Successive approximations, propagation algorithms and the inverse obstacle problem

Approximate backpropagation (ABP) methods have been used to identify the shape of axially symmetric acoustic scatterers in the resonance region from full aperture data [C1]. More recently one such method has been applied to the electromagnetic case [C2], where the Ipswich data [MK1] are available. ABP methods have relied on a heuristic relation between the expansion coefficients, which represent the scattered wave in the far zone and, respectively, on the obstacle boundary, $\Gamma$, and have led to minimization algorithms. In spite of satisfactory computational results, the well - posedness of ABP remains an open problem. A pertaining result, which may justify the method, is the following. Let $\lambda, \mu$ be multi-indices, $\{ v_{\mu} \}$ be e.g., the family of outgoing cylindrical (n = 2) wave functions and $\{ f_{\mu} \}:= {\bf f }$ be the sequence of far field scattering coefficients. Denote outward differentiation on $\Gamma$ by $\partial_N$. Assume both series $ \sum_{\mu} f_{\mu} v_{\mu} $, and $ \sum_{\mu} f_{\mu} \partial_N v_{\mu} $ converge uniformly on $ \Gamma $. Let $ {\bf b } $ be a sequence of inner products in $ L^2( \Gamma ) $, which depend on the incident wave, $ u $, and consider the operator $ {\cal R}{\bf L} := - (i/4)[ \langle u_{\lambda} |_{\Gamma} \partial_N v_{\mu} \rangle ] $, where $ u_{\lambda} := \real [v_{\lambda}] $. Also, let $ L $ be the approximation order and $ \Lambda [L] $ the corresponding set of indices. Denote e.g., by $ {\bf b}^{(L)} $ the finite sequence derived from $ {\bf b} $ and by $ {\bf c}^{(L)} $ the vector of least squares boundary coefficients, which solve $ || u + \sum_{\lambda \in \Lambda [L]} c_{\lambda}^{(L)} v_{\lambda}||_{L^2 (\Gamma)} ^2 = {\rm min} $. \textsc{Theorem}. Assume $ {\bf f}, {\bf b}\in \ell^2 $ and $ {\cal R}{\bf L} : \ell^2 \rightarrow \ell^2 $ is bounded. Let the spectral radius $ r_{\sigma} $ of $ {\cal R}{\bf L} $ satisfy $ r_{\sigma} < 1 $. Then a) $ \forall{\bf b}\in\ell^2 $ there exists a unique fixed point $ {\bf f} $ for the map $ {\bf p} [t+ l] = {\bf b} + {\cal R}{\bf L}\cdot{\bf p} [t], t = 0, 1, 2,... $, obtained by successive approximations, started with an arbitrary $ {\bf p}[0] \in\ell^2 $; b) let $ \bar{\bf c}^{(L)} $ be the fixed point of $ {\bf p}^{(L)} [ t+1] = {\bf b}^{(L)}+{\cal R}{\bf L}^{(L)}\cdot{\bf p}^{(L)}[ t ] $, with $ {\bf p}^{(L)}[0] \equiv {\bf c}^{(L)} $ and $ {\bf p}^{(L)}[ 1 ] = {\bf p}^{(L)} $; if $ | f_\lambda - \bar c_\lambda ^{(L)} | < \epsilon_1 | \bar c_\lambda ^{(L)} | $ and $ | c_\lambda ^{(L)} - \bar c_\lambda | - | p_\lambda ^{(L)} - \bar c_\lambda ^{(L)} | > 2 \epsilon_1 | \bar c_\lambda ^{(L)} |, \forall \lambda \in \Lambda [L] $ then forward propagation is effective i.e., $| f_\lambda ^{(L)} - p_\lambda ^{(L)} | < | f_\lambda ^{(L)} - c_\lambda ^{(L)} |, \forall \lambda \in \Lambda [L] $. These results will be applied to a class of numerical problems and their practical repercussions on shape identification will be discussed. References [Cl] G F CROSTA, The Backpropagation Method in Inverse Acoustics, in Tomography, Impedance Imaging and Integral Geometry, LAM 30 (Edited by M CHENEY, P KUCHMENT, E T QUINTO) pp 35 - 68, AMS: Providence, RI (1994). [C2] G F CROSTA, Scalar and Vector Backpropagation Applied to Shape Identification from Experimental Data: Recent Results and Open Problems to appear in Inverse Problems/Tomography and Image Processing, (Edited by A G Ramm) Plenum: New York, NY [MK1] R V McGahan, R E Kleinman, Special Session on Image Reconstruction Using Real Data, IEEE Antennas and Propagation Magazine 38 39 - 40 (1996)