Convergence analysis of Krylov subspace methods †

Preconditioned Generalized Minimal Residual Method for Solving Fractional Advection-Diffusion Equation

Introduction Fractional differential equations (FDEs)  have  attracted much attention and have been widely used in the fields of finance, physics, image processing, and biology, etc. It is not always possible to find an analytical solution for such equations. The approximate solution or numerical scheme  may be a good approach, particularly, the schemes in numerical linear algebra for solving ...

Krylov Subspace Acceleration of Waveform Relaxation

In this paper we describe and analyze Krylov subspace techniques for accelerating the convergence of waveform relaxation for solving time dependent problems. A new class of accelerated waveform methods, convolution Krylov subspace methods, is presented. In particular, we give convolution variants of the conjugate gradient algorithm and two convolution variants of the GMRES algorithm and analyze...

Nonlinear problems in analysis of Krylov subspace methods

Consider a system of linear algebraic equations Ax = b where A is an n by n real matrix and b a real vector of length n. Unlike in the linear iterative methods based on the idea of splitting of A, the Krylov subspace methods, which are used in computational kernels of various optimization techniques, look for some optimal approximate solution xn in the subspaces Kn(A, b) = span{b, Ab, . . . , A...

Superlinear Convergence of Krylov Subspace Methods for Self-Adjoint Problems in Hilbert Space

On the Superlinear Convergence of Exact and Inexact Krylov Subspace Methods

We present a general analytical model which describes the superlinear convergence of Krylov subspace methods. We take an invariant subspace approach, so that our results apply also to inexact methods, and to non-diagonalizable matrices. Thus, we provide a unified treatment of the superlinear convergence of GMRES, Conjugate Gradients, block versions of these, and inexact subspace methods. Numeri...

On the Convergence of Krylov Subspace Methods for Matrix Mittag-Leffler Functions

In this paper we analyze the convergence of some commonly used Krylov subspace methods for computing the action of matrix Mittag-Leffler functions. As it is well known, such functions find application in the solution of fractional differential equations. We illustrate the theoretical results by some numerical experiments.