Iterative Methods Based on Krylov Subspaces
نویسنده
چکیده
with a symmetric and positive definite operator A defined on a Hilbert space V with dimV = N and its preconditioned version PCG. We derive the CG from the projection in A-inner product point of view. We shall use (·, ·) for the standard inner product and (·, ·)A for the inner product introduced by the SPD operator A. When we say ‘orthogonal’ vectors, we refer to the default (·, ·) inner product and emphasize the orthogonality in (·, ·)A by ‘A-orthogonal’.
منابع مشابه
New variants of the global Krylov type methods for linear systems with multiple right-hand sides arising in elliptic PDEs
In this paper, we present new variants of global bi-conjugate gradient (Gl-BiCG) and global bi-conjugate residual (Gl-BiCR) methods for solving nonsymmetric linear systems with multiple right-hand sides. These methods are based on global oblique projections of the initial residual onto a matrix Krylov subspace. It is shown that these new algorithms converge faster and more smoothly than the Gl-...
متن کاملTime stepping free numerical solution of linear differential equations: Krylov subspace versus waveform relaxation
The aim of this paper is two-fold. First, we propose an efficient implementation of the continuous time waveform relaxation method based on block Krylov subspaces. Second, we compare this new implementation against Krylov subspace methods combined with the shift and invert technique.
متن کاملA Projection Method Based on Extended Krylov Subspaces for Solving Sylvester Equations
In this paper we study numerical methods for solving Sylvester matrix equations of the form AX +XB +CD = 0. A new projection method is proposed. The union of Krylov subspaces in A and its inverse and the union of Krylov subspaces in B and its inverse are used as the right and left projection subspaces, respectively. The Arnoldi-like process for constructing the orthonormal basis of the projecti...
متن کاملDeflated and Augmented Krylov Subspace Techniques
We present a general framework for a number of techniques based on projection methods onàugmented Krylov subspaces'. These methods include the deeated GM-RES algorithm, an inner-outer FGMRES iteration algorithm, and the class of block Krylov methods. Augmented Krylov subspace methods often show a signiicant improvement in convergence rate when compared with their standard counterparts using the...
متن کاملIfeast
The FEAST eigenvalue algorithm is a subspace iteration algorithm that uses contour integration in the complex plane to obtain the eigenvectors of a matrix for the eigenvalues that are located in any user-defined search interval. By computing small numbers of eigenvalues in specific regions of the complex plane, FEAST is able to naturally parallelize the solution of eigenvalue problems by solvin...
متن کاملIntroduction to Krylov Subspace Methods in Model Order Reduction
In recent years, Krylov subspace methods have become popular tools for computing reduced order models of high order linear time invariant systems. The reduction can be done by applying a projection from high order to lower order space using the bases of some subspaces called input and output Krylov subspaces. The aim of this paper is to give an introduction into the principles of Krylov subspac...
متن کامل