We consider a reconstruction problem of the shape of an unknown open set D in a two-dimensional bounded domain from the Cauchy data on ∂ of a nonconstant solution u of the equation u = 0 in \ D. We assume that the Neumann derivative of u vanishes on ∂D and that D is a convex open polygon. We give a formula for the calculation of the support function of D from such data.

We present in this work a novel approach for the reconstruction of wired network topologies from reflection measurements. Existing approaches state the network reconstruction as discrete optimization problem, which is difficult to solve. The (discrete) topology is optimized while the cable lengths are a secondary result. The contribution of this paper is the formulation of the topology reconstruction as a continuous problem. The idea is to rather optimize the (continuous) cable lengths and automatically obtain the topology as a secondary result. Further we present a heuristic algorithm to solve the optimization approximately. Using simulated reflectometry data, we demonstrate the performance of our approach.

No abstract

We report two new mathematical inversion algorithms for the electric impedance tomography. An application to the reconstruction problem of the unknown boundary on which the Neumann derivative of the solution of the Helmholtz equation vanishes is included.

This paper concerns the estimation of the size(Lebesgue measure) of an unknown inclusion imbedded in a known reference conductor or elasctic body. First, we point out two improvements of the esatimate obtained by Kang H., Seo J. and Sheen D. in their paper "The inverse conductivity problem with one measurement: Stability and Estimation of Size", to appear in SIAM J. Math. Anal.. Second, we establish a system of integral inequalities in the similar problem for an anisotropic elastic body and get similar estimates. AN INEQUALITY BY KANG, SEO AND SHEENLet Ω be a bounded domain of R n (n > 2) with Lipschitz boundary and D a subdomain compactly contained in Ω. Let k be a positive number such that k^l.Direct Problem Find u such that r\ n V ·

No abstract

A simple method for some class of inverse obstacle scattering problems is introduced. The observation data are given by a wave field measured on a known surface surrounding unknown obstacles over a finite time interval. The wave is generated by an initial data with compact support outside the surface. The method yields the distance from a given point outside the surface to obstacles and thus more than the convex hull. AMS: 35R30 KEY WORDS: enclosure method, inverse obstacle scattering problem, sound hard obstacle, penetrable obstacle, wave equation let u = u(x, t) satisfy the initial boundary value problem:(1.1)Here we denote the unit outward normal to ∂D by the symbol ν.Let Ω be a bounded domain with smooth boundary such that D ⊂ Ω and R 3 \ Ω is connected. We denote the unit outward normal to ∂Ω by ν again. The ∂Ω is considered as the location of the receivers of the acoustic wave produced by an emitter located at the support of f . In this paper first we consider the following problem.Inverse Problem I. Assume that D is unknown. Extract information about the location and shape of D from u on ∂Ω×]0, T [ for some fixed known f satisfying supp f ∩ Ω = ∅ and T < ∞. This is a quite natural problem, however, to my best knowledge, it seems that no attempt has been done. Clearly the main obstruction is the finiteness of T and f is fixed.Note that u in (R 3 \ Ω)×]0, T [ can be computed from u on ∂Ω× ]0, T [ by the formulawhere z solves the initial boundary value problem in R 3 \ Ω: ∂ 2 t z − △z = 0 in (R 3 \ Ω)× ]0, T [, z = u on ∂Ω× ]0, T [, z(x, 0) = 0 in R 3 \ Ω,

We consider an inverse source problem for the Helmholtz equation in a bounded domain. The problem is to reconstruct the shape of the support of a source term from the Cauchy data on the boundary of the solution of the governing equation. We prove that if the shape is a polygon, one can calculate its support function from such data. An application to the inverse boundary value problem is also included.