We introduce a new geometric approach for the homogenization and inverse homogenization of the divergence form elliptic operator with rough conductivity coefficients $σ(x)$ in dimension two. We show that conductivity coefficients are in one-to-one correspondence with divergence-free matrices and convex functions $s(x)$ over the domain $Ω$. Although homogenization is a non-linear and non-injective operator when applied directly to conductivity coefficients, homogenization becomes a linear interpolation operator over triangulations of $Ω$ when re-expressed using convex functions, and is a volume averaging operator when re-expressed with divergence-free matrices. Using optimal weighted Delaunay triangulations for linearly interpolating convex functions, we obtain an optimally robust homogenization algorithm for arbitrary rough coefficients. Next, we consider inverse homogenization and show how to decompose it into a linear ill-posed problem and a well-posed non-linear problem. We apply this new geometric approach to Electrical Impedance Tomography (EIT). It is known that the EIT problem admits at most one isotropic solution. If an isotropic solution exists, we show how to compute it from any conductivity having the same boundary Dirichlet-to-Neumann map. It is known that the EIT problem admits a unique (stable with respect to $G$-convergence) solution in the space of divergence-free matrices. As such we suggest that the space of convex functions is the natural space in which to parameterize solutions of the EIT problem.
We look at the effective Hamiltonian $\bar H$ associated with the Hamiltonian $H(p,x)=H(p)+V(x)$ in the periodic homogenization theory. Our central goal is to understand the relation between $V$ and $\bar H$. We formulate some inverse problems concerning this relation. Such type of inverse problems are in general very challenging. In the paper, we discuss several special cases in both convex and nonconvex settings.
We consider the homogenization for time-fractional diffusion equations in a periodic structure and we derive the homogenized time-fractional diffusion equation. Then we discuss the determination of the constant diffusion coefficient by minimum data. Combining the results obtained above, we investigate the inverse problems of determining the diffusion coefficient for the homogenized equations from the data in the periodic structure and vice versa, that is, we investigate the inverse problem of determining the diffusion coefficient for the periodic equations from the data in the homogenized stru…
We study the inverse of the divergence operator on a domain $Ω\subset R^3$ perforated by a system of tiny holes. We show that such inverse can be constructed on the Lebesgue space $L^p(Ω)$ for any $1< p < 3$, with a norm independent of perforation, provided the holes are suitably small and their mutual distance suitably large. Applications are given to problems arising in homogenization of steady compressible fluid flows.
Sediment Concentration Estimation via Multiscale Inverse Problem and Stochastic Homogenization
Tsinghua University（清华大学）；Yau Mathematical Sciences Center, Tsinghua University（丘成桐数学科学中心）；Yanqi Lake Beijing Institute of Mathematical Sciences and Applications（北京雁栖湖应用数学研究院）；Ningxia University（宁夏大学）；Xiangtan University（湘潭大学）；City University of Hong Kong（香港城市大学）
We analyze behavior of weak solutions to compressible fluid flows in a bounded domain in $\mathbb{R}^3$, randomly perforated by tiny balls with random size. Assuming the radii of the balls scale like $\varepsilon^α$, $α> 3$, with $\varepsilon$ denoting the average distance between the balls, the problem homogenize with the same limiting equation. Our main contribution is a construction of the Bogovski\uı operator, uniformly in $\varepsilon$, without any assumptions on the minimal distance between the balls.
