Solution of linear systems in high spatial dimensions

Hackbusch, Wolfgang

doi:10.1007/s00791-015-0252-0

Cited by 12 publications

(9 citation statements)

References 30 publications

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“…Concerning adapting the multigrid iteration to the tensor case, we refer to Hackbusch [199]. The critical points are the coarse-grid, the cost, and the smoothing iteration.…”

Section: Multigrid Approachmentioning

confidence: 99%

Iterative Solution of Large Sparse Systems of Equations

Hackbusch

2016

Applied Mathematical Sciences

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141

169

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“…Concerning adapting the multigrid iteration to the tensor case, we refer to Hackbusch [199]. The critical points are the coarse-grid, the cost, and the smoothing iteration.…”

Section: Multigrid Approachmentioning

confidence: 99%

Iterative Solution of Large Sparse Systems of Equations

Hackbusch

2016

Applied Mathematical Sciences

Self Cite

141

169

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“…In this case, the TT format separates not "physical" dimensions of tensors but rather their multilevel structure, and adaptive low-rank approximation allows to resolve this structure in vectors and matrices. In this setting, the TT decomposition is known as the quantized tensor train (QTT) decomposition [21,37,43,44]. This idea is further explained in Section 3.7.…”

Section: Tensor Train Decompositionmentioning

confidence: 99%

Stability of Low-Rank Tensor Representations and Structured Multilevel Preconditioning for Elliptic PDEs

Bachmayr

Kazeev

2020

Found Comput Math

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Folding grid value vectors of size 2 L into Lth order tensors of mode size 2 × · · · × 2, combined with low-rank representation in the tensor train format, has been shown to result in highly efficient approximations for various classes of functions. These include solutions of elliptic PDEs on nonsmooth domains or with oscillatory data. This tensor-structured approach is attractive because it leads to highly compressed, adaptive approximations based on simple discretizations. Standard choices of the underlying bases, such as piecewise multilinear finite elements on uniform tensor product grids, entail the well-known matrix ill-conditioning of discrete operators. We demonstrate that, for low-rank representations, the use of tensor structure itself additionally introduces representation ill-conditioning, a new effect specific to computations in tensor networks. We analyze the tensor structure of a BPX preconditioner for a second-order linear elliptic operator and construct an explicit tensor-structured representation of the preconditioner, with ranks independent of the number L of discretization levels. The straightforward application of the preconditioner yields discrete operators whose matrix conditioning is uniform with respect to the discretization parameter, but in decompositions that suffer from representation ill-conditioning. By additionally eliminating certain redundancies in the representations of the preconditioned discrete operators, we obtain reduced-rank decompositions that are free of both matrix and representation ill-conditioning. For an iterative solver based on soft thresholding of low-rank tensors, we obtain convergence and complexity estimates and demonstrate its reliability and efficiency for discretizations with up to 2 50 nodes in each dimension.

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“…Concerning tensor methods for solving linear systems with high spatial dimensions, we refer to Hackbusch [28].…”

Section: Lemmamentioning

confidence: 99%

Survey on the Technique of Hierarchical Matrices

Hackbusch

2015

Vietnam J. Math.

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Usually, one avoids numerical algorithms involving operations with large, fully populated matrices. Instead, one tries to reduce all algorithms to matrix-vector multiplications involving only sparse matrices. The reason is the large number of floating point operations; e.g., O(n 3 ) for the multiplication of two general n×n matrices. The hierarchical matrix (H-matrix) technique provides tools to perform the matrix operations approximately in almost linear work O(n log * n). The approximation errors are nevertheless acceptable, since large-scale matrices are usually obtained from discretisations which anyway contain a discretisation error. Adjusting the approximation error to the discretisation error yields the factor O(log * n). The operations enabled by the H-matrix technique are not only the matrix addition and multiplication but also the matrix inversion and the LU or Cholesky decomposition. The positive statements from above do not hold for all matrices, but they are valid for the important class of matrices originating from standard discretisations of elliptic partial differential equations or the related integral equations. An important aspect is the fact that the algorithms can be applied in a black-box fashion. Having all matrix operations available, a much larger class of problems can be treated than by the restriction to matrix-vector multiplications. The LU decomposition can be used to construct fast iterations for solving linear systems. Also eigenvalue problems can be treated. The computation of matrix-valued functions is possible (e.g., the matrix exponential function) as well as the solution of matrix equations (e.g., of the Riccati equation).Mathematics Subject Classification (2010) 65F05 · 65F30 · 65F50 · 65F10 · 15A09 · 15A24 · 15A99 · 15A18 · 47A56 · 68P05 · 65N22 · 65N38 · 65R20 · 45B05

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Solution of linear systems in high spatial dimensions

Abstract: We give an overview of various methods based on tensor structured techniques for the solution of linear systems in high spatial dimensions. In particular, we discuss the role of multi-grid variants.

Cited by 12 publications

References 30 publications

Iterative Solution of Large Sparse Systems of Equations

Iterative Solution of Large Sparse Systems of Equations

Stability of Low-Rank Tensor Representations and Structured Multilevel Preconditioning for Elliptic PDEs

Survey on the Technique of Hierarchical Matrices

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