Marco Donatelli — Page 3

arXiv2009

Fast transforms for high order boundary conditions

We study strategies for increasing the precision in the blurring models by maintaining a complexity in the related numerical linear algebra procedures (matrix-vector product, linear system solution, computation of eigenvalues etc.) of the same order of the celebrated Fast Fourier Transform. The key idea is the choice of a suitable functional basis for representing signals and images. Starting from an analysis of the spectral decomposition of blurring matrices associated to the antireflective boundary conditions introduced in [S. Serra Capizzano, SIAM J. Sci. Comput. 25-3 pp. 1307--1325], we extend the model for preserving polynomials of higher degree and fast computations also in the nonsymmetric case. We apply the proposed model to Tikhonov regularization with smoothing norms and the generalized cross validation for choosing the regularization parameter. A selection of numerical experiments shows the effectiveness of the proposed techniques.

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arXiv2008

A note on grid transfer operators for multigrid methods

The Local Fourier analysis (LFA) is a classic tool to prove convergence theorems for multigrid methods (MGMs). In particular, we are interested in optimality that is a convergence speed independent of the size of the involved matrices. For elliptic partial differential equations (PDEs), a well known optimality result requires that the sum of the orders of the grid transfer operators is not lower than the order of the PDE to solve. Analogously, when dealing with MGMs for Toeplitz matrices in the literature an optimality condition on the position and on the order of the zeros of the symbols of the grid transfer operators has been found. In this work we show that in the case of elliptic PDEs with constant coefficients, the two different approaches lead to an equivalent condition. We argue that the analysis for Toeplitz matrices is an algebraic generalization of the LFA, which allows to deal not only with differential problems but also for instance with integral problems. The equivalence of the two approaches gives the possibility of using grid transfer operators with different orders also for MGMs for Toeplitz matrices. We give also a class of grid transfer operators related to the B-spline's refinement equation and we study their geometric properties. This analysis suggests further links between wavelets and multigrid methods. A numerical experimentation confirms the correctness of the proposed analysis.

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