Scale-integration and scale-disintegration in nonlinear homogenization

This work is devoted to scale transformations of stationary nonlinear problems. A class of coarse-scale problems is first derived by integrating a family of two-scale minimization problems (scale-integration), in presence of appropriate orthogonality conditions. The equivalence between the two formulations is established by showing that conversely any solution of the coarse-scale problem can be represented as the fine-scale average of a solution of the two-scale problem (scale-disintegration). This procedure may be applied to the homogenization of several quasilinear problems, and is related to De Giorgi’s notion of Γ-convergence. As an example the homogenization of a simple nonlinear model of magnetostatics is illustrated: a two-scale minimization problem is first derived via Nguetseng’s notion of two-scale convergence, and afterwards the equivalence with a coarse-scale problem is proved.     

Positron Emission Tomography and Random Coefficients Regression

We further explore the relation between random coefficients regression (RCR) and computerized tomography. Recently, Beran et al. (1996, Ann. Statist., 24, 2569–2592) explored this connection to derive an estimation method for the non-parametric RCR problem which is closely related to image reconstruction methods in X-ray computerized tomography. In this paper we emphasize the close connection of the RCR problem with positron emission tomography (PET). Specifically, we show that the RCR problem can be viewed as an idealized (continuous) version of a PET experiment, by demonstrating that the nonparametric likelihood of the RCR problem is equivalent to that of a specific PET experiment. Consequently, methods independently developed for either of the two problems can be adapted from one problem to the other. To demonstrate the close relation between the two problems we use the estimation method of Beran, Feuerverger and Hall for image reconstruction in PET.     

On families of invariant lines of a Brouwer homeomorphism

ABSTRACT

We present properties of equivalence classes of the codivergency relation defined for a Brouwer homeomorphism for which there exists a family of invariant pairwise disjoint lines covering the plane. In particular, using the codivergency relation we describe the sets of regular and irregular points for such Brouwer homeomorphisms. Moreover, we show that under this assumption the interior of each equivalence class of this relation is invariant and simply connected.     

Linearized viscoelastic Oldroyd fluid motion in an almost periodic environment

In most of the linear homogenization problems involving convolution terms so far studied, the main tool used to derive the homogenized problem is the Laplace transform. Here we propose a direct approach enabling one to tackle both linear and nonlinear homogenization problems that involve convolution sequences without using Laplace transform. To illustrate this, we investigate in this paper the asymptotic behavior of the solutions of a Stokes–Volterra problem with rapidly oscillating coefficients describing the viscoelastic fluid flow in a fixed domain. Under the almost periodicity assumption on the coefficients of the problem, we prove that the sequence of solutions of our ?‐problem converges in L² to a solution of a rather classical Stokes system. One important fact is that the memory disappears in the limit. To achieve our goal, we use some very recent results about the sigma‐convergence of convolution sequences. Copyright © 2013 John Wiley & Sons, Ltd.     

An Equivalence Relation between Multiresolution Analysis ofL(R)

This paper is concerned with the development of an equivalence relation between two multiresolution analysis ofL²(R). The relation called unitary equivalence is created by the action of a unitary operator in such a way that the multiresolution structure and the decomposition and reconstruction algorithms remain invariant. A characterization in terms of the scaling functions of the multiresolution analysis is given. Distinct equivalence classes of multiresolution analysis are derived. Finally, we prove that B-splines give rise to nonequivalent examples.     

Viscosity method for homogenization of parabolic nonlinear equations in perforated domains

In this paper, we develop a viscosity method for homogenization of Nonlinear Parabolic Equations constrained by highly oscillating obstacles or Dirichlet data in perforated domains. The Dirichlet data on the perforated domain can be considered as a constraint or an obstacle. Homogenization of nonlinear eigen value problems has been also considered to control the degeneracy of the porous medium equation in perforated domains. For the simplicity, we consider obstacles that consist of cylindrical columns distributed periodically and perforated domains with punctured balls. If the decay rate of the capacity of columns or the capacity of punctured ball is too high or too small, the limit of u? will converge to trivial solutions. The critical decay rates of having nontrivial solution are obtained with the construction of barriers. We also show the limit of u? satisfies a homogenized equation with a term showing the effect of the highly oscillating obstacles or perforated domain in viscosity sense.     

