GMRES is one of the most popular iterative methods for the solution of large linear systems of equations. However, GMRES generally does not perform well when applied to the solution of linear systems of equations that arise from the discretization of linear ill-posed problems with error-contaminated data represented by the right-hand side. Such linear systems are commonly referred to as linear ...

Abstract. This paper studies admissible convergence curves for restarted GMRES and their relation to the curves for full GMRES. It shows that stagnation at the end of a restart cycle is mirrored at the beginning of the next cycle. Otherwise, any non-increasing convergence curve is possible and pairs {A, b} are constructed such that when restarted GMRES is applied to Ax = b, prescribed residual ...

The Conjugate Gradient method (CG), the Minimal Residual method (MINRES), or more generally, the Generalized Minimal Residual method (GMRES) are widely used to solve a linear system Ax = b. The choice of a method depends on A’s symmetry property and/or definiteness), and MINRES is really just a special case of GMRES. This paper establishes error bounds on and sometimes exact expressions for res...

We study how the Newton-GMRES iteration can enable dynamic simulators (time-steppers) to perform fixed-point and path-following computations. For a class of dissipative problems, whose dynamics are characterized by a slow manifold, the Jacobian matrices in such computations are compact perturbations of the identity. We examine the number of GMRES iterations required for each nonlinear iteration...

ML(n)BiCGStab is a Krylov subspace method for the solution of large, sparse and non-symmetric linear systems. In theory, it is a method that lies between the well-known BiCGStab and GMRES/FOM. In fact, when n = 1, ML(1)BiCGStab is BiCGStab and when n = N, ML(N)BiCGStab is GMRES/FOM where N is the size of the linear system. Therefore, ML(n)BiCGStab is a bridge that connects the Lanczos-based BiC...