A direct identification method for current dipoles from electromagnetic fields

We discuss the identification problem for current dipoles in a spherically symmetric conductor. This mathematical model is used for a biomedical inverse problem such as the source current identification for the human brain activity. We have already proposed a direct identification method for this inverse source problem using observations of the magnetic fields outside of the conductor. One of the difficulties of current dipole identification using the magnetic fields is caused by the fact that magnetic field does not include any information about the radial component of dipole moments. In this paper, we consider an improvement of the direct method to identify both radial and tangential components of current dipole moments by combining electric and magnetic observation data. Furthermore, our approach is effective in the case where the number of dipoles is unknown.     

Reconstruction and stability in acousto-optic imaging for absorption maps with bounded variation

The aim of this paper is to propose for the first time a reconstruction scheme and a stability result for recovering from acousto-optic data absorption distributions with bounded variation. The paper extends earlier results in [3] and [5] on smooth absorption distributions. It opens a door for a mathematical and numerical framework for imaging, from internal data, parameter distributions with high contrast in biological tissues.     

Transient anomaly imaging by the acoustic radiation force

This paper is devoted to provide a solid mathematical foundation for a promising imaging technique based on the acoustic radiation force, which acts as a dipolar source. From the rigorously established asymptotic expansions of near- and far-field measurements of the transient wave induced by the anomaly, we design asymptotic imaging methods leading to a quantitative estimation of physical and geometrical parameters of the anomaly.     

Approximate radial symmetry for solutions of a class of boundary value problems in ring-shaped domains

This paper deals with some classical boundary value problems defined in the ring-shaped region $\Omega \backslash\overline{B(R_0)} \subset \Bbb{R}^N$, where B(R ₀) is the N-ball of radius R ₀ centered at the origin and contained in . The authors investigate how the geometry of is affected by some overdertermined data. In particular they show that must be nearly spherical if the overdetermined data are nearly constants.     

A MUSIC-type algorithm for detecting internal corrosion from electrostatic boundary measurements

We establish an asymptotic representation formula for the steady state current perturbations caused by internal corrosive boundary parts of small surface measure. Based on this formula we design a non-iterative method of MUSIC (multiple signal classification) type for localizing the corrosive parts from voltage-to-current observations. We perform numerical experiments to test the viability of the algorithm and the results clearly demonstrate that the algorithm works well even in the presence of relatively high noise ratios. H. Ammari is partially supported by the Brain Pool Korea Program at Seoul National University, H. Kang is partially supported by KOSEF grant R01-2006-000-10002-0, E. Kim is supported by BK21 Math. Division at Seoul National University, and M.S. Vogelius is partially supported by NSF grant DMS-0604999.     

Convergence from an electromagnetic fluid system to the full compressible MHD equations

We are concerned with the zero dielectric constant limit for the full electro-magneto-fluid dynamics in this article. This singular limit is justified rigorously for global smooth solution for both well-prepared and ill-prepared initial data. The explicit convergence rate is also obtained by a elaborate energy estimate. Moreover, we show that for the well-prepared initial data, there is no initial layer, and the electric field always converges strongly to the limit function. While for the ill-prepared data case, there will be an initial layer near t=0. The strong convergence results only hold outside the initial layer.     

Convergence of a Brinkman-type penalization for compressible fluid flows

We show convergence of a Brinkman-type penalization of the compressible Navier-Stokes equation. In particular, the existence of weak solutions for the system in domains with boundaries varying in time is established.     

Uniqueness and stability in determining the heat radiative coefficient,the initial temperature and a boundary coefficient in a parabolic equation

We consider an inverse parabolic problem. We prove that the heat radiative coefficient, the initial temperature and a boundary coefficient can be simultaneously determined from the final overdetermination, provided that the heat radiative coefficient is a priori known in a small subdomain. Moreover we establish a stability estimate for this inverse problem.     

Reconstruction of closely spaced small inhomogeneities via boundary measurements for the full time-dependent Maxwell’s equations

We consider for the full time-dependent Maxwell’s equations the inverse problem of identifying locations and certain properties of small electromagnetic inhomogeneities in a homogeneous background medium from dynamic boundary measurements on the boundary for a finite time interval.     

The reconstruction of groundwater parameters from head data in an unconfined aquifer

The estimation of groundwater flow parameters from head measurements and other ancillary data is fundamental to the process of modelling a groundwater system. In an unconfined aquifer, the problem is more complex because the governing equation for the well heads, the Boussinesq equation, is non-linear. We consider here a new method that allows for the simultaneous computation of the unconfined groundwater parameters as the unique minimum of a convex functional.     

