We consider using a discrete network model in combination with continuous nonlinear optimization models to solve the problem of optimizing channels in nanoporous materials. The problem and the hierarchical optimization algorithm are described in [2]. A key feature of the model is the fact that we use the edges of the finite element grid as the locations of the channels. The focus here is on the use of the discrete model within that algorithm. We develop several approximations to the relevant flow and a greedy algorithm for quickly generating a “good” tree connecting all of the nodes in the finite-element mesh to a designated root node. We also consider Metropolis-Hastings (MH) improvements to the greedy result. We consider both a regular triangulation and a Delaunay triangulation of the region, and present some numerical results.

Algebraic Multigrid (AMG) methods were developed originally for numerically solving Partial Differential Equations (PDE), not necessarily on structured grids. In the last two decades solvers inspired by the AMG approach, were developed for non PDE problems, including data and image analysis problems, such as clustering, segmentation, quantization and others. These solvers share a common principle in that there is a crosstalk between fine and coarse representations of the problems, with flow of information in both directions, fine-to-coarse and coarse-to-fine. This paper surveys some of these problems and the AMG-inspired algorithms for their solution.

In topology optimization problems, we are often forced to deal with large-scale numericalproblems, so that the domain decomposition method occurs naturally. Consider a typicaltopology optimization problem, the minimum compliance problem of a linear isotropicelastic continuum structure, in which the constraints are the partial differentialequations of linear elasticity. We subdivide the partial differential equations into twosubproblems posed on non-overlapping sub-domains. In this paper, we consider the resultingproblem as multilevel one and show that it can be written as one level problem