Daniele Bertaccini

Preconditioned HSS methods for the solution of non-Hermitian positive definite linear systems and applications to the discrete convection-diffusion equation

Summary.We study the role of preconditioning strategies recently developed for coercive problems in connection with a two-step iterative method, based on the Hermitian skew-Hermitian splitting (HSS) of the coefficient matrix, proposed by Bai, Golub and Ng for the solution of nonsymmetric linear systems whose real part is coercive. As a model problem we consider Finite Differences (FD) matrix sequences {An(a,p)}n discretizing the elliptic (convection-diffusion) problem with Ω being a plurirectangle of Rd with a(x) being a uniformly positive function and p(x) denoting the Reynolds function: here for plurirectangle we mean a connected union of rectangles in d dimensions with edges parallel to the axes. More precisely, in connection with preconditioned HSS/GMRES like methods, we consider the preconditioning sequence {Pn(a)}n, Pn(a):= Dn1/2(a)An(1,0) Dn1/2(a) where Dn(a) is the suitably scaled main diagonal of An(a,0). If a(x) is positive and regular enough, then the preconditioned sequence shows a strong clustering at unity so that the sequence {Pn(a)}n turns out to be a superlinear preconditioning sequence for {An(a,0)}n where An(a,0) represents a good approximation of Re(An(a,p)) namely the real part of An(a,p). The computational interest is due to the fact that the preconditioned HSS method has a convergence behavior depending on the spectral properties of {Pn-1(a)Re(An(a,p))}n≈ {Pn-1(a)An(a,0)}n: therefore the solution of a linear system with coefficient matrix An(a,p) is reduced to computations involving diagonals and to the use of fast Poisson solvers for {An(1,0)}n.Some numerical experimentations confirm the optimality of the discussed proposal and its superiority with respect to existing techniques.

Approximate inverse preconditioning for shifted linear systems

In this paper we consider the problem of preconditioning symmetric positive definite matrices of the form Aα=A+αI where α>0. We discuss how to cheaply modify an existing sparse approximate inverse preconditioner for A in order to obtain a preconditioner for Aα. Numerical experiments illustrating the performance of the proposed approaches are presented.

A Circulant Preconditioner for the Systems of LMF-Based ODE Codes

In this paper, a recently introduced block circulant preconditioner for the linear systems of the codes for ordinary differential equations (ODEs) is investigated. Most ODE codes based on implicit formulas, at each integration step, need the solution of one or more unsymmetric linear systems that are often large and sparse. Here, the boundary value methods, a class of implicit methods for the numerical integration of ODEs based on linear multistep formulas, are considered more in detail for initial value problems. Theoretical and practical arguments are given to show that the block circulant preconditioner can give fast preconditioned iterations for various classes of differential problems. Moreover, the P-circulants, a recently introduced circulant approximation for unsymmetric Toeplitz matrices, are shown to be more suitable sometimes than other circulant matrices for the underlying block preconditioner.

Nonsymmetric Preconditioner Updates in Newton-Krylov Methods for Nonlinear Systems

Newton-Krylov methods, a combination of Newton-like methods and Krylov subspace methods for solving the Newton equations, often need adequate preconditioning in order to be successful. Approximations of the Jacobian matrices are required to form preconditioners, and this step is very often the dominant cost of Newton-Krylov methods. Therefore, working with preconditioners may destroy the “Jacobian-free” (or matrix-free) setting where the single Jacobian-vector product can be provided without forming and storing the element of the true Jacobian. In this paper, we propose and analyze a preconditioning technique for sequences of nonsymmetric Jacobian matrices based on the update of an earlier preconditioner. The proposed strategy can be implemented in a matrix-free manner. Numerical experiments on popular test problems confirm the effectiveness of the approach in comparison with the standard ILU-preconditioned Newton-Krylov approaches.

Skew-Circulant Preconditioners for Systems of LMF-Based ODE Codes

We consider the solution of ordinary differential equations (ODEs) using implicit linear multistep formulae (LMF). More precisely, here we consider Boundary Value Methods. These methods require the solution of one or more unsymmetric, large and sparse linear systems. I n [6], Chan et al. proposed using Strang block-circulant preconditioners for solving these linear systems. However, as observed in [1], Strang preconditioners can be often ill-conditioned or singular even when the given system is well-conditioned. In this paper, we propose a nonsingular skew-circulant preconditioner for systems of LMF-based ODE codes. Numerical results are given to illustrate the effectiveness of our method.

Reliable preconditioned iterative linear solvers for some numerical integrators

Implicit time-step numerical integrators for ordinary and evolutionary partial di erential equations need, at each step, the solution of linear algebraic equations that are unsymmetric and often large and sparse. Recently, a block preconditioner based on circulant approximations for the linear systems arising in the boundary value methods (BVMs) was introduced by the author. Here, some circulant approximations are compared and a further new type is considered. Numerical experiments are presented to check the e ectiveness of the various approximations that can be used in the underlying block preconditioner. Copyright ? 2001 John Wiley & Sons, Ltd.

THE CONVERGENCE RATE OF BLOCK PRECONDITIONED SYSTEMS ARISING FROM LMF-BASED ODE CODES ∗

AbstractThe solution of ordinary an partial differential equations using implicit linear multi-step formulas (LMF)is considered. More precisely, boundary value methods (BVMs), a class of methods based on implicit formulas will be taken into account in this paper. These methods require the solution of large and sparse linear systems

Spectral Analysis of a Preconditioned Iterative Method for the Convection‐Diffusion Equation

\hat M