Convergence analysis of Krylov subspace methods

Abstract

Key words Krylov subspace methods, convergence analysis. MSC (2000) 15A06, 65F10, 41A10One of the most powerful tools for solving large and sparse systems of linear algebraic equations is a class of iterative methods called Krylov subspace methods. Their significant advantages like low memory requirements and good approximation properties make them very popular, and they are widely used in applications throughout science and engineering. The use of the Krylov subspaces in iterative methods for linear systems is even counted among the "Top 10" algorithmic ideas of the 20th century. Convergence analysis of these methods is not only of a great theoretical importance but it can also help to answer practically relevant questions about improving the performance of these methods. As we show, the question about the convergence behavior leads to complicated nonlinear problems. Despite intense research efforts, these problems are not well understood in some cases. The goal of this survey is to summarize known convergence results for three well-known Krylov subspace methods (CG, MINRES and GMRES) and to formulate open questions in this area.

Recent computational developments in Krylov subspace methods for linear systems

“…Note that this estimate does not take into account the action of r 0 on the matrix polynomial, therefore it is not sharp in most cases. For detailed studies on worst-case convergence of certain methods applied to some problems or for particular right-hand sides, see [107], [103], [216], [217], [218], [240], [241], [325], [349].…”

P-CSI v1.0, an accelerated barotropic solver for the high-resolution ocean model component in the Community Earth System Model v2.0

“…Both PCG and ChronGear have the same theoretical convergence rate because they are different implementations of the same numerical algorithm (D'Azevedo et al, 1999). Their relative residual in the kth iteration has an upper bound as follows (Liesen and Tichý, 2004):…”

Using the preconditioned Generalized Minimum RESidual (GMRES) method to solve the sea‐ice momentum equation

“…Preconditioning is a technique for accelerating the convergence rate of an iterative method by solving a transformed system that has the same solution as the original problem but which is easier to solve. The convergence rate depends on the spectrum of the transformed system [Liesen and Tichy, 2004]. Typically, a preconditioner reduces the number of distinct eigenvalues of the transformed system and clusters them together.…”

“…Indeed, the Krylov subspace forms a basis for the space containing these polynomials. The GMRES iteration computes the optimal coefficients for this polynomial in order to minimize the residual [Liesen and Tichy, 2004]. In this sense, GMRES works on all the eigenvalues simultaneously.…”

“…Instead, only the results of matrix-vector multiplications are required. Second, when the system is suitably preconditioned, Krylov methods exhibit very good convergence properties [Liesen and Tichy, 2004]. Finally, these methods are well-suited for parallelization [e.g., da Cunha and Hopkins, 1994;DeLong and Ortega, 1998].…”