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426
Preconditioning techniques for large linear systems: A survey
 J. COMPUT. PHYS
, 2002
"... This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization i ..."
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Cited by 189 (5 self)
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This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization issues, and block and multilevel extensions. Some of the challenges ahead are also discussed. An extensive bibliography completes the paper.
A restricted additive Schwarz preconditioner for general sparse linear systems
 SIAM J. Sci. Comput
, 1999
"... Abstract. We introduce some cheaper and faster variants of the classical additive Schwarz preconditioner (AS) for general sparse linear systems and show, by numerical examples, that the new methods are superior to AS in terms of both iteration counts and CPU time, as well as the communication cost w ..."
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Cited by 127 (22 self)
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Abstract. We introduce some cheaper and faster variants of the classical additive Schwarz preconditioner (AS) for general sparse linear systems and show, by numerical examples, that the new methods are superior to AS in terms of both iteration counts and CPU time, as well as the communication cost when implemented on distributed memory computers. This is especially true for harder problems such as indefinite complex linear systems and systems of convectiondiffusion equations from threedimensional compressible flows. Both sequential and parallel results are reported. Key words. Overlapping domain decomposition, preconditioner, iterative method, sparse matrix AMS(MOS) subject classifications. 65N30, 65F10
The heterogeneous multiscale method: A review
 COMMUN. COMPUT. PHYS
, 2007
"... This paper gives a systematic introduction to HMM, the heterogeneous multiscale method, including the fundamental design principles behind the HMM philosophy and the main obstacles that have to be overcome when using HMM for a particular problem. This is illustrated by examples from several applic ..."
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Cited by 100 (5 self)
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This paper gives a systematic introduction to HMM, the heterogeneous multiscale method, including the fundamental design principles behind the HMM philosophy and the main obstacles that have to be overcome when using HMM for a particular problem. This is illustrated by examples from several application areas, including complex fluids, microfluidics, solids, interface problems, stochastic problems, and statistically selfsimilar problems. Emphasis is given to the technical tools, such as the various constrained molecular dynamics, that have been developed, in order to apply HMM to these problems. Examples of mathematical results on the error analysis of HMM are presented. The paper ends with a discussion on some of
Numerical mathematics
, 2000
"... Abstract. In this paper we introduce some basic differential models for the description of blood flow in the circulatory system. We comment on their mathematical properties, their meaningfulness and their limitation to yield realistic and accurate numerical simulations, and their contribution for a ..."
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Cited by 99 (12 self)
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Abstract. In this paper we introduce some basic differential models for the description of blood flow in the circulatory system. We comment on their mathematical properties, their meaningfulness and their limitation to yield realistic and accurate numerical simulations, and their contribution for a better understanding of cardiovascular physiopathology. Mathematics Subject Classification (2000). 92C50,96C10,76Z05,74F10,65N30,65M60. Keywords. Cardiovascular mathematics; mathematical modeling; fluid dynamics; Navier– Stokes equations; numerical approximation; finite element method; differential equations. 1.
Optimized Schwarz methods
 SIAM Journal on Numerical Analysis
, 2006
"... Abstract. Optimized Schwarz methods are a new class of Schwarz methods with greatly enhanced convergence properties. They converge uniformly faster than classical Schwarz methods and their convergence rates dare asymptotically much better than the convergence rates of classical Schwarz methods if th ..."
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Cited by 91 (18 self)
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Abstract. Optimized Schwarz methods are a new class of Schwarz methods with greatly enhanced convergence properties. They converge uniformly faster than classical Schwarz methods and their convergence rates dare asymptotically much better than the convergence rates of classical Schwarz methods if the overlap is of the order of the mesh parameter, which is often the case in practical applications. They achieve this performance by using new transmission conditions between subdomains which greatly enhance the information exchange between subdomains and are motivated by the physics of the underlying problem. We analyze in this paper these new methods for symmetric positive definite problems and show their relation to other modern domain decomposition methods like the new Finite Element Tearing and Interconnect (FETI) variants.
