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139
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 82 (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.
A nonoverlapping domain decomposition method for Maxwell’s equations in three dimensions
 SIAM J. Numer. Anal
"... Abstract. We propose a substructuring preconditioner for solving threedimensional elliptic equations with strongly discontinuous coefficients. The new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by Bramble, Pasiack and Schatz (1989), but with muc ..."
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Cited by 46 (16 self)
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Abstract. We propose a substructuring preconditioner for solving threedimensional elliptic equations with strongly discontinuous coefficients. The new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by Bramble, Pasiack and Schatz (1989), but with much simpler coarse solvers. Though the condition number of the preconditioned system may not have a good bound, we are able to show that the convergence rate of the PCG method with such substructuring preconditioner is nearly optimal, and also robust with respect to the (possibly large) jumps of the coefficient in the elliptic equation. 1.
Fusion frames and distributed processing
 Appl. and Comp. Harmonic Anal
"... Abstract. Let {Wi}i∈I be a (redundant) sequence of subspaces of a Hilbert space each being endowed with a weight vi, and let H be the closed linear span of the Wi’s, a composite Hilbert space. {(Wi, vi)}i∈I is called a fusion frame provided it satisfies a certain property which controls the weighted ..."
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Cited by 31 (11 self)
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Abstract. Let {Wi}i∈I be a (redundant) sequence of subspaces of a Hilbert space each being endowed with a weight vi, and let H be the closed linear span of the Wi’s, a composite Hilbert space. {(Wi, vi)}i∈I is called a fusion frame provided it satisfies a certain property which controls the weighted overlaps of the subspaces. These systems contain conventional frames as a special case, however they reach far “beyond frame theory”. In case each subspace Wi is equipped with a spanning frame system {fij}j∈Ji, we refer to {(Wi, vi, {fij}j∈Ji)}i∈I as a fusion frame system. The focus of this article is on computational issues of fusion frame reconstructions, unique properties of fusion frames important for applications with particular focus on those superior to conventional frames, and on centralized reconstruction versus distributed reconstructions and their numerical differences. The weighted and distributed processing technique described in this article is not only a natural fit to distributed processing systems such as sensor networks, but also an efficient scheme for parallel processing of very large frame systems. Another important component of this article is an extensive study of the robustness of fusion frame systems. 1.
Generalized Multiscale Finite Element Methods (GMsFEM)
, 2013
"... In this paper, we propose a general approach called Generalized Multiscale Finite Element Method (GMsFEM) for performing multiscale simulations for problems without scale separation over a complex input space. As in multiscale finite element methods (MsFEMs), the main idea of the proposed approach i ..."
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Cited by 28 (10 self)
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In this paper, we propose a general approach called Generalized Multiscale Finite Element Method (GMsFEM) for performing multiscale simulations for problems without scale separation over a complex input space. As in multiscale finite element methods (MsFEMs), the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space. In the proposed approach, we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage. The enrichment in the online stage is performed based on a spectral decomposition of the offline space. In the online stage, for any input parameter, a multiscale space is constructed to solve the global problem on a coarse grid. The online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues. The computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be reused for solving the forward problem with any forcing and boundary condition. Compared with the other approaches where global snapshots are used, the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarsegrid level. We present various examples in the paper and some numerical results to demonstrate the effectiveness of our method. 1
An overlapping Schwarz algorithm for almost incompressible elasticity
 SIAM J. Numer. Anal
"... Abstract. Overlapping Schwarz methods are extended to mixed finite element approximations of linear elasticity which use discontinuous pressure spaces. The coarse component of the preconditioner is based on a lowdimensional space previously developed for scalar elliptic problems and a domain decomp ..."
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Cited by 22 (6 self)
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Abstract. Overlapping Schwarz methods are extended to mixed finite element approximations of linear elasticity which use discontinuous pressure spaces. The coarse component of the preconditioner is based on a lowdimensional space previously developed for scalar elliptic problems and a domain decomposition method of iterative substructuring type, i.e., a method based on nonoverlapping decompositions of the domain, while the local components of the preconditioner are based on solvers on a set of overlapping subdomains. A bound is established for the condition number of the algorithm which grows in proportion to the square of the logarithm of the number of degrees of freedom in individual subdomains and the third power of the relative overlap between the overlapping subdomains, and which is independent of the Poisson ratio as well as jumps in the Lamé parameters across the interface between the subdomains. A positive definite reformulation of the discrete problem makes the use of the standard preconditioned conjugate gradient method straightforward. Numerical results, which include a comparison with problems of compressible elasticity, illustrate the findings.
Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: nonoverlapping case
 M2AN Mathematical Modelling and Numercal Analysis
, 2005
"... Summary. We present a twolevel nonoverlapping additive Schwarz method for Discontinuous Galerkin approximations of elliptic problems. In particular, a two levelmethod for both symmetric and nonsymmetric schemes will be considered and some interesting features, which have no analog in the conform ..."
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Cited by 22 (6 self)
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Summary. We present a twolevel nonoverlapping additive Schwarz method for Discontinuous Galerkin approximations of elliptic problems. In particular, a two levelmethod for both symmetric and nonsymmetric schemes will be considered and some interesting features, which have no analog in the conforming case, will be discussed. Numerical experiments on nonmatching grids will be presented. 1
Domain decomposition preconditioners for linear–quadratic elliptic optimal control problems
, 2004
"... ABSTRACT. We develop and analyze a class of overlapping domain decomposition (DD) preconditioners for linearquadratic elliptic optimal control problems. Our preconditioners utilize the structure of the optimal control problems. Their execution requires the parallel solution of subdomain linearquad ..."
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Cited by 19 (4 self)
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ABSTRACT. We develop and analyze a class of overlapping domain decomposition (DD) preconditioners for linearquadratic elliptic optimal control problems. Our preconditioners utilize the structure of the optimal control problems. Their execution requires the parallel solution of subdomain linearquadratic elliptic optimal control problems, which are essentially smaller subdomain copies of the original problem. This work extends to optimal control problems the application and analysis of overlapping DD preconditioners, which have been used successfully for the solution of single PDEs. We prove that for a class of problems the performance of the twolevel versions of our preconditioners is independent of the mesh size and of the subdomain size. 1.
Adaptive Selection of Face Coarse Degrees of Freedom in the BDDC and the FETIDP Iterative Substructuring Methods
, 2006
"... We propose a class of method for the adaptive selection of the coarse space of the BDDC and FETIDP iterative substructuring methods. The methods work by adding coarse degrees of freedom constructed from eigenvectors associated with intersections of selected pairs of adjacent substructures. It is as ..."
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Cited by 19 (4 self)
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We propose a class of method for the adaptive selection of the coarse space of the BDDC and FETIDP iterative substructuring methods. The methods work by adding coarse degrees of freedom constructed from eigenvectors associated with intersections of selected pairs of adjacent substructures. It is assumed that the starting coarse degrees of freedom are already sufficient to prevent relative rigid body motions in any selected pair of adjacent substructures. A heuristic indicator of the the condition number is developed and a minimal number of coarse degrees of freedom is added to decrease the indicator under a given threshold. It is shown numerically on 2D elasticity problems that the indicator based on pairs of substructures with common edges predicts the actual condition number reasonably well, and that the method can select adaptively the hard part of the problem and concentrate computational work there to achieve good convergence of the iterations at a modest cost.
Tallec, “Filtering for distributed mechanical systems using position measurements: perspectives
 in medical imaging,” Inverse Problems
"... We propose an effective filtering methodology designed to perform estimation in a distributed mechanical system using position measurements. As in a previously introduced method, the filter is inspired from robust control feedback, but here we take full advantage of the estimation specificity to cho ..."
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Cited by 19 (5 self)
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We propose an effective filtering methodology designed to perform estimation in a distributed mechanical system using position measurements. As in a previously introduced method, the filter is inspired from robust control feedback, but here we take full advantage of the estimation specificity to choose a feedback law that can act on displacements instead of velocities and still retain the same kind of dissipativity property which guarantees robustness. This is very valuable in many applications for which positions are more readily available than velocities, as in medical imaging. We provide an indepth analysis of the proposed procedure, as well as detailed numerical assessments using a test problem inspired from cardiac biomechanics, as medical diagnosis assistance is an important perspective for this approach. The method is formulated first for measurements based on Lagrangian displacements, but we then derive a nonlinear extension allowing to instead consider segmented images, which of course is even more relevant in medical applications. 1
Analysis of a twolevel Schwarz method with coarse spaces based on local DirichlettoNeumann maps
 Comput. Methods Appl. Math
"... Schwarz method with coarse spaces based on local Dirichlet–to–Neumann maps. computer ..."
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Cited by 15 (7 self)
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Schwarz method with coarse spaces based on local Dirichlet–to–Neumann maps. computer