Preconditioning of iterative methods ‐ theory and applications

The collection of five papers [2–6] published in this issue of Numerical Linear Algebra with Applications originates from the talks delivered at the conference Preconditioning of Iterative Methods Theory and Applications (PIM 2013), organized in honor of Professor Ivo Marek on the occasion of his 80th birthday and represents a tribute to his work and influence in the numerical linear algebra community, compare with [1]. We provide a short summary of the presented papers in order of their submission. Šístek et al. [2] deal with the solution of saddle point systems arising from the mixed-hybrid finite element discretization of flow in porous media including highly permeable fractures. The fractures are modeled by 1D or 2D elements inside three-dimensional domains. The paper deals with an application of the balancing domain decomposition by constraints method to the described problems and provides both theoretical results and numerical experiments, which illustrate the efficiency and the scalability of the balancing domain decomposition by constraints method. Axelsson et al. [3] consider a particular type of a high-order strongly stable time integration method that is applicable to evolution and differential-algebraic problems. To solve the block matrix systems arising within the corresponding time stepping procedure, an efficient preconditioning method is presented and analyzed. The method is applied to a consolidation problem arising in poroelasticity. Kužel and Vaněk [4] propose a special type of nonlinear multigrid scheme in which the prolongator (P ) is updated at each iteration so that the current approximation lies in range of P . The update of the prolongator is carried out in a simple way by adding the current approximation as a new column in the prolongation matrix. An advantage of this scheme constitutes improved convergence with easy implementation. The method is applicable for the solution of both linear and nonlinear problems. In particular, the paper describes application to generalized eigenvalue problems. Kraus et al. [5] present a non-variational auxiliary space multigrid algorithm for general symmetric positive definite problems. The method is based on exact two-by-two block factorization of local (finite element) matrices that correspond to a sequence of coverings of the domain by overlapping or non-overlapping subdomains. The coarse grid matrix is defined as an additive-type approximation of the Schur complement. Its sparsity is controlled by the size and overlap of the subdomains. The coarse grid construction is used in the framework of the Algebraic Multilevel Iterative (AMLI) method, and the efficiency and the robustness of the method when solving multiscale problems with oscillating coefficients are shown theoretically as well as by numerical experiments. Dostál et al. [6] consider Finite Element Tearing and Interconnecting (FETI) domain decomposition for the solution of variational inequalities with varying coefficients. For the solution of the dual problem, they investigate a symmetric stiffness-based preconditioning of the Hessian. They present and analyze a simple variant of the stiffness scaling of the Hessian of the dual function based on reorthogonalization or renormalization of constraint matrices. The efficiency of the method is demonstrated. The conference Preconditioning of Iterative Methods Theory and Applications (PIM 2013) was held at the Faculty of Civil Engineering, Czech Technical University in Prague on July 1–5, 2013. The main scientific themes included preconditioning of sparse symmetric and non-symmetric matrix problems arising in large-scale real-world applications, multilevel preconditioning and multigrid techniques, algebraic multilevel methods, domain decomposition methods for partial differential equations, and multilevel solution of characteristics of Markov chains. The conference program consisted of 15 plenary lectures, 28 contributed talks, and 8 posters. More than 90 participants from 19 countries (Algeria, Austria, Bulgaria, Czech Republic, France,