In this paper we address the solution of linear systems of equations with multiple right-hand sides given at once. When the dimension of the problem is known to be large, preconditioned block Krylov subspace methods are usually considered as the method of choice. Nevertheless to be effective in terms of computational operations it is known that these methods must incorporate a strategy for dete...

We introduce a conservative fixup strategy to remedy negative fluxes within a linear discontinuous spatially discretized radiation transport solver. The strategy is similar in principle to the classical setto-zero fixup used with diamond difference schemes. We also discuss difficulties in creating an effective solver for such an approach and propose a hybrid source iteration/Krylov solver. Nume...

We present a Krylov-W-code ROWMAP for the integration of stii initial value problems. It is based on the ROW-methods of the code ROS4 of Hairer and Wanner and uses Krylov techniques for the solution of linear systems. A special multiple Arnoldi process ensures order p = 4 already for fairly low dimensions of the Krylov subspaces independently of the dimension of the diierential equations. Numer...

There are many examples where non-orthogonality of a basis for Krylov subspace methods arises naturally. These methods usually require less storage or computational effort per iteration than methods using an orthonormal basis (optimal methods), but the convergence may be delayed. Truncated Krylov subspace methods and other examples of non-optimal methods have been shown to converge in many situ...

Recent computational and theoretical studies have shown that the matrix-vector product occurring at each step of a Krylov subspace method can be relaxed as the iterations proceed, i.e., it can be computed in a less exact manner, without degradation of the overall performance. In the present paper a general operator treatment of this phenomenon is provided and a new result further explaining its...

The action of the matrix exponential and related φ functions on vectors plays an important role in the application of exponential integrators to ordinary differential equations. For the efficient evaluation of linear combinations of such actions we consider a new Krylov subspace algorithm. By employing Cauchy’s integral formula an error representation of the numerical approximation is given. Th...

By parameterizing all reduced order models matching some of the first characteristic parameters (moments or Markov parameters or both) of the original system, the poles of the reduced system are prescribed as a key to find a stable reduced order system. Two procedures based on Lanczos and Arnoldi algorithms are proposed to approximate the dominant poles of the original system and to choose them...

Based on the general inner-outer (GIO) iteration method [5,34] and framework [6], we present a multi-step matrix splitting (GMMS) for computing PageRank, analyze its overall convergence property. Moreover, same idea can be used as preconditioning technique accelerating Krylov subspace methods, such GMRES method. Finally, several numerical examples are given to illustrate effectiveness of propos...

In this paper, we present an effective preconditioning technique for solving nonsymmetric saddle-point problems. In particular, we consider those saddlepoint problems that arise in the numerical simulation of particulate flows—flow of solid particles in incompressible fluids, using mixed finite element discretization of the Navier–Stokes equations. These indefinite linear systems are solved usi...

We study methods for the numerical solution of the Helmholtz equation for twodimensional applications in geophysics. The common framework of the iterative methods in our study is a combination of an inner iteration with a geometric multigrid method used as a preconditioner and an outer iteration with a Krylov subspace method. The preconditioning system is based on either a pure or shifted Helmh...