On the Construction of Deflation-Based Preconditioners

Frank, Jason; Vuik, C.

doi:10.1137/s1064827500373231

Cited by 122 publications

(160 citation statements)

References 27 publications

(34 reference statements)

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“…Further, P is a projection and A is symmetric positive definite, hence from Lemma 2.1. by Frank and Vuik [4], follows that P A is positive semi-definite. 2…”

Section: Corollarymentioning

confidence: 93%

“…Note thatx is the solution of equation (4). In this paragraph we first establish uniqueness of the above P Tx , given anyx that satisfies equation (4).…”

Section: Deflated Incomplete Choleski Preconditioned Conjugate Gradientsmentioning

confidence: 96%

“…In this paragraph we first establish uniqueness of the above P Tx , given anyx that satisfies equation (4). Therefore, we first need to establish that for the projected solution we have P T x = P Tx and that hence P Tx is unique.…”

Section: Deflated Incomplete Choleski Preconditioned Conjugate Gradientsmentioning

confidence: 99%

“…A related method is recently presented in [11]. In [4] deflation is used to accelerate block-IC preconditioners combined with Krylov subspace methods in a parallel computing environment.…”

Section: Introductionmentioning

confidence: 99%

“…In our previous work we always use non-overlapping deflation vectors. In [17,18] the interface vertices are only elements of the high permeability subdomains, whereas in [4] no interface vertices occur due to a cell centered discretization. The topic of this paper is: how to choose the value of the deflation vectors at interface points in order to obtain an efficient, robust and parallelizable black-box deflation method.…”

Section: Introductionmentioning

confidence: 99%

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Deflation in Preconditioned Conjugate Gradient Methods for Finite Element Problems

Vermolen

Vuik

Segal

2004

Scientific Computation

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We investigate the influence of the value of deflation vectors at interfaces on the rate of convergence of preconditioned conjugate gradient methods applied to a Finite Element discretization for an elliptic equation. Our set-up is a Poisson problem in two dimensions with continuous or discontinuous coefficients that vary in several orders of magnitude. In the continuous case we are interested in the convergence acceleration of deflation on block preconditioners. For the discontinuous case the Finite Element discretization gives a symmetric matrix with very large condition number and hence many iterations are needed to obtain a solution using conjugate gradients. We use an incomplete Choleski decomposition for the symmetric discretization matrix. Subsequently, deflation is applied to eliminate the disadvantageous effects to convergence caused by the remaining small eigenvalues. Here we investigate the effects of several variants of deflation and we propose an optimal choice for the deflation vectors. Finally, we apply deflation to a parallel preconditioned conjugate method to accellerate the calculation.

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“…Further, P is a projection and A is symmetric positive definite, hence from Lemma 2.1. by Frank and Vuik [4], follows that P A is positive semi-definite. 2…”

Section: Corollarymentioning

confidence: 93%

“…Note thatx is the solution of equation (4). In this paragraph we first establish uniqueness of the above P Tx , given anyx that satisfies equation (4).…”

Section: Deflated Incomplete Choleski Preconditioned Conjugate Gradientsmentioning

confidence: 96%

Section: Deflated Incomplete Choleski Preconditioned Conjugate Gradientsmentioning

confidence: 99%

“…A related method is recently presented in [11]. In [4] deflation is used to accelerate block-IC preconditioners combined with Krylov subspace methods in a parallel computing environment.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Deflation in Preconditioned Conjugate Gradient Methods for Finite Element Problems

Vermolen

Vuik

Segal

2004

Scientific Computation

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Linear Algebraic Solvers and Eigenvalue Analysis

Vorst

Vuik

2017

Encyclopedia of Computational Mechanics Second Edition

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This chapter gives an overview of the most widely used numerical methods for the solution of linear systems of equations and for eigenproblems, including direct methods and iterative methods. Iterative methods are often used in combination with the so‐called preconditioning operators. We will give a brief overview of the various preconditioners that exist. For eigenproblems of the type , the QR method is discussed. The QR method is expensive for larger values of , and for these larger values, a number of iterative methods, including the Lanczos method, Arnoldi's method, and the Jacobi–Davidson method are shown.

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Combined modality therapy of cT2N0M0 esophageal cancer

Kountourakis

Correa

Hofstetter

et al. 2010

Cancer

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BACKGROUND:Treatment strategy for patients with adequately staged cT2N0M0 carcinoma of the thoracic esophagus is currently a subject of debate. This study analyzed the largest series of consecutive cT2N0M0 esophageal cancer patients treated with preoperative chemoradiotherapy. METHODS: Data from all patients with cT2N0M0 (assessment included endoscopic ultrasonography and computed tomography of the chest and abdomen) thoracic esophageal cancer who were treated with preoperative chemoradiation between 1997 and 2009 were analyzed. The Cox regression model and Kaplan-Meier plots were used to analyze the data. RESULTS: Data from 49 patients were analyzed. The median follow-up was 28.46 months. Male sex and adenocarcinoma histology predominated. Pathologic complete response was observed 19 (39%) patients. The 10-year actuarial overall survival (OS) for adenocarcinoma patients was >60%. In the univariate analysis for OS, squamous histology (P ¼ .006), smoking (P ¼ .015), and alcohol consumption (P ¼ .032) were found to be associated with poor OS. In the univariate analysis for disease-free survival (DFS), squamous histology (P ¼ .009) and smoking (P ¼ .014) were associated with poor DFS. In the multivariate analysis for OS, smoking was an independent prognosticator (P ¼ .02). In the multivariate analysis for DFS, advanced pathologic stage (P ¼ .05) and lymph node metastases (P ¼ .006) were independent prognosticators. Patients with adenocarcinoma (P ¼ .002) and those with pathologic N0 disease had better OS and DFS. Upward stage migration occurred in only 10% of patients. CONCLUSIONS: These data suggest that smoking and alcohol influence the long-term outcome of patients with cT2N0M0 disease. Adenocarcinoma patients treated with trimodality therapy had an excellent actuarial 10-year OS and a high rate of pathologic complete response. Trimodality therapy should be prospectively compared with primary surgery in these patients. Cancer 2011;117:925-30.

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On the Construction of Deflation-Based Preconditioners

Cited by 122 publications

References 27 publications

Deflation in Preconditioned Conjugate Gradient Methods for Finite Element Problems

Deflation in Preconditioned Conjugate Gradient Methods for Finite Element Problems

Linear Algebraic Solvers and Eigenvalue Analysis

Combined modality therapy of cT2N0M0 esophageal cancer

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