Volume 16, Issue 6

The brief survey of Ivanov's results concerning quasi-solution is given. Recent effective algorithms for iterative approximation of quasi-solutions to ill-posed problems are described. A general scheme of discrete approximation and examples of its applications to quasi-solution method are presented.

We consider an ill-posed Cauchy problem for an abstract linear differential equation of the first order in a Hilbert space. Both the operator and the initial state are supposed to be noisy. For solving the problem, a class of finite difference methods is used. Coordinating the mesh width with error levels we obtain logarithmic error estimates for the methods of this class.

One of the fundamental results in the theory of ill-posed inverse problems asserts that these problems can become conditionally well-posed when restricting the domain of the forward operator in an appropriate manner. This leads to the study of certain moduli of continuity for the associated restricted inverse operator. The authors systematically study this modulus of continuity and highlight its intimate connection to error bounds of various regularizing procedures. The contributions of V. K. Ivanov and his concept of quasi-solutions are fundamental for such analysis.

A dynamical inversion problem is considered. A regularizing solving algorithm oriented to a quite long time interval of system's functioning is designed. The algorithm is stable with respect to informational noises.

In this paper we derive convergence results for regularized solutions of linear inverse problems obtained by the Bayesian approach in the Ky Fan metric. We show that the convergence rate is order optimal in finite dimensional spaces. Moreover, we prove that order optimal rates can be obtained for weighted Bayesian solutions when the dimension goes to infinity.

For the isotropic elasticity equations we consider ill-posed Cauchy problem with data on a lateral boundary of a cylindrical domain. A stability estimate for solution to this problem is stated. We give exact estimate for the time interval on which the Cauchy data should be given in order to obtain the estimate in the whole cylindrical domain.

In this paper solutions of ill-posed problems with some a priori information about the exact solution are considered. For the first group of such problems it is supposed that the exact solution is a bounded piecewise convex function on some bounded segment [ a, b ]. It is shown that the set of these functions is a compact set in L P [ a, b ] and an approximate solution tends to the exact one uniformly on some subset of [ a, b ]. Sourcewise represented functions form the second group of the problems. For this case it is possible to find a so-called a posteriori error estimation of an approximate solution. The method of extending compacts may help to estimate this a posteriori error.