Serra-Capizzano recently introduced anti-reflecting boundary conditions (AR-BC) for blurring models: the idea seems promising both from the computational and approximation viewpoint. The key point is that, under certain symmetry conditions, the AR-BC matrices can be essentially simultaneously diagonalized by the (fast) sine transform DST I and, moreover, a C1 continuity at the border is guaranteed in the 1D case. Here we give more details for the 2D case and we perform extensive numerical simulations which illustrate that the AR-BC can be superior to Dirichlet, periodic and reflective BCs in certain applications.
We briefly describe a multigrid strategy for unilevel and two-level linear systems whose coefficient matrix An belongs either to the Toeplitz class or to the cosine algebra of type III and
such that An can be naturally associated, in the spectral sense, with a polynomial function f. The interest of the technique is due to its optimal cost of O(N) arithmetic operations, where N is the size of the algebraic problem. We remark that these structures arise in certain 2D image restoration problems or can be used as preconditioners
for more complicated image restoration problems.
Optimal preconditioners are of paramount importance for cg-like methods since they make them converge superlinearly. In preceding papers, we proved that any preconditioner belonging to partially equimodular spaces is not optimal for multilevel Toeplitz matrices where the aforementioned class of spaces includes all the known and used trigonometric matrix algebras. Here we survey and refine these results by focusing our attention on the more difficult case in which the multilevel Toeplitz matrices are Hermitian.
Let {An(f)} be a sequence of nested n X n Toeplitz matrices generated by a Lebesgue integrable real-function f defined on (-(pi) , (pi) ). In this paper, we first present some results about the spectral properties of An(f) (density, range, behavior of the extreme eigenvalues etc.), then we apply these results to the preconditioning problem. We analyze in detail the preconditioned conjugate gradient method, where the proposed poreconditioners An1(g)An(f): we obtain new results about the range, the density and the extremal properties of their spectra. In particular we deal with the critical case where the matrices An(f) are asymptotically ill-conditioned, i.e., zero belongs to the convex hull of the essential range of f. We consider positive definite Toeplitz linear systems (f >= 0), nondefinite Toeplitz linear systems (f with nondefinite sign), with zeros of generic orders. Moreover, these analyses and techniques are partially extended to the block case, too.
Conference Committee Involvement (1)
Advanced Signal Processing Algorithms, Architectures, and Implementations XIII
6 August 2003 | San Diego, California, United States
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