In this paper, we consider the solution of ordinary diierential equations using boundary value methods. These methods require the solutions of one or more unsymmetric, large and sparse linear systems. Krylov subspace methods with the Strang block-circulant preconditioners are proposed for solving these linear systems. We prove that our preconditioners are invertible and all the eigenvalues of t...

In this paper we discuss preconditioners for the incompressible Navier-Stokes equations. In combination with Krylov subspace methods, they give a fast convergence for the solution of the Navier -Stokes equations. With the help of numerical experiments, we report some new findings regarding the convergence of these preconditioners. Besides that, a renumbering scheme for direct solvers and ILU pr...

In this note we present a theoretical analysis of some Krylov subspace approximations to the matrix exponential operation exp(A)v and establish a priori and a posteriori error estimates. Several such approximations are considered. The main idea of these techniques is to approximately project the exponential operator onto a small Krylov subspace and carry out the resulting small exponential matr...

We present low complexity, quickly converging robust adaptive beamformers that combine robust Capon beamformer (RCB) methods and data-adaptive Krylov subspace dimensionality reduction techniques. We extend a recently proposed reduced-dimension RCB framework, which ensures proper combination of RCBs with any form of dimensionality reduction that can be expressed using a full-rank dimension reduc...

The aim of this paper is two-fold. First, we propose an efficient implementation of the continuous time waveform relaxation method based on block Krylov subspaces. Second, we compare this new implementation against Krylov subspace methods combined with the shift and invert technique.

We propose a time-exact Krylov-subspace-based method for solving linear ordinary differential equation systems of the form y D Ay C g.t/ and y D Ay C g.t/, where y.t/ is the unknown function. The method consists of two stages. The first stage is an accurate piecewise polynomial approximation of the source term g.t/, constructed with the help of the truncated singular value decomposition. The se...

The exponential time integration, i.e., time integrationwhich involves thematrix exponential, is an attractive tool for time domain modeling involving Maxwell’s equations. However, its application in practice often requires a substantial knowledge of numerical linear algebra algorithms, such as Krylov subspace methods. In this note we discuss exponential Krylov subspace time integrationmethods ...

IDR(s) is one of the most efficient methods for solving large sparse nonsymmetric linear systems of equations. We present two useful extensions of IDR(s), namely a flexible variant and a multi-shift variant. The algorithms exploit the underlying Hessenberg decomposition computed by IDR(s) to generate basis vectors for the Krylov subspace. The approximate solution vectors are computed using a Qu...

We analyze the convergence of an extended Krylov subspace method for the approximation of operator functions that appear in exponential integrators. For operators, the size of the polynomial part of the extended Krylov subspace is restricted according to the smoothness of the initial data. This restriction for the continuous operator has a significant influence on the approximation of matrix fu...