Abstract
This paper presents a constructive algorithm for solving the problem of reconstructing the shape of a scattering object from measurements of the scattered far field when the unknown object is illuminated by a known incident wave. The problem is recast as an optimization problem with a penalty term. The cost functional consists of a term which assesses the difference between the measured far field and the far field of the solution of the field equation to a particular surface and the penalty term which measures the error in satisfying the boundary conditions on that surface. A complete family of radiating solutions of the Helmholtz equation is employed to construct approximate solutions by solving finite dimensional minimization problems. Existence of solutions of the original problem as well as the finite dimensional approximations is established. Moreover convergence of the approximate solutions to a solution of the original problem is proven. Some preliminary numerical results are presented to indicate the viability of the method.
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