Recent computational developments in Krylov subspace methods for linear systems
 NUMER. LINEAR ALGEBRA APPL
, 2007
"... Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are metho ..."
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Cited by 86 (12 self)
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Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are methods specifically tailored to systems with special properties such as special forms of symmetry and those depending on one or more parameters.
Optimized Schwarz methods without overlap for the Helmholtz equation
 SIAM J. Sci. Comput
"... Abstract. The classical Schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel. Convergence is achieved through overlap of the subdomains. We study in this paper a variant of the Schwarz method which converges without overlap for the Helmholtz eq ..."
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Cited by 64 (16 self)
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Abstract. The classical Schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel. Convergence is achieved through overlap of the subdomains. We study in this paper a variant of the Schwarz method which converges without overlap for the Helmholtz equation. We show that the key ingredients for such an algorithm are the transmission conditions. We derive optimal transmission conditions which lead to convergence of the algorithm in a finite number of steps. These conditions are however nonlocal in nature and we introduce local approximations which we optimize for performance of the Schwarz method. This leads to an algorithm in the class of optimized Schwarz methods. We present an asymptotic analysis of the optimized Schwarz method for two types of transmission conditions, Robin conditions and transmission conditions with second order tangential derivatives. Numerical results illustrate the effectiveness of the optimized Schwarz method on a model problem and on a problem from industry.
Coupling methods for continuum model with molecular model
 International Journal for Multiscale Computational Engineering 2003
"... Coupling methods for continuum models with molecular models are developed. Two methods are studied here: an overlapping domain decomposition method, which has overlapping domain, and an edgetoedge decomposition method, which has an interface between two models. These two methods enforce the compat ..."
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Cited by 48 (6 self)
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Coupling methods for continuum models with molecular models are developed. Two methods are studied here: an overlapping domain decomposition method, which has overlapping domain, and an edgetoedge decomposition method, which has an interface between two models. These two methods enforce the compatibility on the overlapping domain or interface nodes/atoms by the Lagrange multiplier method or the augmented Lagrangian method. 1.
A comparison of Deflation and Coarse Grid Correction applied to porous media flow
, 2003
"... In this paper we compare various preconditioners for the numerical solution of partial dierential equations. We compare a coarse grid correction preconditioner used in domain decomposition methods with a socalled de
ation preconditioner. We prove that the eective condition number of the deflated pr ..."
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Cited by 48 (22 self)
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In this paper we compare various preconditioners for the numerical solution of partial dierential equations. We compare a coarse grid correction preconditioner used in domain decomposition methods with a socalled de
ation preconditioner. We prove that the eective condition number of the deflated preconditioned system is always, i.e. for all deflation vectors and all restrictions and prolongations, below the condition number of the system preconditioned by the coarse grid correction. This implies that the Conjugate Gradient method applied to the deflated preconditioned system converges always faster than the Conjugate Gradient method applied to the system preconditioned by the coarse grid correction. Numerical results for porous media flows emphasize the theoretical results.
A quasiNewton algorithm based on a reduced model for fluidstructure interaction problems in blood flows
 M2AN Math. Model. Numer. Anal
"... Abstract. We propose a quasiNewton algorithm for solving fluidstructure interaction problems. The basic idea of the method is to build an approximate tangent operator which is cost effective and which takes into account the socalled added mass effect. Various test cases show that the method allow ..."
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Cited by 46 (6 self)
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Abstract. We propose a quasiNewton algorithm for solving fluidstructure interaction problems. The basic idea of the method is to build an approximate tangent operator which is cost effective and which takes into account the socalled added mass effect. Various test cases show that the method allows a significant reduction of the computational effort compared to relaxed fixed point algorithms. We present 2D and 3D fluidstructure simulations performed either with a simple 1D structure model or with shells in large displacements. Mathematics Subject Classification. 65M60, 74K25, 76D05, 76Z05. 1.