In a recent work Ng, Chan, and Tang introduced reflecting (Neumann) boundary conditions (BCs) for blurring models and proved that the resulting choice leads to fast algorithms for both deblurring and detecting the regularization parameters in the presence of noise. The key point is that Neumann BC matrices can be simultaneously diagonalized by the (fast) cosine transform DCT III. Here we propose antireflective BCs that can be related to $\tau$ structures, i.e., to the algebra of the matrices that can be simultaneously diagonalized by the (fast) sine transform DST I. We show that, in the generic case, this is a more natural modeling whose features are (a) a reduced analytical error since the zero (Dirichlet) BCs lead to discontinuity at the boundaries, the reflecting (Neumann) BCs lead to C0 continuity at the boundaries, while our proposal leads to C1 continuity at the boundaries; (b) fast numerical algorithms in real arithmetic for both deblurring and estimating regularization parameters. Finally, simple yet significant 1D and 2D numerical evidence is presented and discussed.

A Note on Antireflective Boundary Conditions and Fast Deblurring Models

Spectral analysis and structure preserving preconditioners for fractional diffusion equations

In this paper we are interested in the solution by multigrid strategies of multilevel linear systems whose coefficient matrices belong to the circulant, Hartley, or $\tau$ algebras or to the Toeplitz class and are generated by (the Fourier expansion of) a nonnegative multivariate polynomial f. It is well known that these matrices are banded and have eigenvalues equally distributed as f, so they are ill-conditioned whenever f takes the zero value; they can even be singular and need a low-rank correction.
We prove the V-cycle multigrid iteration to have a convergence rate independent of the dimension even in presence of ill-conditioning. If the (multilevel) coefficient matrix has partial dimension nr at level r, r=1,...,d, then the size of the algebraic system is $N(n)=\prod_{r=1}^d n_r$, O(N(n)) operations are required by our technique, and therefore the corresponding method is optimal.
Some numerical experiments concerning linear systems arising in applications, such as elliptic PDEs with mixed boundary conditions and image restoration problems, are considered and discussed.

/pdf/v-cycle-optimal-convergence-for-certain-multilevel-es1zjir7mb.pdf

V-cycle Optimal Convergence for Certain (Multilevel) Structured Linear Systems

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.

https://www.mat.uniroma2.it/bertaccini/papers/BGST-NM.pdf

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

Singular values and eigenvalues of non-Hermitian block Toeplitz matrices

Superlinear preconditioners (those that provide a proper cluster at 1) are very important for the cg-like methods since they make these methods converge superlinearly. As is well known, for Toeplitz matrices generated by a continuous symbol, many circulant and circulant-like (related to different matrix algebras) preconditioners were proved to be superlinear. In contrast, for multilevel Toeplitz matrices there has been no proof of the superlinearity of any multilevel circulants. In this paper we show that such a proof is not possible since any multilevel circulant preconditioner is not superlinear, in the general case of multilevel Toeplitz matrices. Moreover, for matrices not necessarily Toeplitz, we present some general results proving that many popular structured preconditioners cannot be superlinear.

S. Serra Capizzano

Papers

Any Circulant-Like Preconditioner for Multilevel Matrices Is Not Superlinear