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Inverse scattering via nonlinear integral equations for a Neumann crack. (English) Zbl 1105.35143
Summary: We present a new method for solving the time-harmonic acoustic inverse scattering problem for a sound-hard crack in ${ℝ}^{2}$. Using the integral equation method to solve the inverse scattering problem, one obtains a Fredholm integral equation of the first kind. Instead of applying regularized Newton’s method directly to this integral equation, we derive an equivalent system of two nonlinear integral equations for the inverse problem. In this setting, not only can the regularized Newton’s method still be used to solve the inverse problem numerically, but also has the advantage of removing the need to solve a related direct problem at every iteration.

##### MSC:
 35R30 Inverse problems for PDE 35J25 Second order elliptic equations, boundary value problems 76Q05 Hydro- and aero-acoustics 35P25 Scattering theory (PDE) 45B05 Fredholm integral equations 74J25 Inverse problems (waves in solid mechanics)