Rate of convergence of stochastic iteration procedures in ill-posed problems (Q1106544)

The author considers the iteration scheme \[ u_{n+1}=u_ n- g(A^*_{\eta} A_{\eta}) A^*_{\eta} (Au_ n-f_{\delta})+ w_ n,\quad n\geq 0, \] for solving the (ill-posed) operator equation \(Au=f\). Here A and \(A_{\eta}\) are linear (not necessarily bounded) operators acting from the Hilbert space H to the Hilbert space F, H is separable; f and \(f_{\delta}\) are elements of F, and the \(w_ n\) \((n\geq 0)\) are random variables on a probability space that take their values in H; \(A^*_{\eta}\) is the adjoint of \(A_{\eta}\); and g is a bounded nonnegative Borel function on the nonnegative real numbers. It is assumed that \(\| A_{\eta}-A\| \leq \eta\), \(\| f_{\eta}- f\| \leq \delta\), \(E^ w_ n=0\) and \(E(\| w_ n\|^{\alpha})\leq C(\alpha)\epsilon^{\alpha}\), for any \(\alpha >0\), where \(C(\alpha)\) is a constant depending on \(\alpha\), and E is the expectation in the measure used. The random variables \(w_ n\), \(n\geq 0\), are assumed to be independent in their totality. Under additional technical assumptions, the author studies the rate of convergence of the above iterative scheme in the norm of \(L_ 1\). He determines conditions which ensure an optimal order of convergence as well as an a posteriori stopping rule based on the deviation (\(\tau\) being the stopping time): \[ \tau =\inf \{n\geq 0:\quad \| A_{\eta}u_ n-f_{\delta}\| \leq \mu \quad or\quad n\geq d\mu^{-2}\},\quad d\in (0,\infty). \]