Homogenization of a Ginzburg–Landau model for a nematic liquid crystal with inclusions

We consider a nonlinear homogenization problem for a Ginzburg–Landau functional with a (positive or negative) surface energy term describing a nematic liquid crystal with inclusions. Assuming that inclusions are separated by distances of the same order ɛ as their size, we find a limiting functional as ɛ approaches zero. We generalize the variational method of mesocharacteristics to show that a corresponding homogenized problem for arbitrary, periodic or non-periodic geometries is described by an anisotropic Ginzburg–Landau functional. We obtain computational formulas for material characteristics of an effective medium. As a byproduct of our analysis, we show that the limiting functional is a Γ-limit of a sequence of Ginzburg–Landau functionals. Furthermore, we prove that a cross-term corresponding to interactions between the bulk and the surface energy terms does not appear at the leading order in the homogenized limit.     

A note on the sufficient initial condition ensuring the convergence of directly extended 3-block ADMM for special semidefinite programming

ABSTRACT

In this note, we consider three types of problems, H-weighted nearest correlation matrix problem and two types of important doubly non-negative semidefinite programming, derived from the binary integer quadratic programming and maximum cut problem. The dual of these three types of problems is a 3-block separable convex optimization problem with a coupling linear equation constraint. It is known that, the directly extended 3-block alternating direction method of multipliers (ADMM3d) is more efficient than many of its variants for solving these convex optimization, but its convergence is not guaranteed. By choosing initial points properly, we obtain the convergence of ADMM3d for solving the dual of these three types of problems. Furthermore, we simplify the iterative scheme of ADMM3d and show the equivalence of ADMM3d to the 2-block semi-proximal ADMM for solving the dual's reformulation, under these initial conditions.     

Covering location problem of emergency service facilities in an uncertain environment

In practical location problems on networks, the response time between any pair of vertices and the demands of vertices are usually indeterminate. This paper employs uncertainty theory to address the location problem of emergency service facilities under uncertainty. We first model the location set covering problem in an uncertain environment, which is called the uncertain location set covering model. Using the inverse uncertainty distribution, the uncertain location set covering model can be transformed into an equivalent deterministic location model. Based on this equivalence relation, the uncertain location set covering model can be solved. Second, the maximal covering location problem is investigated in an uncertain environment. This paper first studies the uncertainty distribution of the covered demand that is associated with the covering constraint confidence level α. In addition, we model the maximal covering location problem in an uncertain environment using different modelling ideas, namely, the (α, β)-maximal covering location model and the α-chance maximal covering location model. It is also proved that the (α, β)-maximal covering location model can be transformed into an equivalent deterministic location model, and then, it can be solved. We also point out that there exists an equivalence relation between the (α, β)-maximal covering location model and the α-chance maximal covering location model, which leads to a method for solving the α-chance maximal covering location model. Finally, the ideas of uncertain models are illustrated by a case study.     

Equivalence of Equilibrium Problems and Least Element Problems

In this paper, we introduce the concept of feasible set for an equilibrium problem with a convex cone and generalize the notion of a Z-function for bifunctions. Under suitable assumptions, we derive some equivalence results of equilibrium problems, least element problems, and nonlinear programming problems. The results presented extend some results of [Riddell, R.C.: Equivalence of nonlinear complementarity problems and least element problems in Banach lattices. Math. Oper. Res. 6, 462–474 (1981)] to equilibrium problems. This work was supported by the National Natural Science Foundation of China (10671135) and Specialized Research Fund for the Doctoral Program of Higher Education (20060610005) and the Educational Science Foundation of Chongqing (KJ051307).     

Isomorphisms and non-isomorphisms of at actions

Measure-theoretic classification of ergodic actions of the integers and reals was developed through the use of von Neumann algebras by Krieger and later by Connes and Woods. In particular, the latter showed that AT (approximatively transitive) actions are classified by their Poisson boundary obtained from an inverse limit of (binomial) polynomials. Recently (2008), Giordano and Handelman [GH] showed that the classification can be divorced from the von Neumann algebra and inverse limit constructions, by instead examining (what amounts to the predual) direct limits. This results in a measure-theoretic version of dimension groups (used in C^*-algebras, especially in classification programmes), and a corresponding equivalence relation on diagrams which amounts to determining isomorphism classes. The next step is to translate the equivalence relation to easily (or relatively easily) computable criteria, usually numerical, and this is the thrust of this paper. An ergodic action whose corresponding factor is said to have pure point spectrum if it is isomorphic to its tensor square, or equivalently, the ergodic action satisfies the analogous property. Giordano and Skandalis obtained a numerical sufficient condition in [GS]. Here expressed in terms of a direct limit of either binomial or truncated Poisson distributions (actually more general distributions, using variance), we obtain numerical conditions that are an improvement on those of [GS] in the cases in which the latter apply, and extend them to a much wider class, called relatively absorbing. Some surprises occur, particularly when the gaps in the corresponding random walk increase unboundedly. We also deal with invariants for isomorphism in the measure-theoretic dimension group setting. An old one, developed in [CW], is the T-set. We extend the idea to create new invariants that are effective when the T-set is not; again, these involve numerical data (involving computations of $ l^{\underset{\raise0.3em\hbox{$ l^{\underset{\raise0.3em\hbox{, that is, total variation, distances) that are fairly tractible. We also give effective criteria for the T-set to be trivial modulo roots of unity.     

