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Blaheta, R. (author)In this paper, we show that the deflation method can be viewed as a possible implementation of the CG method with multilevel preconditioner. Further, we demonstrate efficiency and robustness of different implementations of multilevel preconditioners with different "coarse grid" spaces by solving a simple model problem.conference paper 2006
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Nabben, R. (author), Vuik, C. (author)The balancing Neumann-Neumann (BNN) and the additive coarse grid correction (BPS) preconditioner are fast and successful preconditioners within domain decomposition methods for solving partial differential equations. For certain elliptic problems these preconditioners lead to condition numbers which are independent of the mesh sizes and are...conference paper 2006
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Tang, J.M. (author), Vuik, C. (author)Simulating bubbly flows is a very popular topic in CFD. These bubbly flows are governed by the Navier-Stokes equations. In many popular operator splitting formulations for these equations, solving the linear system coming from the discontinuous diffusion equation takes the most computational time, despite of its elliptic origins. Sometimes these...conference paper 2006
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Tang, J.M. (author), Vuik, C. (author)Simulating bubbly flows is a very popular topic in CFD. These bubbly flows are governed by the Navier-Stokes equations. In many popular operator splitting formulations for these equations, solving the linear system coming from the discontinuous diffusion equation takes the most computational time, despite of its elliptic origins. Sometimes these...conference paper 2006
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Nabben, R. (author), Vuik, C. (author)The balancing Neumann-Neumann (BNN) and the additive coarse grid correction (BPS) preconditioner are fast and successful preconditioners within domain decomposition methods for solving partial differential equations. For certain elliptic problems these preconditioners lead to condition numbers which are independent of the mesh sizes and are...conference paper 2006
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Jönsthövel, T.B. (author)Simulations with composite materials often involve large jumps in the coefficients of the underlying stiffness matrix. These jumps can introduce unfavorable eigenvalues in the spectrum of the stiffness matrix. We show that the rigid body modes; the translations and rotations, of the disjunct rigid bodies in the composite material correspond to...doctoral thesis 2012
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Gupta, R. (author)In this work we study the implementations of deflation and preconditioning techniques for solving ill-conditioned linear systems using iterative methods. Solving such systems can be a time-consuming process because of the jumps in the coefficients due to large difference in material properties. We have developed implementations of the iterative...doctoral thesis 2015
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Sheikh, A.H. (author)The Helmholtz equation is the simplest possible model for the wave propagation. Perhaps this is the reason, despite denying traditional iterative methods like Krylov sub-space methods, Multigrids, etcetera, numerical solution of the Helmholtz equation has been an interesting and abundant problem to researchers since years. The work in this...doctoral thesis 2014
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Collignon, T.P. (author)This dissertation deals mainly with the design, implementation, and analysis of efficient iterative solution methods for large sparse linear systems on distributed and heterogeneous computing systems as found in Grid computing. First, a case study is performed on iteratively solving large symmetric linear systems on both a multi–cluster and a...doctoral thesis 2011
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Tang, J.M. (author)The Preconditioned Conjugate Gradient (PCG) method is one of the most popular iterative methods for solving large linear systems with a symmetric and positive semi-definite coefficient matrix. However, if the preconditioned coefficient matrix is ill-conditioned, the convergence of the PCG method typically deteriorates. Instead, a two-level PCG...doctoral thesis 2008
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Xu, S. (author)This thesis deals with two research problems. The first research problem is motivated by the numerical computation involved in the Time Domain Simulation (TDS) of Power Grids. Due to the ever growing size and complexity of Power Grids such as the China National Grid, accelerating TDS has become a stringent need for the online analysis of these...doctoral thesis 2011
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Dwarka, V.N.S.R. (author)The bottleneck in designing iterative solvers for the Helmholtz equation lies in balancing the trade-off between accuracy and scalability. Both the accuracy of the numerical solution and the number of iterations to reach convergence deteriorate in higher dimensions and increase with the wavenumber. To address these issues in this dissertation,...doctoral thesis 2022
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Diaz Cortes, G.B. (author)Simulation of flow through highly heterogeneous porous media results in large ill-conditioned systems of equations. In particular, solving the linearized pressure system can be especially time-consuming. Therefore, extensive efforts to find ways to address this issue effectively are required. In this work, we introduce a POD-based deflation...doctoral thesis 2019
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Tang, J.M. (author), Nabben, R. (author), Vuik, C. (author), Erlangga, Y.A. (author)For various applications, it is well-known that a multi-level, in particular two-level, preconditioned CG (PCG) method is an efficient method for solving large and sparse linear systems with a coefficient matrix that is symmetric positive definite. The corresponding two-level preconditioner combines traditional and projection-type...journal article 2009
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Zouros, G.P. (author), Budko, N.V. (author)The domain integral equation method with its FFT-based matrix-vector products is a viable alternative to local methods in free-space scattering problems. However, it often suffers from the extremely slow convergence of iterative methods, especially in the transverse electric (TE) case with large or negative permittivity. We identify very dense...journal article 2012
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Tang, J.M. (author), MacLachlan, S.P. (author), Nabben, R. (author), Vuik, C. (author)It is well known that two-level and multilevel preconditioned conjugate gradient (PCG) methods provide efficient techniques for solving large and sparse linear systems whose coefficient matrices are symmetric and positive definite. A two-level PCG method combines a traditional (one-level) preconditioner, such as incomplete Cholesky, with a...journal article 2010
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Jönsthövel, T.B. (author), Van Gijzen, M.B. (author), Vuik, C. (author), Scarpas, A. (author)Large discontinuities in material properties, such as those encountered in composite materials, lead to ill-conditioned systems of linear equations. These discontinuities give rise to small eigenvalues that may negatively affect the convergence of iterative solution methods such as the preconditioned conjugate gradient method. This paper...journal article 2013
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Jönsthövel, T.B. (author), Van Gijzen, M.B. (author), MacLachlan, S. (author), Vuik, C. (author), Scarpas, A. (author)Many applications in computational science and engineering concern composite materials, which are characterized by large discontinuities in the material properties. Such applications require fine-scale finite-element meshes, which lead to large linear systems that are challenging to solve with current direct and iterative solutions algorithms....journal article 2012
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Van 't Wout, E. (author), Van Gijzen, M.B. (author), Ditzel, A. (author), Van der Ploeg, A. (author), Vuik, C. (author)Ship simulators are used for training purposes and therefore have to calculate realistic wave patterns around the moving ship in real time. We consider a wave model that is based on the variational Boussinesq formulation, which results in a set of partial differential equations. Discretization of these equations gives a large system of linear...journal article 2010
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Dwarka, V.N.S.R. (author), Vuik, Cornelis (author)Recent research efforts aimed at iteratively solving time-harmonic waves have focused on a broad range of techniques to accelerate convergence. In particular, for the famous Helmholtz equation, deflation techniques have been studied to accelerate the convergence of Krylov subspace methods. In this work, we extend the two-level deflation method...journal article 2022