Valeria Simoncini

RECENT COMPUTATIONAL DEVELOPMENTS IN KRYLOV SUBSPACE METHODS FOR LINEAR SYSTEMS

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. Copyright

A FAMILY OF MIMETIC FINITE DIFFERENCE METHODS ON POLYGONAL AND POLYHEDRAL MESHES

A family of inexpensive discretization schemes for diffusion problems on unstructured polygonal and polyhedral meshes is introduced. The material properties are described by a full tensor. The theoretical results are confirmed with numerical experiments.

A New Iterative Method for Solving Large-Scale Lyapunov Matrix Equations

In this paper we propose a new projection method to solve large-scale continuous-time Lyapunov matrix equations. The new approach projects the problem onto a much smaller approximation space, generated as a combination of Krylov subspaces in

Theory of Inexact Krylov Subspace Methods and Applications to Scientific Computing

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Block--diagonal and indefinite symmetric preconditioners for mixed finite element formulations

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An iterative method for nonsymmetric systems with multiple right-hand sides

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Iterative system solvers for the frequency analysis of linear mechanical systems

. The reduced problem is then solved by means of a direct Lyapunov scheme based on matrix factorizations. The reported numerical results show the competitiveness of the new method, compared to a state-of-the-art approach based on the factorized alternating direction implicit iteration.

Computational methods for linear matrix equations

We provide a general framework for the understanding of inexact Krylov subspace methods for the solution of symmetric and nonsymmetric linear systems of equations, as well as for certain eigenvalue calculations. This framework allows us to explain the empirical results reported in a series of CERFACS technical reports by Bouras, Fraysse, and Giraud in 2000. Furthermore, assuming exact arithmetic, our analysis can be used to produce computable criteria to bound the inexactness of the matrix-vector multiplication in such a way as to maintain the convergence of the Krylov subspace method. The theory developed is applied to several problems including the solution of Schur complement systems, linear systems which depend on a parameter, and eigenvalue problems. Numerical experiments for some of these scientific applications are reported.

A quasi-minimal residual variant of the Bi-CGSTAB algorithm for nonsymmetric systems

We are interested in the numerical solution of large structured indefinite symmetric linear systems arising in mixed finite element approximations of the magnetostatic problem; in particular, we analyze definite block--diagonal and indefinite symmetric preconditioners. Relating the algebraic characteristics of the resulting preconditioned matrix to the properties of the continuous problem and of its finite element discretization, we show that the considered preconditioning strategies make the used Krylov subspace solver insensitive to the mesh refinement parameter, in terms of number of iterations. In order to achieve computational efficiency, we also analyze algebraic approximations to the optimal preconditioners, and discuss their performance on real two and three dimensional application problems.

On the eigenvalues of a class of saddle point matrices

We propose a method for the solution of linear systems