Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions

Dolgov, Sergey; Savostyanov, Dmitry

doi:10.1137/140953289

Cited by 188 publications

(235 citation statements)

References 54 publications

(57 reference statements)

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“…In this work, we use the alternating minimal energy (AMEN) and the density matrix renormalization group (DMRG) methods as proposed in [OD12,DS13a,DS13b]. Another such method is the Newton-Hotelling-Schulz algorithm [Hac+08,Ols+08].…”

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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“…[23]. Alternating iterative tensor algorithms avoid this problem by seeking directly the elements of the tensor format for the solution.…”

Section: Alternating Solversmentioning

confidence: 99%

“…Instead of one block y (k) , we update both y (k) and y (k+1) at each step and adapt the TT rank r k . However, for linear systems with non-symmetric matrices, even the two-block DMRG method may converge to an incorrect solution [23]. A better way to increase TT ranks is to enrich TT blocks explicitly.…”

Section: Alternating Solversmentioning

confidence: 99%

“…This can not only increase the TT rank, but also facilitate the convergence, if we select the augmentation z (k) k properly. The idea of the AMEn algorithm proposed in [23] is to combine the alternating iteration with the steepest descent method. The latter updates the solution by adding the scaled residual, y := y + zh, where z ≈ f − Ay and h is a scalar weight.…”

Section: Alternating Solversmentioning

confidence: 99%

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Fast tensor product solvers for optimization problems with fractional differential equations as constraints

Dolgov

Pearson

Savostyanov

et al. 2016

Applied Mathematics and Computation

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Fractional differential equations have recently received much attention within computational mathematics and applied science, and their numerical treatment is an important research area as such equations pose substantial challenges to existing algorithms. An optimization problem with constraints given by fractional differential equations is considered, which in its discretized form leads to a high-dimensional tensor equation. To reduce the computation time and storage, the solution is sought in the tensor-train format. We compare three types of solution strategies that employ sophisticated iterative techniques using either preconditioned Krylov solvers or tailored alternating schemes. The competitiveness of these approaches is presented using several examples with constant and variable coefficients.

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“…This procedure can determine the TT-ranks adaptively if K > 1 for the block TT format. Dolgov and Savostyanov [9] and Kressner et al [25] further developed an ALS type method which adds rank-adaptivity to the block TT-based ALS method even if K = 1.…”

Section: Introductionmentioning

confidence: 99%

Estimating a Few Extreme Singular Values and Vectors for Large-Scale Matrices in Tensor Train Format

Lee¹,

Cichocki²

2015

SIAM J. Matrix Anal. & Appl.

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We propose new algorithms for singular value decomposition (SVD) of very large-scale matrices based on a low-rank tensor approximation technique called the tensor train (TT) format. The proposed algorithms can compute several dominant singular values and corresponding singular vectors for large-scale structured matrices given in a TT format. The computational complexity of the proposed methods scales logarithmically with the matrix size under the assumption that both the matrix and the singular vectors admit low-rank TT decompositions. The proposed methods, which are called the alternating least squares for SVD (ALS-SVD) and modified alternating least squares for SVD (MALS-SVD), compute the left and right singular vectors approximately through block TT decompositions. The very large-scale optimization problem is reduced to sequential small-scale optimization problems, and each core tensor of the block TT decompositions can be updated by applying any standard optimization methods. The optimal ranks of the block TT decompositions are determined adaptively during iteration process, so that we can achieve high approximation accuracy. Extensive numerical simulations are conducted for several types of TT-structured matrices such as Hilbert matrix, Toeplitz matrix, random matrix with prescribed singular values, and tridiagonal matrix. The simulation results demonstrate the effectiveness of the proposed methods compared with standard SVD algorithms and TT-based algorithms developed for symmetric eigenvalue decomposition.

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Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions

Cited by 188 publications

References 54 publications

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

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

Fast tensor product solvers for optimization problems with fractional differential equations as constraints

Estimating a Few Extreme Singular Values and Vectors for Large-Scale Matrices in Tensor Train Format

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