3 research outputs found
Asymptotic spectra of large matrices coming from the symmetrization of Toeplitz structure functions and applications to preconditioning
No full textThe singular value distribution of the matrix-sequence {YnTn[f]}n, with Tn[f] generated by (Formula presented.), was shown in [J. Pestana and A.J. Wathen, SIAM J Matrix Anal Appl. 2015;36(1):273-288]. The results on the spectral distribution of {YnTn[f]}n were obtained independently in [M. Mazza and J. Pestana, BIT, 59(2):463-482, 2019] and [P. Ferrari, I. Furci, S. Hon, M.A. Mursaleen, and S. Serra-Capizzano, SIAM J. Matrix Anal. Appl., 40(3):1066-1086, 2019]. In the latter reference, the authors prove that {YnTn[f]}n is distributed in the eigenvalue sense as (Formula presented.) under the assumptions that f belongs to (Formula presented.) and has real Fourier coefficients. The purpose of this paper is to extend the latter result to matrix-sequences of the form {h(Tn[f])}n, where h is an analytic function. In particular, we provide the singular value distribution of the sequence {h(Tn[f])}n, the eigenvalue distribution of the sequence {Ynh(Tn[f])}n, and the conditions on f and h for these distributions to hold. Finally, the implications of our findings are discussed, in terms of preconditioning and of fast solution methods for the related linear systems
Preconditioners for fractional diffusion equations based on the spectral symbol
No full textIt is well known that the discretization of fractional diffusion equations with fractional derivatives , using the so-called weighted and shifted Grünwald formula, leads to linear systems whose coefficient matrices show a Toeplitz-like structure. More precisely, in the case of variable coefficients, the related matrix sequences belong to the so-called generalized locally Toeplitz class. Conversely, when the given FDE has constant coefficients, using a suitable discretization, we encounter a Toeplitz structure associated to a nonnegative function, called the spectral symbol, having a unique zero at zero of real positive order between one and two. For the fast solution of such systems by preconditioned Krylov methods, several preconditioning techniques have been proposed in both the one- and two-dimensional cases. In this article we propose a new preconditioner denoted bywhich belongs to the -algebra and it is based on the spectral symbol. Comparing with some of the previously proposed preconditioners, we show that although the low band structure preserving preconditioners are more effective in the one-dimensional case, the new preconditioner performs better in the more challenging multi-dimensional setting.</p
A note on eigenvalues and singular values of variable Toeplitz matrices and matrix-sequences, with application to variable two-step BDF approximations to parabolic equations
No full textHere, we consider a more general class of matrix-sequences and we prove that they belong to the maximal -algebra of generalized locally Toeplitz (GLT) matrix-sequences. Then, we identify the associated GLT symbols and GLT momentary symbols in the general setting and in the specific case, by providing in both cases a spectral and singular value analysis. More specifically, we use the GLT tools in order to study the asymptotic behaviour of the eigenvalues and singular values of the considered BDF matrix-sequences, in connection with the given non-uniform grids. Numerical examples, visualizations, and open problems end the present work.Corrected locations of images and reference
