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In this paper, we consider an inverse backward problem for a nonlinear singularly perturbed parabolic equation of the Burgers' type. We demonstrate how a method of asymptotic analysis of the direct problem allows developing a rather simple algorithm for solving the inverse problem in comparison with minimization of the cost functional. Numerical experiments demonstrate the effectiveness of this approach. [ABSTRACT FROM AUTHOR]

We investigate the mathematical modeling of the 2D acoustic waves propagation, based on the conservation laws. The hyperbolic first-order system of partial differential equations is considered and solved by the method of S. K. Godunov. The inverse problem of reconstructing the density and the speed of sound of the medium is considered. We apply the gradient method to reconstruct the parameters of the medium. The gradient of the functional is obtained. Numerical results are presented. [ABSTRACT FROM AUTHOR]

The coefficient inverse problem for the acoustic equation is considered. We propose the method for reconstructing the density based on the N-approximation by the finite system of one-dimensional problems and the two-dimensional M. G. Krein approach. The two-dimensional analogue of the M. G. Krein approach is applied to reduce the non-linear inverse problem to a family of linear integral equations. We consider the fast algorithm for solving the relevant linear system, based on using the block-Toeplitz structure of the matrix. The algorithm applied to the M. G. Krein equation allows to obtain the solution of the whole family of the integral equations by solving only one linear system. Results of numerical calculations are presented. [ABSTRACT FROM AUTHOR]

The coefficient inverse problem for the two-dimensional wave equation is solved. We apply the Gelfand-Levitan approach to transform the nonlinear inverse problem to a family of linear integral equations. We consider the Monte Carlo method for solving the Gelfand-Levitan equation. We obtain the estimation of the solution of the Gelfand-Levitan equation in one specific point, due to the properties of the method. That allows the Monte Carlo method to be more effective in terms of span cost, compared with regular methods of solving linear system. Results of numerical simulations are presented. [ABSTRACT FROM AUTHOR]