Regularization of ill-posed problems by stochastic iteration procedures in the mean
The author studies a general iteration method of Vainikko for an ill- posed linear operator equation in Hilbert space with approximately given operator and data. In addition, it is assumed that independent random errors influence each iteration. A stopping rule for the iteration which depends on knowledge of an upper bound for the true solution and bounds for the error in the data and operator as well as the satisfaction of a certain inequality involving the expected norm of the random errors, is proposed. The author proves that the approximate solution using this stopping rule converges in expectation to the exact solution as the various error levels go to zero, i.e., the method using the stopping rule is a regularization procedure in the sense of Tikhonov.
scientific article; zbMATH DE number 4011803