V-cycle optimal convergence for DCT-III matrices

The paper analyzes a two-grid and a multigrid method for matrices belonging to the DCT-III algebra and generated by a polynomial symbol. The aim is to prove that the convergence rate of the considered multigrid method (V-cycle) is constant independent of the size of the given matrix. Numerical examples from differential and integral equations are considered to illustrate the claimed convergence properties.Comment: 19 page

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

Multigrid for Qk finite element matrices using a (block) Toeplitz symbol approach

In the present paper, we consider multigrid strategies for the resolution of linear systems arising from the Qk Finite Elements approximation of one-and higher-dimensional elliptic partial differential equations with Dirichlet boundary conditions and where the operator is div (-a(x) 07\u2022), with a continuous and positive over \u3a9, \u3a9 being an open and bounded subset of R2. While the analysis is performed in one dimension, the numerics are carried out also in higher dimension d 65 2, showing an optimal behavior in terms of the dependency on the matrix size and a substantial robustness with respect to the dimensionality d and to the polynomial degree k

Positive representation formulas for finite difference discretizations of (elliptic) second order PDEs

In the present paper we are interested in the Finite Difference (FD) discretization of elliptic second order PDEs of the form $$-\sum_{i,j=1}^d {\frac{\partial }{\partial x_i}} \left(a_{i,j}(x) {\frac{\partial }{\partial x_j}} u(x)\right)=b(x)$$ over $(0,1)^d$ with Dirichlet boundary conditions. The resulting sequence of algebraic systems is characterized by matrices expressible in terms of weighted sums of dyads. The considered representation formulas are then used in order to find deep relationships with Generalized Locally Toeplitz sequences \cite{tilliloc2} and matrix-valued Linear Positive Operators \cite{Sergo}. As a direct consequence we obtain a complete understanding of the distributional spectral properties of these FD matrix sequences, that are used in order to devise an efficient numerical (iterative) solution of the associated linear systems

A non-ideal CSTR: A high codimension bifurcation analysis

none2Pellegrini, LAURA ANNAMARIA; Tablino Possio, C.Pellegrini, LAURA ANNAMARIA; C., Tablino Possi

Multigrid methods for multilevel circulant matrices

We introduce a multigrid technique for the solution of multilevel circulant linear systems whose coefficient matrix has eigenvalues of the form $f(x_j^{[n]})$, where $f$ is continuous and independent of $n=(n_1,\ldots,n_d)$, and $x_j^{[n]} \equiv 2\pi j/n = (2\pi j_1/n_1, \ldots, 2\pi j_d/n_d)$, $0 \le j_r \le n_r - 1$. The interest of the proposed technique pertains to the multilevel banded case, where the total cost is optimal, i.e., $O(N)$ arithmetic operations (ops), $N=\prod_{r=1}^d n_r$, instead of $O(N\log N)$ ops arising from the use of FFTs. In fact, multilevel banded circulants are used as preconditioners for elliptic and parabolic PDEs (with Dirichlet or periodic boundary conditions) and for some two-dimensional image restoration problems where the point spread function (PSF) is numerically banded, so that the overall cost is reduced from $O(k(\varepsilon,n)N \log N)$ to $O(k(\varepsilon,n)N)$, where $k(\varepsilon,n)$ is the number of PCG iterations to reach the solution within an accuracy of $\varepsilon$. Several numerical experiments concerning one-rank regularized circulant discretization of elliptic $2q$-differential operators over one-dimensional and two-dimensional square domains with mixed boundary conditions are performed and discussed