Browsing by Author "Rozložník, Miroslav"

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    A note on adaptivity in factorized approximate inverse preconditioning
    (OVIDIUS UNIV PRESS, FAC MATHEMATICS & COMPUTER SCIENCE, BULEVARDUL MAMAIA 124, CONSTANTA, 900527, ROMANIA, 2020-01-01) Kopal, Jiří; Rozložník, Miroslav; Tůma, Miroslav
    The problem of solving large-scale systems of linear algebraic equations arises in a wide range of applications. In many cases the preconditioned iterative method is a method of choice. This paper deals with the approximate inverse preconditioning AINV/SAINV based on the incomplete generalized Gram-Schmidt process. This type of the approximate inverse preconditioning has been repeatedly used for matrix diagonalization in computation of electronic structures but approximating inverses is of an interest in parallel computations in general. Our approach uses adaptive dropping of the matrix entries with the control based on the computed intermediate quantities. Strategy has been introduced as a way to solve difficult application problems and it is motivated by recent theoretical results on the loss of orthogonality in the generalized Gram-Schmidt process. Nevertheless, there are more aspects of the approach that need to be better understood. The diagonal pivoting based on a rough estimation of condition numbers of leading principal submatrices can sometimes provide inefficient preconditioners. This short study proposes another type of pivoting, namely the pivoting that exploits incremental condition estimation based on monitoring both direct and inverse factors of the approximate factorization. Such pivoting remains rather cheap and it can provide in many cases more reliable preconditioner. Numerical examples from real-world problems, small enough to enable a full analysis, are used to illustrate the potential gains of the new approach.
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    Analýza limitní přesnosti iteračních metod
    (Technická Univerzita v Liberci, 2008-01-01) Jiránek, Pavel; Rozložník, Miroslav
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    Dual variable methods for mixed-hybrid finite element approximation of the potential fluid flow problem in porous media
    (Technická Univerzita v Liberci, 2006-01-01) Arioli, M.; Maryška, Jiří; Rozložník, Miroslav; Tůma, Miroslav
    Mixed-hybrid finite element discretization of Darcy's law and the continuity equation that describe the potential fluid flow problem in porous media leads to symmetric indefinite saddle-point problems. In this paper we consider solution techniques based on the computation of a null-space basis of the whole or of a part of the left lower off-diagonal block in the system matrix and on the subsequent iterative solution of a projected system. This approach is mainly motivated by the need to solve a sequence of such systems with the same mesh but different material properties. A fundamental cycle null-space basis of the whole off-diagonal block is constructed using the spanning tree of an associated graph. It is shown that such a basis may be theoretically rather ill-conditioned. Alternatively, the orthogonal null-space basis of the sub-block used to enforce continuity over faces can be easily constructed. In the former case, the resulting projected system is symmetric positive definite and so the conjugate gradient method can be applied. The projected system in the latter case remains indefinite and the preconditioned minimal residual method (or the smoothed conjugate gradient method) should be used. The theoretical rate of convergence for both algorithms is discussed and their efficiency is compared in numerical experiments. Copyright © 2006, Kent State University.
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    Iteračné riešenie rozsiahlych sústav sedlového bodu v matematickom modelovaní
    (Technická Univerzita v Liberci, 2003-01-01) Rozložník, Miroslav