Matrix valued adaptive cross approximation

Rjasanow, Sergej; Weggler, Lucy

doi:10.1002/mma.4174

Cited by 11 publications

(9 citation statements)

References 13 publications

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“…Since B-splines are piecewise polynomials, it clearly holds that Since the dimension of S S S * p,m (Γ) is larger than the dimension of S S S 1 p p p,Ξ Ξ Ξ m (Γ), the matrix V V V * κ,h is larger than the matrix V V V κ,h . However, it has been shown in [43] for the case of classical higher order Raviart-Thomas elements that the superspace approach can achiever better compression rates and, thus, better computation times. In this particular case, the non-zero elements in T T T were either 1 or −1.…”

Section: Assembly Of the System Matrixmentioning

confidence: 99%

See 1 more Smart Citation

Isogeometric Boundary Elements in Electromagnetism: Rigorous Analysis, Fast Methods, and Examples

Dölz¹,

Kurz²,

Schöps³

et al. 2019

SIAM J. Sci. Comput.

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We analyze a new approach to three-dimensional electromagnetic scattering problems via fast isogeometric boundary element methods. Starting with an investigation of the theoretical setting around the electric field integral equation within the isogeometric framework, we show existence, uniqueness, and quasi-optimality of the isogeometric approach. For a fast and efficient computation, we then introduce and analyze an interpolation-based fast multipole method tailored to the isogeometric setting, which admits competitive algorithmic and complexity properties. This is followed by a series of numerical examples of industrial scope, together with a detailed presentation and interpretation of the results.

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Section: Assembly Of the System Matrixmentioning

confidence: 99%

“…In this paper, we follow the approach of [20,43], which allows for a fast method with element-wise quadrature and fully avoids redundant evaluations of geometry and fundamental solution. Our method exploits the isogeometric structure and yields a simplified implementation based on interpolation on the unit square.…”

mentioning

confidence: 99%

Isogeometric Boundary Elements in Electromagnetism: Rigorous Analysis, Fast Methods, and Examples

Dölz¹,

Kurz²,

Schöps³

et al. 2019

SIAM J. Sci. Comput.

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show abstract

“…In order to achieve convergence and to avoid singular pivots, the safest approach consists in choosing the pivot corresponding to the subblock with the largest σ 3 . A similar approach is used for electromagnetism in [32].…”

Section: Extension To Problems With Vector Unknownsmentioning

confidence: 99%

Theory and implementation of H-matrix based iterative and direct solvers for Helmholtz and elastodynamic oscillatory kernels

Chaillat

Desiderio

Ciarlet

2017

Journal of Computational Physics

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In this work, we study the accuracy and efficiency of hierarchical matrix (H-matrix) based fast methods for solving dense linear systems arising from the discretization of the 3D elastodynamic Green's tensors. It is well known in the literature that standard H-matrix based methods, although very efficient tools for asymptotically smooth kernels, are not optimal for oscillatory kernels. H 2 -matrix and directional approaches have been proposed to overcome this problem. However the implementation of such methods is much more involved than the standard H-matrix representation. The central questions we address are twofold. (i) What is the frequency-range in which the H-matrix format is an efficient representation for 3D elastodynamic problems? (ii) What can be expected of such an approach to model problems in mechanical engineering? We show that even though the method is not optimal (in the sense that more involved representations can lead to faster algorithms) an efficient solver can be easily developed. The capabilities of the method are illustrated on numerical examples using the Boundary Element Method.

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“…One of the reasons is high computational intensity of system matrix assemblers, which fits well with today's design of HPC clusters, where memory accesses are much more costly than arithmetic operations. To overcome the quadratic complexity of BEM, several methods have been proposed including the fast multipole [4,15], or adaptive cross approximation (ACA) [2,16,17] methods, with the latter one utilized in this paper.…”

Section: Introductionmentioning

confidence: 99%

Parallel Adaptive Cross Approximation for the Multi-trace Formulation of Scattering Problems

Kravčenko

Zapletal

Claeys

et al. 2020

Parallel Processing and Applied Mathematics

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We present a highly parallel version of the boundary element method accelerated by the adaptive cross approximation for the efficient solution of scattering problems with composite scatterers. Individual boundary integral operators are treated independently, i.e. the boundary of every homogeneous subdomain is decomposed into clusters of elements defining a block structure of the local matrix. The blocks are distributed across computational nodes by a graph algorithm providing a load balancing strategy. The intra-node implementation further utilizes threading in shared memory and in-core SIMD vectorization to make use of all features of modern processors. The suggested approach is validated on a series of numerical experiments presented in the paper.

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Matrix valued adaptive cross approximation

Cited by 11 publications

References 13 publications

Isogeometric Boundary Elements in Electromagnetism: Rigorous Analysis, Fast Methods, and Examples

Isogeometric Boundary Elements in Electromagnetism: Rigorous Analysis, Fast Methods, and Examples

Theory and implementation of H-matrix based iterative and direct solvers for Helmholtz and elastodynamic oscillatory kernels

Parallel Adaptive Cross Approximation for the Multi-trace Formulation of Scattering Problems

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