Numerical Methods in Inverse Obstacle Scattering
by
Rainer Kress
University of Göttingen

The scattering of   time-harmonic acoustic and electromagnetic waves by  impenetrable  obstacles is modeled by exterior boundary value problems for the reduced wave equation. Given an incident wave uⁱ, the scattered wave   u^s  must satisfy the Helmholtz equation \Delta u + k² u = 0 with wave number k > 0 in the exterior of the scattering obstacle D, the Sommerfeld radiation condition, and a boundary condition such as uⁱ+u^s=0 on \partial D for a sound-soft scatterer. The radiation condition leads to an asymptotic behavior for the scattered wave of the form

us(x) = [\frac {ei k|x|}{|x|}][u\infty(x/|x|)+ O(1/|x|)],     |x| --> \infty,

where the function u_\infty  is the  far field pattern of the scattered wave. The basic  inverse obstacle scattering problem now  is, given the far field pattern u_\infty of the scattered wave u^s for one or several incident plane waves, to determine the shape of D. As opposed to the direct scattering problem which is linear and well-posed, the inverse  problem is nonlinear and severely ill-posed.

Roughly speaking one can distinguish   two different approaches for the approximate solution of the full nonlinear inverse obstacle scattering problem: In a first group of methods the inverse obstacle problem is separated into a linear ill-posed part for the reconstruction of the scattered wave u^s from the far field pattern u_\infty and a nonlinear well-posed part for finding the location of the boundary \partialD from the Dirichlet boundary condition. In a second group of methods  the inverse obstacle problem is either considered as an ill-posed nonlinear operator equation or reformulated as a nonlinear optimization problem and regularized iterative techniques are applied. We will survey the basic ideas of methods from both groups, including numerical examples in  two dimensions.

For an introduction to inverse scattering we refer to Colton, D., \  and Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory. 2nd. ed. Springer-Verlag, Berlin Heidelberg New York 1998.

http://www.num.math.uni-goettingen.de/kress/

Date received: July 19, 1999