Stefano Serra Capizzano

Convergence analysis of two-grid methods for elliptic Toeplitz and PDEs Matrix-sequences

Summary. The solution of large Toeplitz systems with nonnegative generating functions by multigrid methods was proposed in previous papers [13,14,22]. The technique was modified in [6,36] and a rigorous proof of convergence of the TGM (two-grid method) was given in the special case where the generating function has only a zero at

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

x^0=0

Spectral behavior of matrix sequences and discretized boundary value problems

of order at most two. Here, by extending the latter approach, we perform a complete analysis of convergence of the TGM under the sole assumption that f is nonnegative and with a zero at

Spectral and structural analysis of high precision finite difference matrices for elliptic operators

x^0=0

AN ERGODIC THEOREM FOR CLASSES OF PRECONDITIONED MATRICES

of finite order. An extension of the same analysis in the multilevel case and in the case of finite difference matrix sequences discretizing elliptic PDEs with nonconstant coefficients and of any order is then discussed.

Matrix algebra preconditioners for multilevel Toeplitz matrices are not superlinear

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.

Analysis of preconditioning strategies for collocation linear systems

Abstract In this paper we provide theoretical tools for dealing with the spectral properties of general sequences of matrices of increasing dimension. More specifically, we give a unified treatment of notions such as distribution, equal distribution, localization, equal localization, clustering and sub-clustering. As a case study we consider the matrix sequences arising from the finite difference (FD) discretization of elliptic and semielliptic boundary value problems (BVPs). The spectral analysis is then extended to Toeplitz-based preconditioned matrix sequences with special attention to the case where the coefficients of the differential operator are not regular (belong to L 1 ) and to the case of multidimensional problems. The related clustering properties allow the establishment of some ergodic formulas for the eigenvalues of the preconditioned matrices.

Optimal multilevel matrix algebra operators

Abstract In this paper we study the structural properties of matrices coming from high-precision Finite Difference (FD) formulae, when discretizing elliptic (or semielliptic) differential operators L(a,u) of the form (−) k d k d x k a(x) d k d x k u(x) . Strong relationships with Toeplitz structures and Linear Positive Operators (LPO) are highlighted. These results allow one to give a detailed analysis of the eigenvalues localisation/distribution of the arising matrices. The obtained spectral analysis is then used to define optimal Toeplitz preconditioners in a very compact and natural way and, in addition, to prove Szego-like and Widom-like ergodic theorems for the spectra of the related preconditioned matrices. A wide numerical experimentation, confirming the theoretical results, is also reported.

Korovkin theorems and linear positive Gram matrix algebra approximations of Toeplitz matrices

Abstract We consider abstract classes of matrices { A } satisfying some structural conditions and, in particular, satisfying a crucial assumption about the asymptotic distribution of eigenvalues. We prove a similar distribution property for classes of preconditioned matrices constructed by using representants of { A }. As a particular case, this result applies to preconditioned matrices coming from several important contexts: Finite Differences and Faedo-Ritz-Galerkin linear systems associated with elliptic and semielliptic boundary value problems, very general Hermitian Toeplitz structures generated by multivariate Ll functions. This result answers in the positive some structural questions raised by Tyrtyshnikov [E. Tyrtyshnikov, Linear Algebra Appl. 207 (1994) 225–249] and by the author [S. Serra, Linear Algebra Appl. 267 (1997) 139–161; S. Serra, SIAM J. Numer. Anal., in press] in the Toeplitz context.

Superlinear Preconditioners for Finite Differences Linear Systems

Let f be a d-variate 2π periodic continuous function and let {Tn(f )}n, n = (n1 ,...,n d ), be the multiindexed sequence of multilevel N × N Toeplitz matrices (N = N(n) = � i ni ) generated by f. Let A ={ AN }N be a sequence of matrix algebras simultaneously diagonalized by unitary transforms. We show that there exist infinitely many linearly independent trigonometric polynomials (and continuous nonpolynomial functions) f such that rank� (Tn(f ) − PN )/ = o(N (n) � d=1 n −1 i ) for any matrix sequence P ={ PN }∈ A. This implies that no superlinear matrix algebra preconditioner exists in the multilevel Toeplitz case. The above mentioned result improves the analysis of the author and E. Tyrtyshnikov [SIAM J. Matrix Anal. Appl. 21 (2) (1999) 431] where the same was proved under the assumption that the involved algebras are of circulant type.