ABSTRACT
Figure 8.5. Reconstruction of the mask›type aw in elastic medium: kpx1(t) = cos t + 0.65 (cos 2t – 1); kpx2(t) = 1.5 sin t; 0 ≤ t ≤ 2pi
Helpful remarks 1◦. In the case of acoustically hard contour, when the boundary condition has the form ∂p/∂n|l = 0, the problem reduces to a system of two nonlinear integro›differential equations. However, this is not an obstacle to the application of the method discussed below also to this case. 2◦. The methods proposed here differ from other known methods: the Gauss–Newton and Lowenberg–Marquardt methods (see Gill et al., 1981), the smoothing functional method (Tikhonov and Arsenin, 1977; and Section 8.2), and others. In the Lowenberg–Marquardt procedure, for example, a unit step along the direction of descent is always xed. And in the smoothing functional method, α cannot depend on the iteration number (if an iterative procedure is used to minimize the smoothing functional). 3◦. The method discussed in the present section at rst sight seems to coincide with the method proposed by Roger (1981) (see our brief survey in Section 8.4), but this is not so. Roger uses the Newton–Kantorovich method to solve a nonlinear functional equation which, strictly speaking, does not converge in the case of the rst›kind operator equation with compact operators (both linear and nonlinear). In contrast with this, our approach is based on the steepest descent method whose convergence is strictly proved (see Section 8.3), at least for quadratic discrepancy functionals, corresponding to linear operator equations. Locally each functional is quadratic, so although we cannot strictly prove the convergence of our method in the nonlinear case, there is a good chance, starting from the proved convergence for quadratic functional, that the proposed algorithm converges also in the more general nonlinear case considered here.