Two complexity results on c-optimality in experimental design

Finding a c-optimal design of a regression model is a basic optimization problem in statistics. We study the computational complexity of the problem in the case of a finite experimental domain. We formulate a decision version of the problem and prove its NP\boldsymbol{\mathit{NP}}-completeness. We provide examples of computationally complex instances of the design problem, motivated by cryptography. The problem, being NP\boldsymbol{\mathit{NP}}-complete, is then relaxed; we prove that a decision version of the relaxation, called approximate c-optimality, is P-complete. We derive an equivalence theorem for linear programming: we show that the relaxed c-optimality is equivalent (in the sense of many-one LOGSPACE-reducibility) to general linear programming.     

Tree-edges deletion problems with bounded diameter obstruction sets

We study the following problem: given a tree G and a finite set of trees H, find a subset O of the edges of G such that G-O does not contain a subtree isomorphic to a tree from H, and O has minimum cardinality. We give sharp boundaries on the tractability of this problem: the problem is polynomial when all the trees in H have diameter at most 5, while it is NP-hard when all the trees in H have diameter at most 6. We also show that the problem is polynomial when every tree in H has at most one vertex with degree more than 2, while it is NP-hard when the trees in H can have two such vertices.The polynomial-time algorithms use a variation of a known technique for solving graph problems. While the standard technique is based on defining an equivalence relation on graphs, we define a quasiorder. This new variation might be useful for giving more efficient algorithm for other graph problems.     

A variational problem determined by probability measures

ABSTRACT

An optimization problem of maximizing an integral of a function over a family of probability measures is considered. The problem is a generalization of a well-studied variational problem in mathematical economics, concerning optimal allocations. The specific generalization that we examine arises also in the limit of singularly perturbed optimal control problems. We examine the mathematical problem and allude to the singular perturbation motivation.     

Nonlocal limits in the study of linear elliptic systems arising in periodic homogenization

In the present paper, we obtain the two-scale limit system of a sequence of linear elliptic periodic problems with varying coefficients. We show that this system has not the same structure than the classical one, obtained when the coefficients are fixed. This is due to the apparition of nonlocal effects. Our results give an example showing that the homogenization of elliptic problems with varying coefficients, depending on one parameter, gives in general a nonlocal limit problem.     

Multiphase double‐porosity homogenization for perimeter functionals

In this paper, we study a homogenization problem for perimeter energies in highly contrasted media; the analysis of the previous paper is carried out by removing the hypothesis that the perforated medium Rⁿ ? E is composed of disjoint compact components. Assuming E to be the union of a finite number N of connected components E¹, … ,E^N, the Γ‐limit F is a multiphase energy with a ‘decoupled’ surface part, obtained by homogenization from the surface tensions in each E ^j, a trivial bulk term obtained as a weak limit, and a further interacting term between the phases, involving an asymptotic formula for a family minimum problems on invading an asymptotic formula for a family of minimum problems on invading domains with prescribed boundary conditions. Copyright © 2012 John Wiley & Sons, Ltd.     

The hydrodynamic limit for a system with interactions prescribed by Ginzburg-Landau type random Hamiltonian

Summary As a microscopic model we consider a system of interacting continuum like spin field overR ^d . Its evolution law is determined by the Ginzburg-Landau type random Hamiltonian and the total spin of the system is preserved by this evolution. We show that the spin field converges, under the hydrodynamic space-time scalling, to a deterministic limit which is a solution of a certain nonlinear diffusion equation. This equation describes the time evolution of the macroscopic field. The hydrodynamic scaling has an effect of the homogenization on the system at the same time.