[en] In [l] Brandt describes a general approach for algebraic coarsening. Given fine-grid equations and a prescribed relaxation method, an approach is presented for defining both the coarse-grid variables and the coarse-grid equations corresponding to these variables. Although, these two tasks are not necessarily related (and, indeed, are often performed independently and with distinct techniques) in the approaches of [1] both revolve around the same underlying observation. To determine whether a given set of coarse-grid variables is appropriate it is suggested that one should employ compatible relaxation. This is a generalization of so-called F-relaxation (e.g., [2]). Suppose that the coarse-grid variables are defined as a subset of the fine-grid variables. Then, F-relaxation simply means relaxing only the F-variables (i.e., fine-grid variables that do not correspond to coarse-grid variables), while leaving the remaining fine-grid variables (C-variables) unchanged. The generalization of compatible relaxation is in allowing the coarse-grid variables to be defined differently, say as linear combinations of fine-grid variables, or even nondeterministically (see examples in [1]). For the present summary it suffices to consider the simple case. The central observation regarding the set of coarse-grid variables is the following [1]: Observation 1--A general measure for the quality of the set of coarse-grid variables is the convergence rate of compatible relaxation. The conclusion is that a necessary condition for efficient multigrid solution (e.g., with convergence rates independent of problem size) is that the compatible-relaxation convergence be bounded away from 1, independently of the number of variables. This is often a sufficient condition, provided that the coarse-grid equations are sufficiently accurate. Therefore, it is suggested in [1] that the convergence rate of compatible relaxation should be used as a criterion for choosing and evaluating the set of coarse-grid variables. Once a coarse grid is chosen for which compatible relaxation converges fast, it follows that the dependence of the coarse-grid variables on each other decays exponentially or faster with the distance between them, measured in mesh-sizes. This implies that highly accurate coarse-grid equations can be constructed locally. A method for doing this by solving local constrained minimization problems is described in [1]. It is also shown how this approach can be applied to devise prolongation operators, which can be used for Galerkin coarsening in the usual way. In the present research we studied and developed methods based, in part, on these ideas. We developed and implemented an AMG approach which employs compatible relaxation to define the prolongation operator (hut is otherwise similar in its structure to classical AMG); we introduced a novel method for direct (i.e., non-Galerkin) algebraic coarsening, which is in the spirit of the approach originally proposed by Brandt in [1], hut is more efficient and well-defined; we investigated an approach for treating systems of equations and other problems where there is no unambiguous correspondence between equations and unknowns

[en] FOCal Underdetermined System Solver (FOCUSS) is a useful method through reweighted ℓ_2 minimization for sparse recovery. In this paper, we introduce an improved FOCUSS by enhancing sparsity with two reweighted ℓ_2 minimization. The reweighted FOCUSS method has higher mission success rate and better accuracy than FOCUSS. The simulation results illustrate the advantage of reweighted FOCUSS

[en] The two-dimensional layout optimization problem reinforced by the efficient space utilization demand has a wide spectrum of practical applications. Formulating the problem as a nonlinear minimization problem under planar equality and/or inequality density constraints, we present a linear time multigrid algorithm for solving a correction to this problem. The method is demonstrated in various graph drawing (visualization) instances.

[en] In this report we study space-mapping and manifold-mapping, two multi-level optimization techniques that aim at accelerating expensive optimization procedures with the aid of simple auxiliary models. Manifold-mapping improves in accuracy the solution given by space-mapping. In this report, the two mentioned techniques are basically described and then applied in the solving of two minimization problems. Several coarse models are tried, both from a two and a three level perspective. The results with these simple tests confirm the speed-up expected for the multi-level approach

[en] This study presents a critical review of the Prigogine minimum entropy production principle. The minimum entropy production implies the stationary state of the nonequilibrium system and vica versa: the stationary state of the system implies the minimum entropy production. It was shown that the extension of the principle to the so-called integral case is devoid of this property and, therefore, is less interesting in practical and theoretical terms

[en] Based on the notion of the ε -subgradient, we present a unified technique to establish convergence properties of several methods for nonsmooth convex minimization problems. Starting from the technical results, we obtain the global convergence of: (i) the variable metric proximal methods presented by Bonnans, Gilbert, Lemarechal, and Sagastizabal, (ii) some algorithms proposed by Correa and Lemarechal, and (iii) the proximal point algorithm given by Rockafellar. In particular, we prove that the Rockafellar-Todd phenomenon does not occur for each of the above mentioned methods. Moreover, we explore the convergence rate of { parallel x_k parallel } and {f(x_k) } when {x_k } is unbounded and {f(x_k) } is bounded for the non-smooth minimization methods (i), (ii), and (iii)

[en] We analyze the connection between minimizers with good generalizing properties and high local entropy regions of a threshold-linear classifier in Gaussian mixtures with the mean squared error loss function. We show that there exist configurations that achieve the Bayes-optimal generalization error, even in the case of unbalanced clusters. We explore analytically the error-counting loss landscape in the vicinity of a Bayes-optimal solution, and show that the closer we get to such configurations, the higher the local entropy, implying that the Bayes-optimal solution lays inside a wide flat region. We also consider the algorithmically relevant case of targeting wide flat minima of the (differentiable) mean squared error loss. Our analytical and numerical results show not only that in the balanced case the dependence on the norm of the weights is mild, but also, in the unbalanced case, that the performances can be improved. (paper)

[en] We introduce a new concept allowing the recasting of the reciprocity gap method into a variational method. The reciprocity likelihood functional maximization gives rise to nested approximation properties when performed on minimization spaces with increasing dimensions and leads to direct identification methods grounded on the reciprocity property. Application to the identification of point sources is given for illustration of the solution procedure of identification, and an analysis of the effect of noisy data shows that the proposed methods exhibit very good robustness. (authors)

[en] A polycrystal undergoes microstructural changes to reach a lower energy state. In particular, the system evolves so as to reduce the total grain boundary energy. A simple two-dimensional model of a polycrystal comprised of randomly oriented crystalline grains suggests that energy minimization reduces or eliminates any spatial correlation among high-energy grain boundaries. Thus grain boundary engineering not only reduces the density of high-energy boundaries, but it prevents their organization into a coarse, albeit discontinuous, network

[en] We study metric properties of ring Q-homeomorphisms with respect to the p-modulus, p > 2, in the complex plane and establish lower bounds for the areas of disks. An extremal problem concerning minimization of the area functional is also solved.