New convergence results on the global GMRES method for diagonalizable matrices

In the present paper, we give some new convergence results of the global GMRES method for multiple linear systems. In the case where the coefficient matrix A is diagonalizable, we derive new upper bounds for the Frobenius norm of the residual. We also consider the case of normal matrices and we propose new expressions for the norm of the residual.     

Sylvester Tikhonov-regularization methods in image restoration

In this paper, we consider large-scale linear discrete ill-posed problems where the right-hand side contains noise. Regularization techniques such as Tikhonov regularization are needed to control the effect of the noise on the solution. In many applications such as in image restoration the coefficient matrix is given as a Kronecker product of two matrices and then Tikhonov regularization problem leads to the generalized Sylvester matrix equation. For large-scale problems, we use the global-GMRES method which is an orthogonal projection method onto a matrix Krylov subspace. We present some theoretical results and give numerical tests in image restoration.     

Stein implicit Runge-Kutta methods with high stage order for large-scale ordinary differential equations

The Runge-Kutta method is one of the most popular implicit methods for the solution of stiff ordinary differential equations. For large problems, the main drawback of such methods is the cost required at each integration step for computing the solution of a nonlinear system of equations. In this paper, we propose to reduce the cost of the computation by transforming the linear systems arising in the application of Newton's method to Stein matrix equations. We propose an iterative projection method onto block Krylov subspaces for solving numerically such Stein matrix equations. Numerical examples are given to illustrate the performance of our proposed method.     

A preconditioned block Arnoldi method for large scale Lyapunov and algebraic Riccati equations

In the present paper, we propose a preconditioned Newton–Block Arnoldi method for solving large continuous time algebraic Riccati equations. Such equations appear in control theory, model reduction, circuit simulation amongst other problems. At each step of the Newton process, we solve a large Lyapunov matrix equation with a low rank right hand side. These equations are solved by using the block Arnoldi process associated with a preconditioner based on the alternating direction implicit iteration method. We give some theoretical results and report numerical tests to show the effectiveness of the proposed approach.