Polarization tensor corresponding to near zero volume inhomogeneities was introduced in the pioneering work by Capdeboscq-Vogelius \cite{CV1,CV2}. A beautiful application of the polarization tensor to an inverse problem involving inhomogeneities was also given by them. In this article, we take an approach toward polarization tensor via homogenized tensor. Accordingly, we introduce polarization tensor corresponding to inhomogeneities with positive volume fraction.A relation between this tensor and the homogenized tensor is found. Next, we proceed to examine the sense in which this tensor is con…
We consider the interior transmission problem associated with the scattering by an inhomogeneous (possibly anisotropic) highly oscillating periodic media. We show that, under appropriate assumptions, the solution of the interior transmission problem converges to the solution of a homogenized problem as the period goes to zero. Furthermore, we prove that the associated real transmission eigenvalues converge to transmission eigenvalues of the homogenized problem. Finally we show how to use the first transmission eigenvalue of the period media, which is measurable from the scattering data, to obt…
In this paper, we use the theory of symmetric Dirichlet forms to derive Feynman-Kac formulae for the forward problem of electrical impedance tomography with possibly anisotropic, merely measurable conductivities corresponding to different electrode models on bounded Lipschitz domains. Subsequently, we employ these Feynman-Kac formulae to rigorously justify stochastic homogenization in the case of a stochastic boundary value problem arising from an inverse anomaly detection problem. Motivated by this theoretical result, we prove an estimate for the speed of convergence of the projected mean-squ…
A parameter identification problem in stochastic homogenization
INRIA Rocquencourt（法国国家信息与自动化研究所）；MATHERIALS research-team（MATHERIALS 研究团队）；CNRS, UMR 8234 PHENIX（法国国家科学研究中心）；ANDRA（法国放射性废物管理局）；Laboratoire Navier, École Nationale des Ponts et Chaussées, Université Paris-Est（国立路桥学院拉维埃实验室，巴黎东大学）
In this paper we study the homogenization of a class of energies concentrated on lines. In dimension $2$ (i.e., in codimension $1$) the problem reduces to the homogenization of partition energies studied by \cite{AB}. There, the key tool is the representation of partitions in terms of $BV$ functions with values in a discrete set. In our general case the key ingredient is the representation of closed loops with discrete multiplicity either as divergence-free matrix-valued measures supported on curves or with $1$-currents with multiplicity in a lattice. In the $3$ dimensional case the main motiv…
We study and characterize the optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form. We obtain that the optimal rate of convergence is either $O(\varepsilon)$ or $O(\varepsilon^2)$ depending on the diffusion matrix $A$, source term $f$, and boundary data $g$. Moreover, we show that the set of diffusion matrices $A$ that give optimal rate $O(\varepsilon)$ is open and dense in the set of $C^{2,α}$ periodic, symmetric, and positive definite matrices, which means that generically, the optimal rate is $O(\varepsilon)$.
We consider the discrete eigenvalues of the operator $H_\eps=-Δ+V(\x)+\eps^2Q(\eps\x)$, where $V(\x)$ is periodic and $Q(\y)$ is localized on $\R^d,\ \ d\ge1$. For $\eps>0$ and sufficiently small, discrete eigenvalues may bifurcate (emerge) from spectral band edges of the periodic Schrödinger operator, $H_0 = -Δ_\x+V(\x)$, into spectral gaps. The nature of the bifurcation depends on the homogenized Schrödinger operator $L_{A,Q}=-\nabla_\y\cdot A \nabla_\y +\ Q(\y)$. Here, $A$ denotes the inverse effective mass matrix, associated with the spectral band edge, which is the site of the bifurcation…
We study the homogenization of a diffusion process which takes place in a binary structure formed by an ambiental connected phase surrounding a suspension of very small spheres distributed in an $\veps$-periodic network. The asymptotic distribution of the concentration is determined for both phases, as $\veps\to 0$, assuming that the suspension has mass of unity order and vanishing volume. Three cases are distinguished according to the values of a certain limit capacity. When it is positive and finite, the macroscopic system involves a two-concentration system, coupled through a term accountin…
In this paper we provide converge rates for the homogenization of the Poisson problem with Dirichlet boundary conditions in a randomly perforated domain of $\mathbb{R}^d$, $d \geq 3$. We assume that the holes that perforate the domain are spherical and are generated by a rescaled marked point process $(Φ, \mathcal{R})$. The point process $Φ$ generating the centres of the holes is either a Poisson point process or the lattice $\mathbb{Z}^d$; the marks $\mathcal{R}$ generating the radii are unbounded i.i.d random variables having finite $(d-2+β)$-moment, for $β> 0$. We study the rate of converge…
This paper is concerned with the behavior of the ergodic constant associated with convex and superlinear Hamilton-Jacobi equation in a periodic environment which is perturbed either by medium with increasing period or by a random Bernoulli perturbation with small parameter. We find a first order Taylor's expansion for the ergodic constant which depends on the dimension d. When d = 1 the first order term is non trivial, while for all d $\ge$ 2 it is always 0. Although such questions have been looked at in the context of linear uniformly elliptic homogenization, our results are the first of this…
We consider $σ$-harmonic mappings, that is mappings $U$ whose components $u_i$ solve a divergence structure elliptic equation ${\rm div} (σ\nabla u_i)=0$, for $i=1,\ldots,n $. We investigate whether, with suitably prescribed Dirichlet data, the Jacobian determinant can be bounded away from zero. Results of this sort are required in the treatment of the so-called hybrid inverse problems, and also in the field of homogenization studying bounds for the effective properties of composite materials.
