ИСТИНА

Traditional well-known methods for solving inverse problems encounter some problems and limitations in solving the inverse problems for nonlinear singularly perturbed partial differential equations whose solutions contain stationary or moving fronts. The idea of using asymptotic analysis to construct effective numerical algorithms for solving problems of such kind was first realized for the coefficient inverse problem for a nonlinear singularly perturbed equation of the reaction-diffusion-advection type with data at the final time moment was considered. The iterative gradient method was implemented and for its effective realization a priori information about the location and dynamics of the moving front, extracted by the asymptotic analysis, was used. However, there is another efficient direction for using asymptotic analysis in solving inverse problems of considered classes. An important feature of the asymptotic analysis is that it gives the possibility to reduce the original nonlinear singularly perturbed problem for the partial differential equation to much simpler problems that do not contain small parameters and have less dimension (and sometimes even contain not differential but algebraic equations). Moreover, problem statements reduced using asymptotic analysis often relate explicitly or semi-explicitly the parameters that must be restored in solving the inverse problem (coefficients in the equation, boundary and initial conditions, etc.) with the additional information. For example, it may be information about the location of moving reaction front which is the most natural in real applications (experimental observations of the location of the shock front, reaction or combustion front, etc.).