Shape identification of scatterers Using a time-dependent adjoint method

This paper deals with the inverse problem of accurately identifying the shape, size and location of 2D objects or voids in a given medium via the measurement of waves scattered from them. Examples of scatterers include an obstacle in a 2D acoustic medium or a small hole in a membrane. To fix ideas, the latter example, governed by the scalar wave equation, serves here in demonstrating the method. In the proposed method, a known source generates waves that propagate in the membrane, and these time-dependent waves are measured by a small number of sensors located at chosen points. Based on these measurements, an iterative process is performed to find the closed curve that represents the boundary of the scatterer. Mathematically, the iterative process aims at minimizing a cost functional that represents the difference between the measured wave signals and the wave signals obtained in the presence of a candidate scatterer. The gradient of this functional, which represents its sensitivity to the scatterer geometry, is calculated efficiently using the adjoint method. Finite elements are used to discretize the spatial domain, the Newmark method is used for time-stepping, and a smoothed gradient method is used for the functional minimization. The unknown boundary of the scatterer is discretized from the outset, which allows a simpler derivation of the adjoint procedure compared to a fully variational treatment. The unknown curve is defined by a parametric representation, that is completely general and does not make any preliminary assumptions about the scatterer geometry. Multiple sources are used to enhance the identification. The performance of the proposed method is demonstrated via numerical examples. It is shown that excellent identification is obtained even in the presence of noise, with a small number of sensors, and with no need for regularization (other than that induced by the discretization).