In this article, we investigate the spectrum of the Neumann-Poincaré operator associated to a periodic distribution of small inclusions with size $\varepsilon$, and its asymptotic behavior as the parameter $\varepsilon$ vanishes. Combining techniques pertaining to the fields of homogenization and potential theory, we prove that the limit spectrum is composed of the `trivial' eigenvalues $0$ and $1$, and of a subset which stays bounded away from $0$ and $1$ uniformly with respect to $\varepsilon$. This non trivial part is the reunion of the \textit{Bloch spectrum}, accounting for the collective…
We establish uniform Lipschitz estimates for second-order elliptic systems in divergence form with rapidly oscillating, almost-periodic coefficients. We give interior estimates as well as estimates up to the boundary in bounded $C^{1,α}$ domains with either Dirichlet or Neumann data. The main results extend those in the periodic setting due to Avellaneda and Lin for interior and Dirichlet boundary estimates and later Kenig, Lin, and Shen for the Neumann boundary conditions. In contrast to these papers, our arguments are constructive (and thus the constants are in principle computable) and the …
For a family of systems of linear elasticity with rapidly oscillating periodic coefficients, we establish sharp boundary estimates with either Dirichlet or Neumann conditions, uniform down to the microscopic scale, without smoothness assumptions on the coefficients. Under additional smoothness conditions, these estimates, combined with the corresponding local estimates, lead to the full Rellich type estimates in Lipschitz domains and Lipschitz estimates in $C^{1, α}$ domains. The $C^α$, $W^{1,p}$, and $L^p$ estimates in $C^1$ domains for systems with VMO coefficients are also studied. The appr…
We consider a one-dimensional diffusion process with coefficients that are periodic outside of a finite 'interface region'. The question investigated in this article is the limiting long time / large scale behaviour of such a process under diffusive rescaling. Our main result is that it converges weakly to a rescaled version of skew Brownian motion, with parameters that can be given explicitly in terms of the coefficients of the original diffusion. Our method of proof relies on the framework provided by Freidlin and Wentzell for diffusion processes on a graph in order to identify the generator…
This paper proposes a new approach for solving ill-posed nonlinear inverse problems. For ease of explanation of the proposed approach, we use the example of lung electrical impedance tomography (EIT), which is known to be a nonlinear and ill-posed inverse problem. Conventionally, penalty-based regularization methods have been used to deal with the ill-posed problem. However, experiences over the last three decades have shown methodological limitations in utilizing prior knowledge about tracking expected imaging features for medial diagnosis. The proposed method's paradigm is completely differe…
The present paper deals with the asymptotic behavior of equi-coercive sequences $\{\mathcal{F}_n\}$ of nonlinear functionals defined over vector-valued functions in $W_)^{1,p}(Ω)^M$ , where $p>1$, $M\ge1$, and $Ω$ is a bounded open set of $\mathbb{R}^N$, $N\ge2$. The strongly local energy density $F_n({\cdot}, Du)$ of the functional $\{\mathcal{F}_n\}$ satisfies a Lipschitz condition with respect to the second variable, which is controlled by a positive sequence $\{a_n\}$ which is only bounded in some suitable space $L^r(Ω)$. We prove that the sequence $\{\mathcal{F}_n\}$ $Γ$-converges for the…
This paper is concerned with the homogenization of the Dirichlet eigenvalue problem, posed in a bounded domain $Ω\subset\mathbb R^2$, for a vectorial elliptic operator $-\nabla\cdot A^ε(\cdot)\nabla$ with $ε$-periodic coefficients. We analyse the asymptotics of the eigenvalues $λ^{ε,k}$ when $ε\rightarrow 0$, the mode $k$ being fixed. A first-order asymptotic expansion is proved for $λ^{ε,k}$ in the case when $Ω$ is either a smooth uniformly convex domain, or a convex polygonal domain with sides of slopes satisfying a small divisors assumption. Our results extend those of Moskow and Vogelius r…
We investigate the large-distance asymptotics of optimal Hardy weights on $\mathbb Z^d$, $d\geq 3$, via the super solution construction. For the free discrete Laplacian, the Hardy weight asymptotic is the familiar $\frac{(d-2)^2}{4}|x|^{-2}$ as $|x|\to\infty$. We prove that the inverse-square behavior of the optimal Hardy weight is robust for general elliptic coefficients on $\mathbb Z^d$: (1) averages over large sectors have inverse-square scaling, (2), for ergodic coefficients, there is a pointwise inverse-square upper bound on moments, and (3), for i.i.d.\ coefficients, there is a matching …
The present paper deals with the homogenization of the heat conduction which takes place in a binary three-dimensional medium consisting of an ambiental phase having conductivity of unity order and a rectangular honeycomb structure formed by a set of thin layers crossing orthogonally and periodically. We consider the case when the conductivity of the thin layers is in inverse proportion to the vanishing volume of the rectangular honeycomb structure. We find the system that governs the asymptotic behaviour of the temperature distribution of this binary medium. The dependence with respect to the…
This paper concerns inverse problems for strongly coupled Schrödinger equations. The purpose of this inverse problem is to retrieve a stationary potential in the strongly coupled Schrödinger equations from either boundary or internal measurements. Two stability results are derived from a new Carleman estimate for the strongly coupled Schrödinger equations.
