The convergence proof of the no-response test for localizing an inclusion

Abstract

In this paper, we use the no-response test idea, introduced in ([L-P],[P1]) for the inverse obstacle problem, to identify the interface of the discontinuity of the coefficient \gamma of the equation $\nabla･\gamma(x)\nabla+c(x)$ with piecewise regular \gamma and {\it bounded} function c(x). We use infinitely many Cauchy data as measurement and give a recontructive method to localize the interface. We will base this multiwave version of the no-response test on two different proofs. The first one contains a pointwise estimate as used by the singular sources method. The second one is built on an energy (or an integral) estimate which is the basis of the probe method. As a conclusion of this, the no response can be seen as a unified framework for the probe and the singular sources method. As a further contribution, we provide a formula to reconstruct the valuse of the jump of $\gamma(x), x \in \partial D$ at the boundary.