Linear estimates of accuracy for approximate solutions of inverse problems

Abstract

In this paper, we explore the question of which non-linear inverse problems, which are solved by a selected regularization method, may have so-called linear a priori accuracy estimates – that is, the accuracy of corresponding approximate solutions linearly depends on the error level of the data. In particular, we prove that if such a linear estimate exists, then the inverse problem under consideration is well posed, according to Tikhonov. For linear inverse problems, we find that the existence of linear estimates lead to, under some assumptions, the well-posedness (according to Tikhonov) on the whole space of solutions. Moreover, we consider a method for solving inverse problems with guaranteed linear estimates, called the residual method on the correctness set (RMCS). The linear a priori estimates of absolute and relative accuracy for the RMCS are presented, as well as analogous a posteriori estimates. A numerical illustration of obtaining linear a priori estimates for appropriate parametric sets of solutions using RMCS is given in comparison with Tikhonov regularization. The a posteriori estimates are calculated on these parametric sets as well.

Keywords:

• inverse problems
• well-posedness according to Tikhonov
• linear error estimate

AMS Subject Classifications:

• Class code: 47A52
• 47J06
• 65N20

Additional information

Funding

The work was supported by the Russian Foundation for Basic Research [grant number 14-01-00182-a] and [grant number 14-01-91151-GFEN-a].