Networks & Heterogeneous Media
March 2020 , Volume 15 , Issue 1
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We consider the inverse problem of finding matrix valued edge or node quantities in a graph from measurements made at a few boundary nodes. This is a generalization of the problem of finding resistors in a resistor network from voltage and current measurements at a few nodes, but where the voltages and currents are vector valued. The measurements come from solving a series of Dirichlet problems, i.e. finding vector valued voltages at some interior nodes from voltages prescribed at the boundary nodes. We give conditions under which the Dirichlet problem admits a unique solution and study the degenerate case where the edge weights are rank deficient. Under mild conditions, the map that associates the matrix valued parameters to boundary data is analytic. This has practical consequences to iterative methods for solving the inverse problem numerically and to local uniqueness of the inverse problem. Our results allow for complex valued weights and give also explicit formulas for the Jacobian of the parameter to data map in terms of certain products of Dirichlet problem solutions. An application to inverse problems arising in networks of springs, masses and dampers is presented.
We study time-harmonic Maxwell's equations in meta-materials that use either perfect conductors or high-contrast materials. Based on known effective equations for perfectly conducting inclusions, we calculate the transmission and reflection coefficients for four different geometries. For high-contrast materials and essentially two-dimensional geometries, we analyze parallel electric and parallel magnetic fields and discuss their potential to exhibit transmission through a sample of meta-material. For a numerical study, one often needs a method that is adapted to heterogeneous media; we consider here a Heterogeneous Multiscale Method for high contrast materials. The qualitative transmission properties, as predicted by the analysis, are confirmed with numerical experiments. The numerical results also underline the applicability of the multiscale method.
This paper investigates the incompressible limit of a system modelling the growth of two cells population. The model describes the dynamics of cell densities, driven by pressure exclusion and cell proliferation. It has been shown that solutions to this system of partial differential equations have the segregation property, meaning that two population initially segregated remain segregated. This work is devoted to the incompressible limit of such system towards a free boundary Hele Shaw type model for two cell populations.
By using dimension reduction and homogenization techniques, we study the steady flow of an incompresible viscoplastic Bingham fluid in a thin porous medium. A main feature of our study is the dependence of the yield stress of the Bingham fluid on the small parameters describing the geometry of the thin porous medium under consideration. Three different problems are obtained in the limit when the small parameter
We introduce non-Abelian Kuramoto model on
We briefly address two particular models: Kuramoto models on
Although non-Abelian Kuramoto models can be introduced on various symmetric spaces, we restrict our analysis to the case when underlying manifold is the 3-sphere. Due to geometric and algebraic properties of this specific manifold, variations of this model are meaningful and geometrically well justified.
We explain in this paper the similarity arising in the homogenization process of some composite fibered media with the problem of the reduction of dimension
We propose a finite difference method based on the Lax-Friedrichs scheme for a model of interaction between multiple solid particles and an inviscid fluid. The single-particle version has been studied extensively during the past decade. The model studied here consists of the inviscid Burgers equation with multiple nonconservative moving source terms that are singular and account for drag force interaction between the fluid and the particles. Each particle trajectory satisfies a differential equation that ensures conservation of momentum of the entire system. To deal with the singular source terms we discretize a model that associates with each particle an advection PDE whose solution is a shifted Heaviside function. This alternative model is well known but has not previously been used in numerical methods. We propose a definition of entropy solution which directly generalizes the previously defined single-particle notion of entropy solution. We prove convergence (along a subsequence) of the Lax-Friedrichs approximations, and also prove that if the set of times where the particle paths intersect has Lebesgue measure zero, then the limit is an entropy solution. We also propose a higher resolution version of the scheme, based on MUSCL processing, and present the results of numerical experiments.
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