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We introduce the concept of very weak solution to a Cauchy problem for elliptic equations. The Cauchy problem is regularized by a well-posed non-local boundary value problem whose solution is also understood in a very weak sense. A stable finite difference scheme is suggested for solving the non-local boundary value problem and then applied to stabilizing the Cauchy problem. Some numerical examples are presented for showing the efficiency of the method. [ABSTRACT FROM AUTHOR]

The inverse problem for the acoustic equation is considered. We propose a method of reconstruction of the density approximating 2D inverse acoustic problem by a finite system of one dimensional inverse acoustic problems. The 2D analogy of the Gel'fand--Levitan--Krein method is established. The inverse acoustic problem is formulated and the short outline of the history and development in this field are given in Section 1. In Section 2 we consider the 2D analogy of the Gel'fand--Levitan--Krein equation. The N-approximation of the Gel'fand--Levitan--Krein equation is obtained for inverse acoustic problem in Section 3. The numerical results are presented in Section 4. [ABSTRACT FROM AUTHOR]