Stefano Serra-Capizzano

It is well known that ω-circulant matrices with ω≠0 can be simultaneously diagonalized by a transform matrix, which can be factored as the product of a diagonal matrix, depending on ω, and of the unitary matrix Fn associated to the Fast Fourier Transform. Hence, all the sets of ω-circulants form algebras whose computational power, in terms of complexity, is the same as the classical circulants with ω=1. However, stability is a delicate issue, since the condition number of the transform is equal to that of the diagonal part, tending to max{|ω|,|ω|−1}. For ω=0, the set of related matrices is still an algebra, which is the algebra of lower triangular matrices, but they do not admit a common transform since most of them (all except the multiples of the identity) are non-diagonalizable. In the present work, we review two modern applications, ranging from parallel computing in preconditioning of PDE approximations to algorithms for subdivision schemes, and we emphasize the role of such algebra. For the two problems, few numerical tests are conducted and critically discussed and the related conclusions are drawn.

Topology optimization aims to find the best material layout subject to given constraints. The so‐called material distribution methods cast the governing equation as an extended or fictitious domain problem, in which a coefficient field represents the design. When solving the governing equation using the finite element method, a large number of elements are used to discretize the design domain, and an element‐wise constant function approximates the coefficient field in the considered design domain. This article presents a spectral analysis of the (large) coefficient matrices associated with the linear systems stemming from the finite element discretization of a linearly elastic problem for an arbitrary coefficient field. Based on the spectral information, we design a multigrid method which turns out to be optimal, in the sense that the (arithmetic) cost for solving the related linear systems, up to a fixed desired accuracy, is proportional to the matrix‐vector cost, which is linear in the corresponding matrix size. The method is tested, and the numerical results are very satisfactory in terms of linear cost and number of iterations, which is bounded by a constant independent of the matrix size.