Extracting discontinuity using the probe and enclosure methods

Abstract

This is a review article on the development of the probe and enclosure methods from past to present, focused on their central ideas together with various applications.

Revisiting the probe and enclosure methods

“…Under a condition on the jump of the real or imaginary part of the term across the boundary of the obstacle and some kind of regularity for the obstacle surface including Lipschitz one as a special case, the enclosure method using infinitely many pairs of the Dirichlet and Neumann data is established. This is a full extension of a result in the unpublished manuscript [15] (see also [19]) in which an inverse obstacle problem for a non-absorbing medium governed by the stationary Schrödiner equation was considered. Now let us formulate the two problems mentioned above more precisely and describe statements of the results.…”

“…First we give an L 2 -estimate of the so-called reflected solution in terms of the L 1 -norm of v which is an arbitrary solution of equation (1.7) 6 . The proof is essentially the same as that of Lemma 3.1 in [19] (taken from unpublished manuscript [15]) in which the case V 0 (x) ≡ k 2 is treated. Here for reader's convenience we present its proof.…”

“…See also Remark 3.2 for an explanation about the role of (3.4) in the proof of Theorems 1.2 and 1.3. Next we describe another lemma which is originally stated in the unpublished manuscript [15] and Lemma 3.2 in [19] provided D is an open set of R 3 and ∂D ∈ C 2 and ω satisfies:…”

“…The probe and enclosure methods initiated by the author in [9], [8], [11], [10], [13], [12] and [14] are methodologies in reconstruction issue for inverse obstacle problems governed by partial differential equations. Nowadays we have various applications, see the recent survey paper [19] for the results and references. However, looking back to some of those applications, one has not yet fully developed its possibility.…”

Inverse problems of the wave equation for media with mixed but separated heterogeneous parts

“…Ikehata has conducted many other studies on reconstruction for various equations by the enclosure method(see e.g. [11]). Also for the wave equation, each case of Dirichlet boundary condition, Neumann type boundary condition, dissipative boundary condition and inclusions is investigated in [6,7,8].…”