In order reduction of high order linear time invariant systems based on two-sided Krylov subspace methods, the Lanczos algorithm is commonly used to find the bases for input and output Krylov subspaces and to calculate the reduced order model by projection. However, this method can be numerically unstable even for systems with moderate number of states and can only find a limited number of bior...

Optimal Krylov subspace methods like GMRES and GCR have to compute an orthogonal basis for the entire Krylov subspace to compute the minimal residual approximation to the solution. Therefore, when the number of iterations becomes large, the amount of work and the storage requirements become excessive. In practice one has to limit the resources. The most obvious ways to do this are to restart GM...

By using the matrix exponential operator, solution of the system can be written as y(t) = exp(−tA)v. Numerical algorithms, which are based on this approach, are called exponential time integration methods. The essential point is that not the matrix exponential itself but rather its action on the vector v is computed. An attractive feature of the formula y(t) = exp(−tA)v is that it provides solu...

Krylov subspace methods are used for solving systems of linear equations Hx+ c = 0. We present a general Krylov subspace method that can be applied to a symmetric system of linear equations, i.e., for a system in which H = H . In each iteration, we have a choice of scaling of the orthogonal vectors that successively make the Krylov subspaces available. We define an extended representation of ea...

Abstract. This paper presents a reformulation of Krylov Subspace Spectral (KSS) Methods, which build on Gene Golub’s many contributions pertaining to moments and Gaussian quadrature, to produce high-order accurate approximate solutions to variable-coefficient time-dependent PDE. This reformulation serves two useful purposes. First, it more clearly illustrates the distinction between KSS methods...

This paper presents a new efficient approach for the solution of the lp-lq minimization problem based on the application of successive orthogonal projections onto generalized Krylov subspaces of increasing dimension. The subspaces are generated according to the iteratively reweighted least-squares strategy for the approximation of lp/lq-norms by weighted l2-norms. Computed image restoration exa...

Abstract The objective of this research is to develop and study iterative methods in the Krylov subspaces for solving systems linear algebraic equations (SLAEs) with non-symmetric sparse matrices high orders arising approximation multi-dimensional boundary value problems on unstructured grids. These are also relevant many applications, including diffusion-convection equations. considered algori...

A couple of generalizations of matrix Krylov subspace methods to tensors are presented. It is shown that a particular variant can be interpreted as a Krylov factorization of the tensor. A generalization to tensors of the Krylov-Schur method for computing matrix eigenvalues is proposed. The methods are intended for the computation of lowrank approximations of large and sparse tensors. A few nume...