Proceedings of Inverse Problems in Engineering: Theory and Practice 3rd Int. Conference on Inverse Problems in Engineering June 13-18, 1999, Port Ludlow, WA, USA IM04 ITERATIVE REGULARIZATIONMETHODS IN INVERSE SCATTERING Thorsten Hohage Institute for Industrial Mathematics, Univ. Linz Altenberger Str. 69 A-4040 Linz Austria Email: hohage@indmath.uni-linz.ac.at ABSTRACT The numerical performances of Landweber iteration, the Newton-CG method, the Levenberg-Marquardt algorithm, and the iteratively Regularized Gauß-Newton method are compared for a nonlinear, severely ill-posed inverse scattering problem in two space dimensions. A modification of the Gauß-Newton method is suggested, which compares favorably with the above methods. A convergence proof is presented including the effects of the numerical approximation of the solution operator. Keywords: inverse obstacle scattering, iterative regulariza- tion methods, operator approximations, convergence rates. 1 Introduction We consider the following problem. Let D IR 2 be a star-shaped, smooth domain describing the cross section of a long, cylindrical scattering obstacle. For an incident plane time- harmonic wave u i x : e ik x d , d 1 k 0 the scattered field u s and the total field u : u i u s satisfy ∆u k 2 u 0 in IR 2 ¯ D (1) r ∂u ∂r iku 0 r x ∞ (2) u 0 on ∂D (3) In acoustics, this describes scattering from a sound-soft obsta- cle, and in electromagnetics, it describes scattering of a polar- ized wave from a perfect conductor. The Sommerfeld radiation condition (2) implies the asymptotic behavior ux 1 x u ∞ x x O 1 x x ∞ (Colton, Kreß 97). The function u ∞ : x : x 1 C , called the far-field pattern of u s , is analytic. We want to solve the inverse problem to identify the shape of D from measurements of u ∞ for one fixed incident wave u i . This problem is difficult to solve since it is nonlinear and severely ill-posed. To formulate it as a nonlinear operator equation Fq u ∞ we describe the boundary ∂D by a radial function q :02π IR: ∂D q : qt cos t sin t : t 02π The characterization of the Fr´ echet derivatives of F , which has been accomplished some years ago by Kirsch and others, has paved the way for the application of iterative regularization methods to solve this and related problems numerically. These methods have been intensively investigated recently, and con- vergence results have been obtained under some conditions on the nonlinearity of the operator. Examples include Landwe- ber iteration (Hanke, Neubauer, Scherzer 95), the iteratively 1 Copyright  1999 by ASME