi-dimensional arrays in tensor train format (2011, methodology): “Then it was called the QTT format [24] and applied for some univariate functions with small QTT ranks.”
DMRG Approach to Fast Linear Algebra in the TT-Format (2011, background): “This concept is naturally generalized to high-order problems (see [30, 31, 32, 33, 31, 16, 34]).”
Fast Solution of Parabolic Problems in the Tensor Train/Quantized Tensor Train Format with Initial Application to the Fokker-Planck Equation (2012, methodology): “The QTT-representation of the vector v can be obtained either by using known analytical representations [35, 24] (e.”
Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances (2012, methodology): “In [55] the diadically quantized representation of vectors (called there quantics1) was generalized to q-adic folding applied to the general N -d tensors.”
Quantized-TT-Cayley Transform for Computing the Dynamics and the Spectrum of High-Dimensional Hamiltonians (2011, methodology): “([24]) Introduce the q-adic folding transform of degree 2 6 p 6 L, Fq,d,p : Wn,d →Wm,dp, m = (m1, .”
Superfast Fourier Transform Using QTT Approximation (2012, methodology): “It has the rank-one QTT representation [31] exp ( which plays a key role for the efﬁcient Fourier transform algorithm in the QTT format.”
Low-Rank Explicit QTT Representation of the Laplace Operator and Its Inverse (2012, methodology): “Also, one may adopt the quantized tensor train (QTT) format [30, 19, 31], which is the TT format applied to a vector after virtual dimensions are introduced by reshaping it to smaller mode sizes (for example, nk = 2).”