Algorithm construction and numerical solution based on the gradient method of one inverse problem for the acoustics equation

Abstract

The paper considers the problem of continuation of solutions of hyperbolic equations from a part of the domain boundary. These problems include the Cauchy problem for a hyperbolic equation with data on a timelike surface. In the inverse problems, the inhomogeneities are located at some depth under the medium layer, the parameters of which are known. In this case, an important tool for practitioners are the problems of continuation of geophysical fields from the Earth's surface towards the lay of inhomogeneities. In equations of mathematical physics, solution of the continuation problem from part of the boundary is in many cases strongly ill-posed problems in classes of functions of finite smoothness. The ill-posedness of this problem is considered, that is, the example of Hadamard, a Cauchy problem for a hyperbolic equation, is given. The physical formulation of the continuation problem is considered and reduced to the inverse problem. The definition of the generalized solution is formulated and the correctness of the direct problem is presented in the form of a theorem. The inverse problem is reduced to the problem of minimizing the objective functional. The objective functional is minimized by the Landweber method. By the increment of the functional, we consider the perturbed problem for the direct problem. We multiply the equation of the perturbed problem by some function and integrate by parts, we obtain the formulation of the conjugate problem. After that, we get the gradient of the functional. The algorithm for solving the inverse problem is listed. A finite-difference algorithm for the numerical solution of the problem is presented. The numerical solution of the direct problem is performed by the method of inversion of difference schemes. The results of numerical calculations are presented