A Shape Optimization Problem in Inverse Acoustics

Given the incident field, a plane wave or a superposition thereof, and the scattered far field, the problem addressed herewith consists of determining the shape of a sound soft obstacle, which is axially symmetric, star shaped with respect to the origin and smooth. The unknowns are the parameters, by which shape is represented as the linear combination of e.g., trigonometric functions. Some methods of solution proposed in the past consist in transforming the problem into the constrained minimization of an objective function, which consisted of the boundary defect and a penalty term. The boundary defect is a squared L2 norm of the incident plus approximate scattered field at the obstacle surface. The penalty term usually compares the computed and measured scattered far fields. The method presented herewith retains the minimization approach: it however replaces the constraint contained in the penalty term by a functional relationship between the coefficients, which appear in approximate representations of the far and resp., boundary scattered field. Said relationship is expressed by the approximate back propagation (ABP) operator, a product of some matrices, the entries of which depend on the shape parameters because they are inner products of suitable basis functions on the obstacle surface. As a consequence, a boundary defect alone has to be minimized with respect to the shape parameters. Some properties of the ABP operator are presented: they depend on those of the basis functions (spherical wave functions, their real parts and the normal derivatives of both at the obstacle surface) and are related e.g., to the properties of the T matrix method, used in the solution of forward scattering problems. Two classes of numerical problems are considered herewith: i) shape identification from the boundary coefficients of the scattered field and ii) shape identification from the far field coefficients. In either case the performance of the minimization algorithm agrees with an error estimate based on applying the projection theorem to the boundary defect. All examples have been selected to comply with a known uniqueness result, which would apply if the scattered field were exactly known: uniqueness is however lost because of approximation. The role of supplementary information in improving numerical reconstruction is demonstrated. Some preliminary results are also given for shape identification from the approximate scattering amplitude.