Inverse Acoustic Scattering Problems With Impedance Boundary Conditions

In this paper, we discuss the scattering problem of an acoustic time-harmonic with impedance boundary conditionswhere D ∈ R³ is a bounded domain with sufficiently smooth boundary dD and outward unit normal v, incident plane wave is uⁱ(x) = e^ikx.d with incident direction d and wave number k ∈ C(Im(k) ≥ 0), λ is called surface impedance.As the direct scattering problem indicated above, the existence and the uniqueness of a solution have been well established by D.Colton and R.Kress via potential theory. Recently people pay much attention to inverse acoustic and electromagnetic scattering theory, it is a typical inverse problem of mathematics and physics. The inverse problem, given known solution information, is to find some unknown-quantity of scattering problem, such as the coefficient of differential equation, the scattering domain or some conditions. In this paper, we consider two inverse scattering problems of differential equation partial differential equation which are obtained by problem (*): (I) Given boundary condition, wave number k, incident direction d and far field pattern u_∞(x, d), we want to determine obstacle D(i.e. the shape of dD);(II) Given wave number K, incident direction d and obstacle D, we want to find the surface impedance A(z). Here, we will solve the problem (I) with Newton iterative methods;the problem (II) will be tackled quite successfully by integral equation methods(potential theory) and Tikhonov regularization methods.