Star-mesh transform

Abstract: This paper deals with an efficient technique for computing high-quality approximations of Schur complement matrices to be used in various preconditioners for the iterative solution of finite element discretizations of elliptic boundary value problems. The Schur complements are based on a two-by-two block decomposition of the matrix, and their approximations are computed by assembly of local (macroelement) Schur complements. The block partitioning is done by imposing a particular node ordering following the grid refinement hierarchy in the discretization mesh. For the theoretical derivation of condition number bounds, but not for the actual application of the method, we assume that the corresponding differential operator is self-adjoint and positive definite. The numerical efficiency of the proposed Schur complement approximation is illustrated in the framework of block incomplete factorization preconditioners of multilevel type, which require approximations of a sequence of arising Schur complement matrices. The behavior of the proposed approximation is compared with that of the coarse mesh finite element matrix, commonly used as an approximation of the Schur complement in the context of the above preconditioning methods. Moreover, the influence of refining a coarse mesh using a higher refinement number ($m$) than the customary $m=2$ is analyzed and its efficiency is also illustrated by numerical tests.

Abstract: We describe an application of the Legendre transform to communication networks. The Legendre transform applied to max-plus algebra linear systems corresponds to the Fourier transform applied to conventional linear systems. Hence, it is a powerful tool that can be applied to max-plus linear systems and their identification. Linear max-plus algebra has been already used to describe simple data communication networks. We first extend the Legendre transform as the slope transform to non-concave/non-convex functions. We then use it to analyze a simple communication network. We also propose an identification method for its transfer characteristic, and we confirm the results using the ns-2 network simulator.

Abstract: In a method for a receiver a signal is received at the receiver, where after a Cholesky factor or a block Cholesky factor of the Schur complement or a negative of the Schur complement of a matrix is determined at a processor entity by deriving a generator matrix of the Schur complement or the negative of the Schur complement from the generator matrix of the matrix and then applying a Schur algorithm to the generator matrix of the Schur complement or the negative of the Schur complement. The derivation comprises permutating at least a part of matrix columns below a certain row, shifting these columns downward a predefined number of rows and deleting a number of top rows. A filter for the signal may then be defined by computing filter coefficients for filtering the signal based on the derived Cholesky factor or block Cholesky factor. In another application a received signal is estimated based on similar determinations.