Hierarchical Matrices Based on a Weak Admissibility Criterion

Hackbusch, Wolfgang; Khoromskij, Boris N.; Kriemann, Ronald

doi:10.1007/s00607-004-0080-4

Cited by 81 publications

(88 citation statements)

References 14 publications

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“…§2.1) defined on a rectangle with the only singularity at the origin. The identical problem appeared in the study of weakly admissible clusters in the theory of H-matrices (see [22]). A separable approximation of F with r terms will lead to a Kronecker tensor-product approximation of the matrix with Kronecker rank r. The interior structure of the Kronecker factors will however depend on the method by which they are constructed.…”

Section: Introductionmentioning

confidence: 93%

“…• The matrix-by-vector complexity with the Kronecker-tensor ansatz (1.1) is outperformed asymptotically by the almost 2 linear estimates which are typical for H-matrices (see [17,18,19,22,23]) or the mosaic-skeleton method (cf. [31,32,33]) as well as the earlier well-known methods such as panel clustering [24], multipole [28,30], and interpolation using regular or hierarchical grids (cf.…”

Section: Introductionmentioning

confidence: 99%

“…Note that the matrices U k and V k obtained by the algebraic SVD-based recompression method (cf. [22,33]) lead directly to the H-matrix structure. This is what we observe in our numerical experiments.…”

Section: Introductionmentioning

confidence: 99%

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Hierarchical Kronecker tensor-product approximations

Hackbusch

Khoromskij

Тыртышников

2005

Journal of Numerical Mathematics

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106

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The goal of this work is the presentation of some new formats which are useful for the approximation of (large and dense) matrices related to certain classes of functions and nonlocal (integral, integrodifferential) operators, especially for high-dimensional problems. These new formats elaborate on a sum of few terms of Kronecker products of smaller-sized matrices (cf. [34,35]). In addition to this we need that the Kronecker factors possess a certain data-sparse structure. Depending on the construction of the Kronecker factors we are led to so-called "profile-low-rank matrices" or hierarchical matrices (cf. [17,18]). We give a proof for the existence of such formats and expound a gainful combination of the Kronecker-tensor-product structure and the arithmetic for hierarchical matrices.

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Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

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Hierarchical Kronecker tensor-product approximations

Hackbusch

Khoromskij

Тыртышников

2005

Journal of Numerical Mathematics

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106

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“…However, in numerical tests better results have been obtained according to the computational time using larger η and correspondingly having larger but fewer admissible subblocks. In [40] better results have been obtained as well when even subblocks s × t with s = t were accepted as admissible. In numerical tests η := 50 has been a good choice and this value has been used in the rest of the work.…”

Section: Task (T1): Construction Of the H-matricesmentioning

confidence: 99%

Solving an elliptic PDE eigenvalue problem via automated multi-level substructuring and hierarchical matrices

Gerds

Grasedyck

2013

Comput. Visual Sci.

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To solve an elliptic PDE eigenvalue problem in practice, typically the finite element discretisation is used. From approximation theory it is known that only the smaller eigenvalues and their corresponding eigenfunctions can be well approximated by the finite element discretisation because the approximation error increases with increasing size of the eigenvalue. The number of well approximable eigenvalues or eigenfunctions, however, is unknown. In this work asymptotic estimates of these quantities are derived. For example, it is shown that for three-dimensional problems under certain smoothness assumptions on the data only the smallest Θ(N 2/5 ) eigenvalues and only the eigenfunctions associated to the smallest Θ(N 1/4 ) eigenvalues can be well approximated by the finite element discretisation, when the N -dimensional finite element spaces of piecewise affine functions with uniform mesh refinement are used.To solve the discretised elliptic PDE eigenvalue problem and to compute all well approximable eigenvalues and eigenfunctions, a new method is introduced which combines a recursive version of the automated multi-level substructuring (short AMLS) method with the concept of hierarchical matrices (short H-matrices). AMLS is a domain decomposition technique for the solution of elliptic PDE eigenvalue problems where, after some transformation, a reduced eigenvalue problem is derived whose eigensolutions deliver approximations of the sought eigensolutions of the original problem.Whereas the classical AMLS method is very efficient for elliptic PDE eigenvalue problems posed in two dimensions, it is getting very expensive for three-dimensional problems, due to the fact that it computes the reduced eigenvalue problem via dense matrix operations. This problem is resolved by the use of hierarchical matrices. H-matrices are a data-sparse approximation of dense matrices which, e.g., result from the inversion of the stiffness matrix that is associated to the finite element discretisation of an elliptic PDE operator. The big advantage of H-matrices is that they provide matrix arithmetic with almost linear complexity. This fast H-matrix arithmetic is used for the computation of the reduced eigenvalue problem. Beside this, the size of the reduced eigenvalue problem is bounded by a new recursive version of AMLS which further reduces the costs for the computation and the solution of this problem. Altogether this leads to a new method which is well-suited for three-dimensional problems and which allows us to compute a large amount of eigenpair approximations in optimal complexity.

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“…The covariance matrix C is not sparse and, in general, requires O(n 2 ) units of memory for the storage and O(n 2 ) FLOPS for the matrix-vector multiplication. In this section it will be shown how to approximate general covariance matrices with the H-matrix format [17,18,20,22]. The H-matrix technique is nothing but a hierarchical division of a given matrix into subblocks and further approximation of the majority of them by low-rank matrices (Fig.…”

Section: H-matrix Techniquementioning

confidence: 99%

Application of hierarchical matrices for computing the Karhunen–Loève expansion

Khoromskij

Litvinenko²,

Matthies³

2008

Computing

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Realistic mathematical models of physical processes contain uncertainties. These models are often described by stochastic differential equations (SDEs) or stochastic partial differential equations (SPDEs) with multiplicative noise. The uncertainties in the right-hand side or the coefficients are represented as random fields. To solve a given SPDE numerically one has to discretise the deterministic operator as well as the stochastic fields. The total dimension of the SPDE is the product of the dimensions of the deterministic part and the stochastic part. To approximate random fields with as few random variables as possible, but still retaining the essential information, the Karhunen-Loève expansion (KLE) becomes important. The KLE of a random field requires the solution of a large eigenvalue problem. Usually it is solved by a Krylov subspace method with a sparse matrix approximation. We demonstrate the use of sparse hierarchical matrix techniques for this. A log-linear computational cost of the matrix-vector product and a log-linear storage requirement yield an efficient and fast discretisation of the random fields presented.

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Hierarchical Matrices Based on a Weak Admissibility Criterion

Cited by 81 publications

References 14 publications

Hierarchical Kronecker tensor-product approximations

Hierarchical Kronecker tensor-product approximations

Solving an elliptic PDE eigenvalue problem via automated multi-level substructuring and hierarchical matrices

Application of hierarchical matrices for computing the Karhunen–Loève expansion

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