Discretization and iterational approximation of solutions of nonlinear ill-posed problems (Q1317598)

We consider a nonlinear operator equation \[ Fx=y, \qquad F: X\to Y,\tag{1} \] on a pair of Hilbert spaces, where the operator \(F\) is not continuously inverse, i.e., the problem is ill-posed. It is supposed that the right-hand side is defined with an error \(\delta\): \[ \| y_ \delta- y\|\leq\delta.\tag{2} \] We suggest a regular algorithm to solve the problem (1)--(2). This algorithm consists of two steps. In the first step we apply to problem (1) the variational Tikhonov regularization and discrete approximation by the sequence of finite- dimensional extremal problems \[ \min_{x_ n\in X_ n} \{\| F_ n x_ n- Q_ n y_{\delta_ n}\|^ 2+ \alpha_ n\| x_ n- x_ n^ 0\|^ 2\}, \tag{3} \] where \(\{X_ n\}\) is a sequence of finite- dimensional spaces and \(\{F_ n\}\) is a family of approximating operators. The convergence theorem is formulated and proved. In the second step, we use the iterational Gauss-Newton method with quadratic restrictions for approximate determination of an extremal element in (3); we give the proof of its convergence. As an application, we research a nonlinear one-dimensional equation of gravimetry, and based on general results we describe a regular calculational procedure for the solution of this problem. Our method differs profitably from the approach described in [\textit{C. R. Vogel}, SIAM J. Control Optimization 28, No. 1, 34-49 (1990; Zbl 0696.65096)], because here we do not need to know the norm of exact solution and, under natural assumptions, it enables us to construct a regularizing algorithm for gravimetry problem.