We consider the inverse problem in geophysics of imaging the subsurface of the Earth in cases where a region below the surface is known to be formed by strata of different materials and the depths and thicknesses of the strata and the (possibly anisotropic) conductivity of each of them need to be identified simultaneously. This problem is treated as a special case of the inverse problem of determining a family of nested inclusions in a medium $Ω\subset\mathbb{R}^n$, $n \geq 3$.
The ill-posedness of Calderón's inverse conductivity problem, responsible for the poor spatial resolution of Electrical Impedance Tomography (EIT), has been an impetus for the development of hybrid imaging techniques, which compensate for this lack of resolution by coupling with a second type of physical wave, typically modeled by a hyperbolic PDE. Here we show how, using EIT data alone, to efficiently detect interior jumps and other singularities of the conductivity. Analysis of the complex geometrical optics solutions of Astala and Päivärinta [\emph{Ann. Math.}, {\bf 163} (2006)] in 2D makes…
In this paper, we consider the Dirichlet Laplacian operator -Δ on a curved quantum guide in R n (n = 2, 3) with an asymptotically straight reference curve. We give uniqueness results for the inverse problem associated to the reconstruction of the curvature by using either observations of spectral data or a boot-strapping method. keywords: Inverse Problem, Quantum Guide, Curvature
We consider an inverse boundary value problem for a semilinear wave equation on a time-dependent Lorentzian manifold with time-like boundary. The time-dependent coefficients of the nonlinear terms can be recovered in the interior from the knowledge of the Neumann-to-Dirichlet map. Either distorted plane waves or Gaussian beams can be used to derive uniqueness.
We prove a uniqueness theorem for an inverse boundary value problem for the Maxwell system with boundary data assumed known only in part of the bound- ary. We assume that the inaccessible part of the boundary is either part of a plane, or part of a sphere. This work generalizes the results obtained by Isakov for the Schrödinger equation to Maxwell equations.
We consider two inverse boundary value problems for the time-harmonic Maxwell equations in an infinite slab. Assuming that tangential boundary data for the electric and magnetic fields at a fixed frequency is available either on subsets of one boundary hyperplane, or on subsets of different boundary hyperplanes, we show that the electromagnetic material parameters, the conductivity, electric permittivity, and magnetic permeability, are uniquely determined by these partial measurements.
We consider the electrostatic inverse boundary value problem also known as electrical impedance tomography (EIT) for the case where the conductivity is a piecewise linear function on a domain $Ω\subset\mathbb{R}^n$ and we show that a Lipschitz stability estimate for the conductivity in terms of the local Dirichlet-to-Neumann map holds true.
We consider the inverse boundary value problem for operators of the form $-\triangle+q$ in an infinite domain $Ω=\mathbb{R}\timesω\subset\mathbb{R}^{1+n}$, $n\geq3$, with a periodic potential $q$. For Dirichlet-to-Neumann data localized on a portion of the boundary of the form $Γ_1=\mathbb{R}\timesγ_1$, with $γ_1$ being the complement either of a flat or spherical portion of $\partialω$, we prove that a log-type stability estimate holds.
Let $L:=-Δ+V$ be a nonnegative Schrödinger operator on $L^2({\bf R}^N)$, where $N\ge 2$ and $V$ is a radially symmetric inverse square potential. In this paper we assume either $L$ is subcritical or null-critical and we establish a method for obtaining the precise description of the large time behavior of $e^{-tL}φ$, where $φ\in L^2({\bf R}^N,e^{|x|^2/4}\,dx)$.
We consider the Schrodinger operator in n-dimensional rectangular domains with either Dirichlet or Neumann boundary conditions on the faces and study the constraints on the potential imposed by fixing the spectrum of the operator.We study also the asymptotics of the heat kernel when t tends to 0.
This paper presents a mathematical framework for a flexible pressure-sensor model using electrical impedance tomography (EIT). When pressure is applied to a conductive membrane patch with clamped boundary, the pressure-induced surface deformation results in a change in the conductivity distribution. This change can be detected in the current-voltage data ({\it i.e.,} EIT data) measured on the boundary of the membrane patch. Hence, the corresponding inverse problem is to reconstruct the pressure distribution from the data. The 2D apparent conductivity (in terms of EIT data) corresponding to the…
We provide a general scheme, in the combined frameworks of Mathematical Scattering Theory and Factorization Method, for inverse scattering for the couple of self-adjoint operators $(\widetildeΔ,Δ)$, where $Δ$ is the free Laplacian in $L^{2}({\mathbb R}^{3})$ and $\widetildeΔ$ is one of its singular perturbations, i.e., such that the set $\{u\in H^{2}({\mathbb R}^{3})\cap \text{dom}(\widetildeΔ)\, :\, Δu=\widetildeΔu\}$ is dense. Typically $\widetildeΔ$ corresponds to a self-adjoint realization of the Laplace operator with some kind of boundary conditions imposed on a null subset; in particular…
