Iterative algorithms for computing the shape of a hard scattering object: Computing the shape derivative

The problem of determining the shape of an acoustically hard scattering object from remote scattering measurements is considered. An iterative approach is used to find the object shape that minimizes the mean-squared difference between a set of actual and predicted scattering observations. A crucial task in this minimization is the computation of the “shape derivative,” or functional gradient, of the mean-square error with respect to the object’s shape or boundary. The shape derivative tells us how to update the object’s shape to reduce the mean-square error at each iteration. If, for example, the object’s boundary is parameterized with N variables, a brute-force approach to computing the shape derivative using finite-differences would require a minimum of N+1 forward solutions per iteration. We show how the shape derivative can be computed with just two forward solutions: one ordinary forward solution and a suitably constructed adjoint solution. This approach is independent of N and is not only far more efficient, but numerically less error prone, than finite-difference schemes for computing derivatives.