A coupled FEM-BEM algorithm for the inverse acoustic medium problem

We present a numerical scheme for reconstructing the refractive index of an inhomogeneous two dimensional medium using acoustic far field data. The numerical scheme is based only on the mild assumption that the inhomogeneous medium is contained in the unit disk, and does not require axis-symmetry or other similar restrictions. Reconstruction of the refractive index, without the assumption of axis-symmetry, is achieved using an expansion in the high order Logan--Shepp polynomials. The Logan--Shepp expansion coefficients of the refractive index are formulated as the solution of a nonlinear equation, which is solved using a regularised Newton-type solver. Nonlinear function evaluations, which involve solving a forward scattering problem, are performed using an efficient coupled finite-element/\penalty \exhyphenpenalty boundary element method, which ensures that the radiation condition is incorporated exactly. The scheme is demonstrated by reconstructing challenging continuous and discontinous media from noisy far field data. References M. Abramowitz and I. A. Stegun, editors. Handbook of Mathematical Functions. National Bureau of Standards, 1964. doi:10.1119/1.1972842. K. Atkinson, D. Chien, and O. Hansen. A spectral method for elliptic equations: the Dirichlet problem. Adv. Comput. Math., 33:69–1891, 2014. doi:10.1007/s10444-009-9125-8. A. Barnett and L. Greengard. A new integral representation for quasi-periodic fields and its application to two-dimensional band structure calculations. J. Comput. Phys., 229:6898–6914, 2010. doi:10.1016/j.jcp.2010.05.029. D. Colton and R. Kress. Inverse Acoustic and Electromagnetic Scattering Theory. Springer, 2012. doi:10.1007/978-1-4614-4942-3. S. Gutman and M. Klibanov. Regularized quasi-Newton method for inverse scattering problems. Math. Comput. Modelling, 18:5–31, 1993. doi:10.1016/0895-7177(93)90076-B. S. Gutman and M. Klibanov. Iterative method for multi-dimensional inverse scattering problems at fixed frequencies. Inverse Problems, 10:573–599, 1994. doi:10.1088/0266-5611/10/3/006. S. Gutman and M. Klibanov. Two versions of quasi-Newton method for multidimensional inverse scattering problem. J. Comput. Acoust., 1:197–228, 1993. doi:10.1142/S0218396X93000123. M. Hanke. A regularizing Levenburg-Marquardt scheme, with applications to inverse groundwater filtration problems. Inverse Problems, 13:79–95, 1997. doi:10.1088/0266-5611/13/1/007. T. Hohage. On the numerical solution of a three-dimensional inverse medium scattering problem. Inverse Problems, 17:1743–1763, 2001. doi:10.1088/0266-5611/17/6/314. T. Hohage. Fast numerical solution of the electromagnetic medium scattering problem and applications to the inverse problem. J. Comput Phys., 214:224–238, 2006. doi:10.1016/j.jcp.2005.09.025. A. Kirsch. An Introduction to the Mathematical Theory of Inverse Problems. Springer, 2011. doi:10.1007/978-1-4419-8474-6. A. Kirsch and P. Monk. An analysis of the coupling of finite-element and Nystrom methods in acoustic scattering. IMA J. Numer. Anal, 14:523–544, 1994. doi:10.1093/imanum/14.4.523. B. F. Logan and L. A. Shepp. Optimal reconstruction of a function from its projections. Duke Math. J., 42:645–659, 1975. doi:10.1215/S0012-7094-75-04256-8