Superfast Wavelet Transform Using Quantics-TT
Approximation. I. Application to Haar Wavelets

Khoromskij, Boris N.; Miao, Sentao

doi:10.1515/cmam-2014-0016

Cited by 4 publications

(5 citation statements)

References 26 publications

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“…The QTT approximation method enables the multidimensional vector transforms with logarithmic complexity scaling, O(log N). For example, we mention the superfast FFT [15], Laplacian inverse [26] and wavelet [47] transforms.…”

Section: Discussionmentioning

confidence: 99%

Grid-based lattice summation of electrostatic potentials by assembled rank-structured tensor approximation

Khoromskaia

Khoromskij

2014

Computer Physics Communications

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Our recent method for low-rank tensor representation of sums of the arbitrarily positioned electrostatic potentials discretized on a 3D Cartesian grid reduces the 3D tensor summation to operations involving only 1D vectors however retaining the linear complexity scaling in the number of potentials. Here, we introduce and study a novel tensor approach for fast and accurate assembled summation of a large number of lattice-allocated potentials represented on 3D N × N × N grid with the computational requirements only weakly dependent on the number of summed potentials. It is based on the assembled low-rank canonical tensor representations of the collected potentials using pointwise sums of shifted canonical vectors representing the single generating function, say the Newton kernel. For a sum of electrostatic potentials over L × L × L lattice embedded in a box the required storage scales linearly in the 1D grid-size, O(N ), while the numerical cost is estimated by O(N L). For periodic boundary conditions, the storage demand remains proportional to the 1D grid-size of a unit cell, n = N/L, while the numerical cost reduces to O(N ), that outperforms the FFT-based Ewald-type summation algorithms of complexity O (N 3 log N ). The complexity in the grid parameter N can be reduced even to the logarithmic scale O(log N ) by using data-sparse representation of canonical N -vectors via the quantics tensor approximation. For justification, we prove an upper bound on the quantics ranks for the canonical vectors in the overall lattice sum. The presented approach is beneficial in applications which require further functional calculus with the lattice potential, say, scalar product with a function, integration or differentiation, which can be performed easily in tensor arithmetics on large 3D grids with 1D cost. Numerical tests illustrate the performance of the tensor summation method and confirm the estimated bounds on the tensor ranks. [48]. This paper is an essentially improved version of the preprint of the Max-Planck Institute for Mathematics in the Sciences

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Section: Discussionmentioning

confidence: 99%

Grid-based lattice summation of electrostatic potentials by assembled rank-structured tensor approximation

Khoromskaia

Khoromskij

2014

Computer Physics Communications

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“…Various transformations with particular structure have been shown to be efficiently realizable for multilevel representations as in (3.34), including convolutions (Hackbusch 2011), Toeplitz matrices (Kazeev, Khoromskij and Tyrtyshnikov 2013a) and wavelet transforms (Khoromskij and Miao 2014).…”

Section: Multilevel Tensorized Representationsmentioning

confidence: 99%

Low-rank tensor methods for partial differential equations

Bachmayr¹

2023

Acta Numerica

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Low-rank tensor representations can provide highly compressed approximations of functions. These concepts, which essentially amount to generalizations of classical techniques of separation of variables, have proved to be particularly fruitful for functions of many variables. We focus here on problems where the target function is given only implicitly as the solution of a partial differential equation. A first natural question is under which conditions we should expect such solutions to be efficiently approximated in low-rank form. Due to the highly nonlinear nature of the resulting low-rank approximations, a crucial second question is at what expense such approximations can be computed in practice. This article surveys basic construction principles of numerical methods based on low-rank representations as well as the analysis of their convergence and computational complexity.

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“…The efficient low-rank QTT representation for a class of discrete multidimensional operators (matrices) was proven in [51,79]. Moreover, based on the QTT approximation, the important algebraic matrix operations like FFT, convolution and wavelet transforms can be performed in O(log 2 N) complexity [24,52,75] (see also §2.4 below).…”

Section: Quantics Tensor Approximation Of Function Related Vectorsmentioning

confidence: 99%

“…The super-fast QTT wavelet transform of logarithmic complexity by the exact low-rank representation of the wavelet transform matrix was described in [75] (see also [104] for the related discussion).…”

Section: Operators In Quantized Tensor Spacesmentioning

confidence: 99%

“…Several important applications of tensor numerical methods remain beyond the scope of this review. They include multidimensional preconditioning [65,3] and solution of linear systems [86,28,120,30,29,12], parametric/stochastic PDEs [93,81,76,27,13], greedy algorithms [90,9,19,33], PGD model reduction methods [2,31,116], super-fast FFT, convolution and wavelet transforms [24,52,75], tensor-product interpolation and cross approximation [103,8,109,5], integration of singular or highly oscillating functions [80], control problems [115], complexity theory [4,21], etc.…”

Section: Introductionmentioning

confidence: 99%

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Tensor numerical methods for multidimensional PDES: theoretical analysis and initial applications

Khoromskij

2015

ESAIM: Proc.

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Abstract. We present a brief survey on the modern tensor numerical methods for multidimensional stationary and time-dependent partial differential equations (PDEs). The guiding principle of the tensor approach is the rank-structured separable approximation of multivariate functions and operators represented on a grid. Recently, the traditional Tucker, canonical, and matrix product states (tensor train) tensor models have been applied to the grid-based electronic structure calculations, to parametric PDEs, and to dynamical equations arising in scientific computing. The essential progress is based on the quantics tensor approximation method proved to be capable to represent (approximate) function related d-dimensional data arrays of size N d with log-volume complexity, O(d log N ). Combined with the traditional numerical schemes, these novel tools establish a new promising approach for solving multidimensional integral and differential equations using low-parametric rank-structured tensor formats. As the main example, we describe the grid-based tensor numerical approach for solving the 3D nonlinear Hartree-Fock eigenvalue problem, that was the starting point for the developments of tensor-structured numerical methods for large-scale computations in solving real-life multidimensional problems. We also discuss a new method for the fast 3D lattice summation of electrostatic potentials by assembled low-rank tensor approximation capable to treat the potential sum over millions of atoms in few seconds. We address new results on tensor approximation of the dynamical Fokker-Planck and master equations in many dimensions up to d = 20. Numerical tests demonstrate the benefits of the rank-structured tensor approximation on the aforementioned examples of multidimensional PDEs. In particular, the use of grid-based tensor representations in the reduced basis of atomics orbitals yields an accurate solution of the Hartree-Fock equation on large N × N × N grids with a grid size of up to N = 10 5 .

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Superfast Wavelet Transform Using Quantics-TT Approximation. I. Application to Haar Wavelets

Cited by 4 publications

References 26 publications

Grid-based lattice summation of electrostatic potentials by assembled rank-structured tensor approximation

Grid-based lattice summation of electrostatic potentials by assembled rank-structured tensor approximation

Low-rank tensor methods for partial differential equations

Tensor numerical methods for multidimensional PDES: theoretical analysis and initial applications

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