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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]

An inverse problem of reconstructing the two-dimensional coefficient of thewave equation is solved by a stochastic projection method. We apply the Gel'fand-Levitan approach to reduce the nonlinear inverse problem to a family of linear integral equations. The stochastic projection method is applied to solve the relevant linear system. We analyze the structure of the problem to increase the efficiency of the method by constructing an improved initial approximation. A smoothing spline is used to treat the random errors of the method. The method has low cost andmemory requirements. Results of numerical calculations are presented. [ABSTRACT FROM AUTHOR]

• Gradient-based approach is a common scheme for solving the inverse problems. • It is crucial to efficiently compute the gradient of the functional. • The proposed algorithm for the gradient calculation is 10 percent faster. • Low memory requirements are important for large grids. Numerical solution of inverse problem for 2D acoustic system of conservation laws by gradient type method requires storage of O (N 3) elements which is crucial on large grids with O (N) points in single dimension. In this article we present an approach to save twice memory on the stage of adjoint problem and gradient calculation and compare it with usual approach in memory and CPU time cost. Numerical comparison for CPU time and memory of one step of iteration process which consists of direct problem solution, adjoint problem solution and calculation of the gradient are presented. [ABSTRACT FROM AUTHOR]