Global solution to the one-dimensional equations for a self-gravitating viscous radiative and reactive gas

In this paper we consider a system of equations describing a motion of a self-gravitating one-dimensional gaseous medium in the presence of radiation and reacting process. By introducing Lagrangian mass coordinate, this free-boundary problem is reduced to the problem in a fixed domain with an explicit gravitational term. Based on the fundamental local existence result and a priori estimates, we can construct a classical unique global solution.     

Stable determination of conductivity by boundary measurements

We consider the problem of determining the scalar coefficient γ in the elliptic equation div(γ grad u) = 0 in ω when, for every Dirichlet datum u = ? on ?ω , the Neumann datum γ(?/ ? n)u = ∧._γ? is known. We prove a continuous dependence result     

Stability estimates in inverse scattering

An algorithm is given for calculating the solution to the 3D inverse scattering problem with noisy discrete fixed energy data. The error estimates for the calculated solution are derived. The methods developed are of a general nature and can be used in many applications: in nondestructive evaluation and remote sensing, in geophysical exploration, medical diagnostics, and technology.     

An inverse heat conduction problem with a nonlinear source term

In this paper, the nonlinear problem of an inhomogeneous heat equation with linear boundary conditions will be considered. The surface heat flux history of a heated conducting body will be identified. The approach of the proposed method is to approximate the unknown function using linear polynomial pieces which are determined consecutively from the solution of the minimization problem on the basis of overspecified data. Some numerical examples will also be presented.     

Stability Estimates in Inverse Problems for Hyperbolic Equations

We discuss here a method for investigating inverse problems for hyperbolic equations based on combining an asymptotic expansion of solutions to the equations in a neighborhood of a characteristic surface, the connections between coefficients of the expansion and the unknown coefficients of the equation and the estimates of a solution to the Cauchy problem in terms of the data on a time-like surface. We give some applications of this method to a number of inverse problems. Stability results to the inverse problems are given. This work was partly supported by the Russian Foundation for Basic Research (Grant No. 05-01-00171). Received: December 2005     

Marcinkiewicz estimates for degenerate parabolic equations with measure data

We consider solutions to degenerate parabolic equations with measurable coefficients, having on the right-hand side a measure satisfying a suitable density condition; we prove integrability results for the gradient in the Marcinkiewicz scale.     

The quasineutral limit of compressible Navier-Stokes-Poisson system with heat conductivity and general initial data

The quasineutral limit of compressible Navier-Stokes-Poisson system with heat conductivity and general (ill-prepared) initial data is rigorously proved in this paper. It is proved that, as the Debye length tends to zero, the solution of the compressible Navier-Stokes-Poisson system converges strongly to the strong solution of the incompressible Navier-Stokes equations plus a term of fast singular oscillating gradient vector fields. Moreover, if the Debye length, the viscosity coefficients and the heat conductivity coefficient independently go to zero, we obtain the incompressible Euler equations. In both cases the convergence rates are obtained.     

Global solutions and asymptotic behavior for a parabolic degenerate coupled system arising from biology

In this paper we will focus on a parabolic degenerate system with respect to unknown functions u and w on a bounded domain of the two dimensional Euclidean space. This system appears as a mathematical model for some biological processes. Global existence and uniqueness of a nonnegative classical Hölder continuous solution are proved. The last part of the paper is devoted to the study of the asymptotic behavior of the solutions.     

Existence and uniqueness of a solution for a two dimensional nonlinear inverse diffusion problem

The problem of identifying the coefficient in a square porous medium is considered. It is shown that under certain conditions of data f,g, and for a properly specified class A of admissible coefficients, there exists at least one a∈A such that (a,u) is a solution of the corresponding inverse problem.     

Polarization tensors and effective properties of anisotropic composite materials

In this paper, we present a systematic scheme for derivations of asymptotic expansions including higher-order terms, with estimates, of the effective electrical conductivity of periodic dilute composites in terms of the volume fraction occupied by the inclusions. The conductivities of the inclusion and the matrix may be anisotropic. Our derivations are based on layer potential techniques, and valid for high contrast mixtures and inclusions with Lipschitz boundaries. The asymptotic expansion is given in terms of the polarization tensor and the volume fraction of the inclusions. Important properties, such as symmetry and positivity, of the anisotropic polarization tensors are derived.