Superfast Fourier Transform Using QTT Approximation

Dolgov, Sergey; Khoromskij, Boris N.; Savostyanov, Dmitry

doi:10.1007/s00041-012-9227-4

Cited by 59 publications

(84 citation statements)

References 57 publications

(135 reference statements)

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“…To be able to perform computations for these problems, the curse of dimensionality has to be overcome. The observation that certain kinds of structured matrices may be efficiently represented using the QTT format has been made for Toeplitz matrices [Ols+06,Ose+11], the Laplace differential operator and its inverse [KK12,Ose10], general PDEs and eigenvalue problems [Kho11], convolution operators [Hac11], and the FFT [Dol+12]. Recent developments feature its use to solve multi-dimensional integro-differential equations arising in fields such as quantum chemistry, electrostatics, stochastic modeling and molecular dynamics [Kho15].…”

Section: Low-rank Tensor Approximation Of Linear Operatorsmentioning

confidence: 99%

A Tensor-Train accelerated solver for integral equations in complex geometries

Corona

Rahimian

Zorin

2017

Journal of Computational Physics

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We present a framework using the Quantized Tensor Train (QTT) decomposition to accurately and efficiently solve volume and boundary integral equations in three dimensions. We describe how the QTT decomposition can be used as a hierarchical compression and inversion scheme for matrices arising from the discretization of integral equations. For a broad range of problems, computational and storage costs of the inversion scheme are extremely modest O log N and once the inverse is computed, it can be applied in O N log N .We analyze the QTT ranks for hierarchically low rank matrices and discuss its relationship to commonly used hierarchical compression techniques such as FMM and HSS. We prove that the QTT ranks are bounded for translation-invariant systems and argue that this behavior extends to nontranslation invariant volume and boundary integrals.For volume integrals, the QTT decomposition provides an efficient direct solver requiring significantly less memory compared to other fast direct solvers. We present results demonstrating the remarkable performance of the QTT-based solver when applied to both translation and non-translation invariant volume integrals in 3D.For boundary integral equations, we demonstrate that using a QTT decomposition to construct preconditioners for a Krylov subspace method leads to an efficient and robust solver with a small memory footprint. We test the QTT preconditioners in the iterative solution of an exterior elliptic boundary value problem (Laplace) formulated as a boundary integral equation in complex, multiply connected geometries.

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Section: Low-rank Tensor Approximation Of Linear Operatorsmentioning

confidence: 99%

A Tensor-Train accelerated solver for integral equations in complex geometries

Corona

Rahimian

Zorin

2017

Journal of Computational Physics

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“…The QTT approximation method allows us to represent a class of matrices in low QTT rank format [19], and enables the multidimensional FFT and convolution transforms with logarithmic complexity scaling, O(log n), see [7,20].…”

Section: Appendix B Basic Tensor Formats and Operationsmentioning

confidence: 99%

“…However, it was the first step in the development of tensor numerical methods, and a "proof of concept" for their applicability in electronic structure calculations. Besides, these results promoted spreading and further evolution of the tensor-structured methods in the community of numerical analysis [10,38,7,34,31,3,36,28,8,16].…”

Section: Introductionmentioning

confidence: 96%

Black-Box Hartree–Fock Solver by Tensor Numerical Methods

Khoromskaia

2014

Computational Methods in Applied Mathematics

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The Hartree-Fock eigenvalue problem governed by the 3D integro-differential operator is the basic model in ab initio electronic structure calculations. Several years ago the idea to solve the Hartree-Fock equation by fully 3D grid based numerical approach seemed to be a fantasy, and the tensor-structured methods did not exist. In fact, these methods evolved during the work on this challenging problem. In this paper, our recent results on the topic are outlined and the black-box Hartee-Fock solver by tensor numerical methods is presented. The approach is based on the rank-structured calculation of the core hamiltonian and of the two-electron integrals tensor using the problem adapted basis functions discretized on n × n × n 3D Cartesian grids. The arising 3D convolution transforms with the Newton kernel are replaced by a combination of 1D convolutions and 1D Hadamard and scalar products. The approach allows huge spatial grids, with n 3 ≃ 10 15 , yielding high resolution at low cost. The two-electron integrals are computed via multiple factorizations. The Laplacian Galerkin matrix can be computed "on-the-fly" using the quantized tensor approximation of O(log n) complexity. The performance of the black-box solver in Matlab implementation is compatible with the benchmark packages based on the analytical (pre)evaluation of the multidimensional convolution integrals. We present ab initio Hartree-Fock calculations of the ground state energy for the amino acid molecules, and of the "energy bands" for the model examples of extended (quasi-periodic) systems.

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“…The theoretical substantiation of the QTT representation on a class of discretized di erential and integral operators were presented in [18], [9], and later in [1,4,10].…”

Section: Introductionmentioning

confidence: 99%

“…Fourier transform or convolution. The Fourier transform in QTT format was introduced in [4], where a new superfast Fourier transform algorithm was developed, analyzed and veri ed numerically. In particular, the QTT approximation was e ciently applied to an input vector exploiting the multilevel structure of the Cooley-Tukey FFT algorithm.…”

Section: Introductionmentioning

confidence: 99%

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

Khoromskij

Miao

2014

Computational Methods in Applied Mathematics

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We propose a superfast discrete Haar wavelet transform (SFHWT) as well as its inverse, using the low-rank Quantics-TT (QTT) representation for the Haar transform matrices and input-output vectors. Though the Haar matrix itself does not have a low QTT rank approximation, we show that factor matrices used at each step of the traditional multilevel Haar wavelet transform algorithm have explicit QTT representations of low rank. The SFHWT applies to a vector representing a signal sampled on a uniform grid of size = 2 . We develop two algorithms which roughly require square logarithmic time complexity (log 2 ) with respect to the grid size, hence outperforming the traditional fast Haar wavelet transform (FHWT) of linear complexity ( ). Our approach also applies to the FHWT inverse as well as to the multidimensional wavelet transform. Numerical experiments demonstrate that the SFHWT algorithm is robust in keeping low rank of the resulting output vector and it outperforms the traditional FHWT for grid sizes larger than a certain value depending on the spacial dimension.

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Superfast Fourier Transform Using QTT Approximation

Cited by 59 publications

References 57 publications

A Tensor-Train accelerated solver for integral equations in complex geometries

A Tensor-Train accelerated solver for integral equations in complex geometries

Black-Box Hartree–Fock Solver by Tensor Numerical Methods

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

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