A coarse space X endowed with a linear order compatible the structure of is called linearly ordered. We prove that every ordered locally convex and asymptotic dimension either 0 or 1. If metrizable, then family all right bounded subsets has selector.

In this paper, we study the effects of the coarse grid correction process on multigrid convergence for hyperbolic problems in one and two dimensions. We approach this from the perspective of phase error, which allows us to exploit the hyperbolic nature of the underlying PDE. In particular, we consider three combinations of coarse grid operators and coarse grid solution approaches: (1) inexact c...

Networks have become a popular way to concisely represent complex nonlinear systems where the interactions and parameters are imprecisely known. One challenge is how best to describe the associated dynamics, which can exhibit complicated behavior sensitive to small changes in parameters. A recently developed computational approach that we refer to as a database for dynamics provides a robust an...

We extend coarse graining of cellular automata to investigate aspects of emergence. From the total coarse graining approach introduced by Israeli and Goldenfeld, Coarsegraining of cellular automata, emergence, and the predictability of complex systems, Phys. Rev. E, 2006, we devise partial coarse graining, and show qualitative differences in the results of total and partial coarse graining. Mut...

Abstract. The two common approaches to defining coarse operators in multigrid numerical algorithms are discretization coarse approximation (DCA) and (Petrov-)Galerkin coarse approximation (GCA). Here, a new approach called collocation coarse approximation (CCA) is introduced, which—like GCA—is algebraically defined and able to cater to difficult features such as discontinuous coefficients, but,...