Approximation of boundary element matrices

Bebendorf, Mario

doi:10.1007/pl00005410

Cited by 754 publications

(689 citation statements)

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“…In terms of a relative error of approximation, an alternate notion of rank is introduced [19,42] based on a relative approximation error ε. The ε-rank of a matrix A in the norm · is defined as:…”

Section: Low-rank Matricesmentioning

confidence: 99%

“…The implication of the kernel being asymptotically smooth on admissible clusters is that the submatrix corresponding to the blocks τ × σ have exponentially decaying singular values and are well approximated by lowrank matrices. The rank of the resulting matrix is O(k d ) [42], where k is the number of terms retained in the Taylor series expansion.…”

Section: Leaf Nodes Directmentioning

confidence: 99%

“…The idea behind the cross approximation is based on the result described in [42], which states that supposing a matrix A is well approximated by a low-rank matrix, by a clever choice of k columns C ∈ R m×k and k rows R of the matrix A, we can approximate A of the form:…”

Section: Adaptive Cross Approximationmentioning

confidence: 99%

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Application of Hierarchical Matrices to Linear Inverse Problems in Geostatistics

Saibaba

Ambikasaran

et al. 2012

Oil Gas Sci. Technol. – Rev. IFP Energies nouvelles

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Nous commençons par décrire brièvement l'approche géostatistique dans le contexte d'un problème inverse linéaire, puis discutons les difficultés rencontrées dans le cadre d'une implémentation à grande échelle. Ensuite, en utilisant une approche basée sur les matrices hiérarchiques, nous montrons comment réduire le coût de calcul d'un produit matrice-vecteur de O(m 2 ) à O(m log m) dans le cas de matrices de covariance denses ; m désignant ici le nombre d'inconnues. Combinée avec un solveur de Krylov, qui ne requiert pas la construction explicite de la matrice, cette méthode conduit à un algorithme beaucoup plus rapide pour résoudre le système d'équations associé à l'approche géostatistique. Nous illustrons enfin la performance de notre algorithme dans un situation spécifique, surveiller la concentration de CO 2 à l'aide de la tomographie sismique entre puits. Abstract -Application of Hierarchical Matrices to Linear Inverse Problems in Geostatistics -Characterizing the uncertainty in the subsurface is an important step for exploration and extraction of natural resources, the storage of nuclear material and gasses such as natural gas or CO 2 . Imaging the subsurface can be posed as an inverse problem and can be solved using the geostatistical approach [Kitanidis P.K. (2007) Geophys. Monogr. Ser. 171, 19-30, doi:10.1029/171GM04;Kitanidis (2011 Kitanidis ( ) doi: 10.1002

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Section: Low-rank Matricesmentioning

confidence: 99%

Section: Leaf Nodes Directmentioning

confidence: 99%

See 1 more Smart Citation

Application of Hierarchical Matrices to Linear Inverse Problems in Geostatistics

Saibaba

Ambikasaran

et al. 2012

Oil Gas Sci. Technol. – Rev. IFP Energies nouvelles

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“…Recently, it was realized that the Adaptive Cross Approximation (ACA) [26] can be employed to construct excitation-free macro basis functions over the surface of a brick that afford the same level of accuracy as the eigencurrents [27,28]. Moreover, carrying out the ACA of the scattering operator is far faster than determining the eigencurrents.…”

Section: Introductionmentioning

confidence: 99%

Extracting Surface Macro Basis Functions From Low-Rank Scattering Operators With the Aca Algorithm

Lancellotti

2016

PIER M

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Abstract-The Adaptive Cross Approximation (ACA) algorithm has been used to compress the rankdeficient sub-blocks of the matrices that arise in the numerical solution of integral equations (IEs) with the Method of Moments. In the context of the linear embedding via Green's operator (LEGO) method -a domain decomposition technique based on IEs -an electromagnetic problem is modelled by combining "bricks" in turn described by scattering operators which, in many situations, are singular. As a result, macro basis functions defined on the boundary of a brick can be generated by applying the ACA to a scattering operator. Said functions allow compressing the weak form of the LEGO functional equations which then use up less computer memory and are faster to invert.

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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%