Direct and iteration methods for solving inverse and ill-posed problems

In this paper we describe direct and iteration methods for solving inverse and ill-posed problems such as backwards parabolic equation and Cauchy problem for Laplace equation. We as well study the inverse problem for the acoustic equation. The boundary control method and the Gel'fand–Levitan–Krein method are investigated for recovering some characteristics of the density. We describe numerical algorithm for the inverse acoustic problem. The nonlinear inverse problem is reduced to the system of linear algebraic equations. We demonstrate the results of computer simulation.