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An application of shape optimization in the solution of inverse acoustic scattering problems. (English) Zbl 1055.35138
Summary: We consider the problem of determining the shape of an object immersed in an acoustic medium from measurements obtained at a distance from the object. We recast this problem as a shape optimization problem where we search for the domain that minimizes a cost function that quantifies the difference between the measured and expected signals. The measured and expected signals are assumed to satisfy a boundary-value problem given by the Helmholtz equation with the Sommerfeld condition imposed at infinity. Gradient-based algorithms are used to solve this optimization problem. At every step of the algorithm the derivative of the cost function with respect to the parameters that describe the shape of the object is calculated. We develop an efficient method based on the adjoint equations to calculate the derivative and show how this method is implemented in a finite element setting. The predominant cost of each step of the algorithm is equal to one forward solution and one adjoint solution and therefore is independent of the number of parameters used to describe the shape of the object. Numerical examples showing the efficacy of the proposed methodology are presented.
MSC:
35R30Inverse problems for PDE
49M05Numerical methods in calculus of variations based on necessary conditions
76Q05Hydro- and aero-acoustics
49Q10Optimization of shapes other than minimal surfaces
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation