Aitken’s acceleration of the Schwarz process using singular value decomposition for heterogeneous 3D groundwater flow problems

Berenguer, Laurent; Dufaud, Thomas; Tromeur-Dervout, Damien

doi:10.1016/j.compfluid.2012.01.026

Cited by 8 publications

(7 citation statements)

References 21 publications

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“…The general form of the Q is as follows: Step 3: The singular value method [13] is used to obtain the so lution of least square fitting. Furthermore, the principal direction and principal curvature of p, are calculated.…”

Section: Normal Vector Estimationmentioning

confidence: 99%

Algorithm to Construct Line-Shaped Feature Point Clouds of Single-Value Ruled Blades for Gas Turbines

Zhou

Wang

et al. 2014

Journal of Manufacturing Science and Engineering

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A lg o rith m to C onstruct L in e -S h a p e d F e a tu re P o in t C louds of S in g le -V a lu e R u le d B la d e s fo r G as T u rb in e sUsing the geometric features of surfaces, we proposed an algorithm to construct line shaped feature point clouds for the reverse design of a blade surface. Blade surfaces in the gas turbines field require high accuracy and have complex shapes. By quickly recog nizing the border of the scattered point clouds of blades, the back-project method was used to extract the ruled generatrix vector. By solving the multi-objective optimization problem of border point clouds, the annealing algorithm was applied to match border point clouds nonrigidly. Equally divided points were constructed based on the point clouds registration results to get inerratic-orderly line-shaped feature point clouds that are beneficial for generating high quality blade surfaces quickly. The proposed algorithm is very effective for dealing with the point clouds of single-value ruled blades. It is also theoretically significant and may be applied for reconstructing the surface of point clouds of single-value ruled blade surfaces. Experiments verify the feasibility of the proposed algorithm.From the geometric continuity of the curved surface, the border features of blade surfaces [11] can be divided into three types:

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Section: Normal Vector Estimationmentioning

confidence: 99%

Algorithm to Construct Line-Shaped Feature Point Clouds of Single-Value Ruled Blades for Gas Turbines

Zhou

Wang

et al. 2014

Journal of Manufacturing Science and Engineering

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“…This technique use the pure linear convergence of the Schwarz method i.e the error propagation operator does not depend ot he itretation. The pure algebraical version of the acceleration of Schwarz by Aitken's technique (for non separable operator and independent of the mesh at the artificial interface) was given in [TD09] and the first massively parallel implementation was studied in L. Berenguer and D. Tromeur-Dervout [BDTD13]. This technique uses the Singular Values Decomposition (SVD)of the matrix gathering the iterated Schwarz solutions at all artificial interfaces of the domain decomposition to build a low rank approximation of the searched converged solution.…”

Section: Introductionmentioning

confidence: 99%

Sparse Aitken–Schwarz domain decomposition with application to Darcy flow

Berenguer¹,

Tromeur-Dervout²

2022

Computers & Fluids

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“…The number of snapshots is not chosen adaptively, but fixed to ensure that the acceleration is performed after the same number of Schwarz iterations for both algorithms. This comparison is made for two permeability fields, the first one is generated according to the parameters (λ, σ) = (10, 1) and the second one according to (λ, σ) = (20,3). Note that both parameters have been increased in order to slow the convergence rate of the Schwarz method.…”

Section: Comparison Of the Approximations Of Pmentioning

confidence: 99%

“…Moreover, the decreasing of the singular values gives an a priori criterion to select the singular vectors involved in the Aitken's acceleration operator approximation. This allows solving the large 3D computation of the linear Darcy equation where the permeability field follows a random log normal distribution law [3] in section 5.…”

Section: Introductionmentioning

confidence: 99%

Approximating the trace of iterative solutions at the interfaces with Nonuniform Fourier transform and singular value decomposition for cost-effectively accelerating the convergence of Schwarz domain decomposition

Tromeur-Dervout

2013

ESAIM: Proc.

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Résumé. Cet acte traite de la représentation des solutions itérées aux interfaces de la méthode de décomposition de domaine de type Schwarz afin d'approximer de manière adaptative son opérateur d'erreur aux interfaces des sous domaines. Ceci permet de construire de manièreéconomique l'accélération de la convergence de la méthode itérative enétendant la technique d'accélération de la convergence de Aitken au cas vectoriel. La première représentation est fondée sur la construction d'une transformée de Fourier discrète non uniforme définie sur un maillage non régulier. Nous montrons comment construire une base de Fourier de dimension N+1 sur ce maillageà partir de la construction numérique d'une forme sesquilinéaire, son exactitude pour les polynômes trigonométriques de degré N/2, et numériquement sa capacité d'approximation spectrale dépendante de la continuité de la fonctionà approximer. La décroissance des modes de Fourier de la trace de la solution itérée aux interfaces nous fournis une estimation pour sélectionner de manière adaptive les modes intervenant dans l'accélération. Le défaut de cette approche est d'être dépendant de la continuité de la solution itérée aux interfaces. La deuxième représentation, purement algébrique, utilise une décomposition en valeurs singulières des solutions itérées aux interfaces pour fournir un ensemble de vecteurs singuliers orthogonaux dont les valeurs singulières associées fournissent une estimation pour adapter l'accélération. La méthode Aitken-Schwarz résultante est ensuite appliquée au calculà grandeéchelle pour résoudre unécoulement 3D en milieu poreux modélisé par l'équation de Darcy linéaire où la perméabilité suit une distribution aléatoire log-normale. 35Abstract. This paper deals with the representation of the trace of iterative Schwarz solutions at the interfaces of domain decomposition to approximate adaptively the interface error operator. This allows to build a cost-effectively accelerating of the convergence of the iterative method by extending to the vectorial case the Aitken's accelerating convergence technique. The first representation is based on the building of a nonuniform discrete Fourier transform defined on a non-regular grid. We show how to construct a Fourier basis of dimension N+1 on this grid by building numerically a sesquilinear form, its exact accuracy to represent trigonometric polynomials of degree N / 2, and its spectral approximation property that depends on the continuity of the function to approximate. The decay of Fourier-like modes of the approximation of the trace of the iterative solution at the interfaces provides an estimate to adaptively select the modes involved in the acceleration. The drawback of this approach is to be dependent on the continuity of the trace of the iterated solution at the interfaces. The second representation, purely algebraic, uses a singular value decomposition of the trace of the iterative solution at the interfaces to provide a set of orthogonal singular vectors of which the associated singular values provi...

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Aitken’s acceleration of the Schwarz process using singular value decomposition for heterogeneous 3D groundwater flow problems

Cited by 8 publications

References 21 publications

Algorithm to Construct Line-Shaped Feature Point Clouds of Single-Value Ruled Blades for Gas Turbines

Algorithm to Construct Line-Shaped Feature Point Clouds of Single-Value Ruled Blades for Gas Turbines

Sparse Aitken–Schwarz domain decomposition with application to Darcy flow

Approximating the trace of iterative solutions at the interfaces with Nonuniform Fourier transform and singular value decomposition for cost-effectively accelerating the convergence of Schwarz domain decomposition

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