On H 2-Matrices

Hackbusch, Wolfgang; Khoromskij, Boris N.; Sauter, Stéfan A.

doi:10.1007/978-3-642-59709-1_2

Cited by 196 publications

(247 citation statements)

References 5 publications

Supporting

Mentioning

247

Contrasting

Order By: Relevance

“…(For a more complete introduction, see Hackbusch & Sauter, 1993;Hackbusch et al, 1997Hackbusch et al, , 2000Sauter, 2000; DEFINITION 6.1 (Cluster tree) A cluster tree T is a tree † whose vertices (called 'clusters') consist of unions σ = ∪{τ : τ ∈ T } for certain subsets T ⊂ T . These are required to satisfy the following properties: There are standard procedures for constructing cluster trees (see for example , Example 2.1).…”

Section: Panel Clustering In the General Casementioning

confidence: 99%

Finite elements on degenerate meshes: inverse-type inequalities and applications

Graham

Hackbusch

Sauter

2005

IMA Journal of Numerical Analysis

View full text Add to dashboard Cite

In this paper we obtain a range of inverse-type inequalities which are applicable to finite-element functions on general classes of meshes, including degenerate meshes obtained by anisotropic refinement. These are obtained for Sobolev norms of positive, zero and negative order. In contrast to classical inverse estimates, negative powers of the minimum mesh diameter are avoided. We give two applications of these estimates in the context of boundary elements: (i) to the analysis of quadrature error in discrete Galerkin methods and (ii) to the analysis of the panel clustering algorithm. Our results show that degeneracy in the meshes yields no degradation in the approximation properties of these methods.

show abstract

Section: Panel Clustering In the General Casementioning

confidence: 99%

Finite elements on degenerate meshes: inverse-type inequalities and applications

Graham

Hackbusch

Sauter

2005

IMA Journal of Numerical Analysis

View full text Add to dashboard Cite

show abstract

“…We conclude that combining the factorized approximation of admissible blocks (3) with the factorized nested representation (7) of the cluster bases (V τ ) τ ∈T I and (W σ ) σ ∈T I leads to an approximation of G by an H 2 -matrix G [3,11].…”

Section: G I J = ϕ I (X) G(x Y)ϕ J (Y) Dy Dx For All I J ∈ Imentioning

confidence: 99%

Distributed $${{\mathcal H}^2}$$ -matrices for non-local operators

Börm

Bendoraityte

2008

Comput. Visual Sci.

View full text Add to dashboard Cite

H 2 -matrices can be used to approximate dense n × n matrices resulting from the discretization of certain non-local operators (e.g., Fredholm-type integral operators) in O(nk) units of storage, where k is a parameter controlling the accuracy of the approximation. Since typically k n holds, this representation is much more efficient than the conventional representation by a two-dimensional array. For very large problem dimensions, the amount of available storage becomes a limiting factor for practical algorithms. A popular way to provide sufficiently large amounts of storage at relatively low cost is to use a cluster of inexpensive computers that are connected by a network. This paper presents a method for managing an H 2 -matrix on a distributed-memory cluster that can be proven to be of almost optimal parallel efficiency.

show abstract

“…We present this subsection for the readers convenience (see [11], [15], [21] for more details). We restrict ourselves here to the Galerkin boundary element discretisation of boundary integral equations while collocation or Nyström discretisations can be considered as well.…”

Section: Construction Of the Hierarchical Clusteringmentioning

confidence: 99%

Adaptive Galerkin boundary element methods with panel clustering

2006

View full text Add to dashboard Cite

In this paper, we will propose a boundary element method for solving classical boundary integral equations on complicated surfaces which, possibly, contain a large number of geometric details or even uncertainties in the given data. The (small) size of such details is characterised by a small parameter and the regularity of the solution is expected to be low in such zones on the surface (which we call the wire-basket zones). We will propose the construction of an initial discretisation for such type of problems. Afterwards standard strategies for boundary element discretisations can be applied such as the h, p, and the adaptive hp-version in a straightforward way. For the classical boundary integral equations, we will prove the optimal approximation results of our so-called wire-basket boundary element method and discuss the stability aspects. Then, we construct the panel-clustering and -matrix approximations to the corresponding Galerkin BEM stiffness matrix. The method is shown to have an almost linear complexity with respect to the number of degrees of freedom located on the wire basket. The present paper introduces an hp-version of BEM for the Laplace equation in polyhedral domains based on meshes which are concentrated to zones on the surface (wire-basket zones), where the regularity of the solution is expected to be low. For the classical boundary integral equations, we prove the optimal approximation results and discuss the stability aspects. Then, we construct the panel-clustering and H-matrix approximations to the corresponding Galerkin BEM stiffness matrix and prove their linear-logarithmic cost. The method is shown to have an almost linear complexity with respect to the number of degrees of freedom located on the wire basket.

show abstract

On H 2-Matrices

Cited by 196 publications

References 5 publications

Finite elements on degenerate meshes: inverse-type inequalities and applications

Finite elements on degenerate meshes: inverse-type inequalities and applications

Distributed $${{\mathcal H}^2}$$ -matrices for non-local operators

Adaptive Galerkin boundary element methods with panel clustering

Contact Info

Product

Resources

About