Approximation of Integral Operators by Variable-Order Interpolation

Börm, Steffen; Löhndorf, Maike; Melenk, Jens Markus

doi:10.1007/s00211-004-0564-3

Cited by 41 publications

(23 citation statements)

References 21 publications

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“…The following remark recalls the well-known fact that controlling all higher derivatives of a function implies that it belongs to the class of analytic functions A M,ρ (τ ) (see e.g., [3] for the proof in the case τ = I).…”

Section: Function Spacesmentioning

confidence: 99%

“…Due to classical results on the best polynomial approximation we know that for any f ∈ A M,ρ (I), there holds inf 3) where P N (I) is the set of polynomials of degree N on I. Moreover, we have 4) where I N is the polynomial interpolation operator at the N + 1 Chebyshev nodes on I (see, e.g., [24]) and c does not depend on f .…”

Section: Local Polynomial Approximation On τ ∈ G\g Roughmentioning

confidence: 99%

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Adaptive Galerkin boundary element methods with panel clustering

2006

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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.

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Section: Function Spacesmentioning

confidence: 99%

Section: Local Polynomial Approximation On τ ∈ G\g Roughmentioning

confidence: 99%

Adaptive Galerkin boundary element methods with panel clustering

2006

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“…A very simple and robust construction of an H 2 -matrix approximation of G is based on interpolation [4,8]: we fix interpolation points (x τ,ν ) k ν=1 and corresponding Lagrange polynomials (L τ,ν ) k ν=1 for all clusters τ ∈ T I and approximate g by its interpolant…”

Section: Original Algorithmsmentioning

confidence: 99%

“…This strategy ensures that each cluster has either two or no sons, therefore the cluster tree satisfies our assumptions if n is not too small. The partition P I×I is constructed using twodimensional bounding boxes (this is the natural choice since in embedded in R 2 ), and the H 2 -matrix approximation is derived by tensor-product interpolation of the kernel function on the bounding boxes [4].…”

Section: Lemma 7 (Broadcast)mentioning

confidence: 99%

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

Börm

Bendoraityte

2008

Comput. Visual Sci.

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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.

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“…Instead of interpolation [8,9], truncated Taylor expansion [19,20] or adaptive cross approximation [3,5], in our case we can simply use a truncated singular value decomposition, as the computational cost of setting up the H-matrix approximation is negligible as compared to the application of this matrix in the course of the time integration of the MCTDHF equations.…”

Section: Remarkmentioning

confidence: 99%