The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval

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,

The enclosure method for the heat equation using time-reversal invariance for a wave equation

“…However, for future purpose which aims at extending the method presented in this paper to other partial differential equations, including systems, we present here an another proof which is based on the solution formula on (2.1) in (ξ, t)-space not (x, t)-space. It is a combination of Parseval's identity and the residue theorem and completely different from the original proof given in [22] which is a combination of Kirchhoff's formula and explicit computation formula of some volume potentials.…”

“…Then, applying Lemma 2.3 in [22] to the right-hand side on (2.18), we obtain, as τ → ∞ e 2τ T −2 √ τ η w * 1 2…”

“…for all s ≥ 0, which is also a consequence of Kirchhoff's formula. From (2.10) and (2.17) together with Proposition 3.2 and 3.3 in [22] we have the explicit expression of the righthand side on (2.15):…”

“…Quite recently, in [22] the author considered an inverse obstacle problem for the wave governed by the wave equation in a bounded domain and introduced an idea which combines the time-reversal invariance of the wave equation with the newest version of enclosure method [19] which employs a single point on the graph of the response operator associated with the inverse obstacle problem. Note that the idea itself in [19] has been applied also to a system of equations in the linear theory of thermoelasticity in [20].…”

“…The key point of the idea introduced in [22] is: making a time-reversal operation on the solution of the free space wave equation supported on an arbitrary given ball at the initial time. This gives us an explicit input Neumann data depending on the ball such that the corresponding Dirichlet data as the output over a finite time interval yields the minimum sphere having the same center point as the ball and enclosing the obstacle.…”