Abstract: This paper is devoted to the numerical solution of
large-scale linear ill-posed systems. A multilevel regularization
method is proposed. This method is based on a synthesis of
the Arnoldi-Tikhonov regularization technique and the multilevel
technique. We show that if the Arnoldi-Tikhonov method is
a regularization method, then the multilevel method is also a
regularization one. Numerical experiments presented in this paper
illustrate the effectiveness of the proposed method.
Abstract: In this paper we study numerical methods for solving Sylvester matrix equations of the form AX +XBT +CDT = 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 projection subspaces is outlined. We show that the approximate solution is an exact solution of a perturbed Sylvester matrix equation. Moreover, exact expression for the norm of residual is derived and results on finite termination and convergence are presented. Some numerical examples are presented to illustrate the effectiveness of the proposed method.
Abstract: In the present work, we propose a new method for
solving the matrix equation AXB=F . The new method can
be considered as a generalized form of the well-known global full
orthogonalization method (Gl-FOM) for solving multiple linear
systems. Hence, the method will be called extended Gl-FOM (EGl-
FOM). For implementing EGl-FOM, generalized forms of block
Krylov subspace and global Arnoldi process are presented. Finally,
some numerical experiments are given to illustrate the efficiency of
our new method.
Abstract: This article is devoted to the numerical solution of
large-scale quadratic eigenvalue problems. Such problems arise in
a wide variety of applications, such as the dynamic analysis of
structural mechanical systems, acoustic systems, fluid mechanics,
and signal processing. We first introduce a generalized second-order
Krylov subspace based on a pair of square matrices and two initial
vectors and present a generalized second-order Arnoldi process for
constructing an orthonormal basis of the generalized second-order
Krylov subspace. Then, by using the projection technique and the
refined projection technique, we propose a restarted generalized
second-order Arnoldi method and a restarted refined generalized
second-order Arnoldi method for computing some eigenpairs of largescale
quadratic eigenvalue problems. Some theoretical results are also
presented. Some numerical examples are presented to illustrate the
effectiveness of the proposed methods.
Abstract: In the present work, we propose a new projection method for solving the matrix equation AXB = F. For implementing our new method, generalized forms of block Krylov subspace and global Arnoldi process are presented. The new method can be considered as an extended form of the well-known global generalized minimum residual (Gl-GMRES) method for solving multiple linear systems and it will be called as the extended Gl-GMRES (EGl- GMRES). Some new theoretical results have been established for proposed method by employing Schur complement. Finally, some numerical results are given to illustrate the efficiency of our new method.
Abstract: Restarted GMRES methods augmented with approximate eigenvectors are widely used for solving large sparse linear systems. Recently a new scheme of augmenting with error approximations is proposed. The main aim of this paper is to develop a restarted GMRES method augmented with the combination of harmonic Ritz vectors and error approximations. We demonstrate that the resulted combination method can gain the advantages of two approaches: (i) effectively deflate the small eigenvalues in magnitude that may hamper the convergence of the method and (ii) partially recover the global optimality lost due to restarting. The effectiveness and efficiency of the new method are demonstrated through various numerical examples.