We survey Krylov subspace methods for the solution of linear systems with focus on commonly used and recently developed methods. The approach is theoretical and complementary to the engineering-based first article of this special issue. In particular convergence results are derived from a general theoretical framework, compiled and analyzed. I . B a c k g r o u n d The purpose of this paper is ...

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...

Let B be an unknown linear evolution process on C ' `(Zd) driving an unknown initial state x and producing the states {Bx, ` = 0, 1, . . .} at different time levels. The problem under consideration in this paper is to find as much information as possible about B and x from the measurements Y = {x(i), Bx(i), . . . , Bix(i) : i ∈ Ω ⊂ Z}. If B is a “low-pass” convolution operator, we show that we ...

Krylov subspace methods have proved effective for many non-Hermitian eigenvalue problems, yet the analysis of such algorithms is involved. Convergence can be characterized by the angle the approximating subspace forms with a desired invariant subspace, resulting in a geometric framework that is robust to eigenvalue ill-conditioning. This paper describes a new bound on this angle that handles th...

In this paper, we propose a novel reduced-rank adaptive filtering algorithm by blending the idea of the Krylov subspace methods with the set-theoretic adaptive filtering framework. Unlike the existing Krylov-subspace-based reduced-rank methods, the proposed algorithm tracks the optimal point in the sense of minimizing the ‘true’ mean square error (MSE) in the Krylov subspace, even when the esti...

A unified view of several recently introduced reduced-rank adaptive filters is presented. As all considered methods use Krylov subspace for rank reduction, the approach taken in this work is inspired from Krylov subspace methods for iterative solutions of linear systems. The alternative interpretation so obtained is used to study the properties of each considered technique and to relate one red...

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...

We consider the numerical solution of projected Lyapunov equations using Krylov subspace iterative methods. Such equations play a fundamental role in balanced truncation model reduction of descriptor systems. We present generalizations of the extended block and global Arnoldi methods to projected Lyapunov equations and compare these methods with the alternating direction implicit method with re...

In this paper, we describe tensor methods for large systems of nonlinear equations based on Krylov subspace techniques for approximately solving the linear systems that are required in each tensor iteration. We refer to a method in this class as a tensor-Krylov algorithm. We describe comparative testing for a tensor-Krylov implementation versus an analogous implementation based on a Newton-Kryl...