ANALYTICAL AND NUMERICAL METHODS FOR SOLVING THE INVERSE PROBLEM OF DETERMINING THE PROPORTION OF BURNED BIOMASS IN LANDSCAPE FOREST FIRES WITH DATA ON THE POSITION OF THE COMBUSTION FRONT

A model of a landscape forest fire is considered, consisting of two equations: describing the temperature distribution during a forest fire and an equation describing the distribution of burned biomass in a two-dimensional case. Based on this initial model, an inverse problem was posed to determine the proportion of burned biomass in landscape forest fires based on observations of the fire front propagation dynamics in a two-dimensional observation area. It is known that forest fires do not burn all of the forest biomass, but only a certain proportion. Determining the proportion of burned biomass is an important applied problem, the rapid solution of which allows us to draw conclusions about the danger of such fires and, in some cases, the feasibility of extinguishing them. The difficulty in solving these types of problems in practice is due to the fact that the problems under consideration belong to the class of inverse problems for singularly perturbed partial differential equations. Classical methods for solving inverse problems for partial differential equations are based on gradient methods for minimizing the objective functional, which necessitates multiple solutions of the direct and adjoint problems. Asymptotic analysis methods allow the original formulation of the inverse problem to be reduced to a simpler (reduced) problem, the computational complexity of which is significantly lower.However, the approach used has limited applicability conditions, some of which are also thoroughly examined in the paper using model examples.