20 research outputs found
Truncated decompositions and filtering methods with Reflective/Anti-Reflective boundary conditions: a comparison
The paper analyzes and compares some spectral filtering methods as truncated
singular/eigen-value decompositions and Tikhonov/Re-blurring regularizations in
the case of the recently proposed Reflective [M.K. Ng, R.H. Chan, and W.C.
Tang, A fast algorithm for deblurring models with Neumann boundary conditions,
SIAM J. Sci. Comput., 21 (1999), no. 3, pp.851-866] and Anti-Reflective [S.
Serra Capizzano, A note on anti-reflective boundary conditions and fast
deblurring models, SIAM J. Sci. Comput., 25-3 (2003), pp. 1307-1325] boundary
conditions. We give numerical evidence to the fact that spectral decompositions
(SDs) provide a good image restoration quality and this is true in particular
for the Anti-Reflective SD, despite the loss of orthogonality in the associated
transform. The related computational cost is comparable with previously known
spectral decompositions, and results substantially lower than the singular
value decomposition. The model extension to the cross-channel blurring
phenomenon of color images is also considered and the related spectral
filtering methods are suitably adapted.Comment: 22 pages, 10 figure
PRECONDITIONING STRATEGIES FOR 2D FINITE DIFFERENCE MATRIX SEQUENCES
In this paper we are concerned with the spectral analysis of the sequence of preconditioned matrices {P-1 n An(a, m1, m2, k)}n, where n = (n1, n2), N(n) = n1n2 and where An (a, m1, m2, k) 08 \u211dN(n)
7 N(n) is the symmetric two-level matrix coming from a high-order Finite Difference (FD) discretization of the problem (equation presented) with \u3bd denoting the unit outward normal direction and where m1 and m2 are parameters identifying the precision order of the used FD schemes. We assume that the coefficient a(x, y) is nonnegative and that the set of the possible zeros can be represented by a finite collection of curves. The proposed preconditioning matrix sequences correspond to two different choices: the Toeplitz sequence {An(1, m1, m2, k)}n and a Toeplitz based sequence that adds to the Toeplitz structure the informative content given by the suitable scaled diagonal part of An(a, m1, m2 k). The former case gives rise to optimal preconditioning sequences under the assumption of positivity and boundedness of a. With respect to the latter, the main result is the proof of the asymptotic clustering at unity of the eigenvalues of the preconditioned matrices, where the "strength" of the cluster depends on the order k, on the regularity features of a(x, y) and on the presence of zeros of a(x, y)
Multigrid preconditioners for symmetric Sinc systems
The symmetric Sinc-Galerkin method applied to a separable second-order self-adjoint elliptic boundary value problem gives rise to a system of linear equations ( ? x ?D y + D x ?? y ) u = g where ? is the Kronecker product symbol, ? x and ? y are Toeplitz-plus-diagonal matrices, and D x and D y are diagonal matrices. The main contribution of this paper is to present a two-step preconditioning strategy based on the banded matrix approximation and the multigrid iteration for these Sinc-Galerkin systems. Numerical examples show that the multigrid preconditioner is practical and efficient to precondition the conjugate gradient method for solving the above symmetric Sinc-Galerkin linear system
Sistemi dinamici non lineari controllati
Dottorato di ricerca in matematica computazionale e ricerca operativa. 8. ciclo. Supervisori S. Albertoni e G. Biardi. Coordinatore V. CapassoConsiglio Nazionale delle Ricerche - Biblioteca Centrale - P.le Aldo Moro, 7, Rome; Biblioteca Nazionale Centrale - P.za Cavalleggeri, 1, Florence / CNR - Consiglio Nazionale delle RichercheSIGLEITItal
Clustering/Distribution Analysis and Preconditioned Krylov Solvers for the Approximated Helmholtz Equation and Fractional Laplacian in the Case of Complex-Valued, Unbounded Variable Coefficient Wave Number μ
We consider the Helmholtz equation and the fractional Laplacian in the case of the complex-valued unbounded variable coefficient wave number , approximated by finite differences. In a recent analysis, singular value clustering and eigenvalue clustering have been proposed for a preconditioning when the variable coefficient wave number is uniformly bounded. Here, we extend the analysis to the unbounded case by focusing on the case of a power singularity. Several numerical experiments concerning the spectral behavior and convergence of the related preconditioned GMRES are presented