In this paper we address the solution of linear systems of equations with multiple right-hand sid... more In this paper we address the solution of linear systems of equations with multiple right-hand sides given at once. When the dimension of the problem is known to be large, preconditioned block Krylov subspace methods are usually considered as the method of choice. Nevertheless to be effective in terms of computational operations it is known that these methods must incorporate a strategy for detecting when a linear combination of the systems has approximately converged. This strategy usually leads to an explicit block size reduction often called deflation. While initial deflation or deflation at the beginning of a cycle are nowadays popular, block Krylov subspace methods incorporating deflation at each iteration are quite rare. The purpose of this paper is thus to extend the block flexible restarted GMRES method to variants that allow the use of deflation at each iteration when solving multiple right-hand side problems given at once. The main goal of deflation will be then to reduce the cost of each iteration of the block Krylov subspace method by judiciously choosing which information to use and which information to be postponed. For the purpose of analysis we introduce a new minimal block residual method that incorporates block size reduction at each iteration named Deflated Minimal Block Residual method. First we study its main mathematical properties. We notably show that the Frobenius norm of the block residual is always nonincreasing. Second we justify the choice of the deflation strategy based on a nonincreasing behaviour of the singular values of the scaled block true residual. Third we propose a variant of the deflated block residual method that includes truncation at each iteration. Finally we discuss the computational cost of the algorithm and the possibility of breakdown in exact arithmetic. Numerical experiments on two different problems * TOTAL, Centre Scientifique et Technique Jean issued from wave propagation situations requiring the solution of multiple right-hand side problems are then discussed. On these test cases the new block flexible method including deflation at each iteration has proven to be more efficient in terms of both preconditioner applications and computational operations than recent block flexible Krylov subspace methods performing deflation at restart only.
SIAM Journal on Matrix Analysis and Applications, 2011
This work is concerned with the development and study of a minimum residual norm subspace method ... more This work is concerned with the development and study of a minimum residual norm subspace method based on the Generalized Conjugate Residual method with inner Orthogonalization (GCRO) method that allows flexible preconditioning and deflated restarting for the solution of non-symmetric or non-Hermitian linear systems. First we recall the main features of Flexible Generalized Minimum Residual with deflated restarting (FGMRES-DR), a recently proposed algorithm of the same family but based on the GMRES method. Next we introduce the new inner-outer subspace method named FGCRO-DR. A theoretical comparison of both algorithms is then made in the case of flexible preconditioning. It is proved that FGCRO-DR and FGMRES-DR are algebraically equivalent if a collinearity condition is satisfied. While being nearly as expensive as FGMRES-DR in terms of computational operations per cycle, FGCRO-DR offers the additional advantage to be suitable for the solution of sequences of slowly changing linear systems (where both the matrix and right-hand side can change) through subspace recycling. Numerical experiments on the solution of multidimensional elliptic partial differential equations show the efficiency of FGCRO-DR when solving sequences of linear systems.
Novas perspectivas no tratamento da hiperplasia adrenal congênita
A HAC é um distúrbio congênito, caracterizado pela deficiência na biossíntese do cortisol, associ... more A HAC é um distúrbio congênito, caracterizado pela deficiência na biossíntese do cortisol, associado ou não à deficiência de aldosterona e, consequentemente, à superprodução de andrógeno. Essa condição afeta igualmente homens e mulheres. No sexo feminino, ...
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