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This article deals with the question of the unique solvability of systems of linear algebraic equations, to the solution of which many inverse problems of geophysics are reduced as a result of discretization when applying the methods of integral equations or integral representations. Examples are given of degenerate and nondegenerate systems of different dimensions that arise in the processing of magnetometric and gravimetric data from experimental observations. Conclusions are drawn about the methods for constructing the optimal grid of experimental observation points. [ABSTRACT FROM AUTHOR]

This paper considers a model for the accumulation of mutations in a population of mice with a weakened function of polymerases responsible for correcting DNA copying errors during cell division. The model uses the results of the experiment published by Japanese scientists, which contain data on the accumulation of phenotypic differences in three isolated groups of laboratory mice. We have developed a model for the accumulation of negative mutations. Since the accumulation of phenotypic differences in each of the three groups of mice occurred in its own way, we assumed that these differences were associated with genotypic differences in the zeroth generation and set the inverse problem of determining the initial distribution of these differences. Additional information for solving the inverse problem was a set of experimental data on the number of mutant lines and the number of individuals in each group of mice. The results obtained confirmed our assumption. [ABSTRACT FROM AUTHOR]

The work continues a series of articles devoted to the peculiarities of solving coefficient inverse problems for nonlinear singularly perturbed equations of the reaction-diffusion-advection-type with data on the position of the reaction front. In this paper, we place the emphasis on some problems of the numerical solving process. One of the approaches to solving inverse problems of the class under consideration is the use of methods of asymptotic analysis. These methods, under certain conditions, make it possible to construct the so-called reduced formulation of the inverse problem. Usually, a differential equation in this formulation has a lower dimension/order with respect to the differential equation, which is included in the full statement of the inverse problem. In this paper, we consider an example that leads to a reduced formulation of the problem, the solving of which is no less a time-consuming procedure in comparison with the numerical solving of the problem in the full statement. In particular, to obtain an approximate numerical solution, one has to use the methods of the numerical diagnostics of the solution's blow-up. Thus, it is demonstrated that the possibility of constructing a reduced formulation of the inverse problem does not guarantee its more efficient solving. Moreover, the possibility of constructing a reduced formulation of the problem does not guarantee the existence of an approximate solution that is qualitatively comparable to the true one. In previous works of the authors, it was shown that an acceptable approximate solution can be obtained only for sufficiently small values of the singular parameter included in the full statement of the problem. However, the question of how to proceed if the singular parameter is not small enough remains open. The work also gives an answer to this question. [ABSTRACT FROM AUTHOR]

The paper considers the features of numerical reconstruction of the advection coefficient when solving the coefficient inverse problem for a nonlinear singularly perturbed equation of the reaction-diffusion-advection type. Information on the position of a reaction front is used as data of the inverse problem. An important question arises: is it possible to obtain a mathematical connection between the unknown coefficient and the data of the inverse problem? The methods of asymptotic analysis of the direct problem help to solve this question. But the reduced statement of the inverse problem obtained by the methods of asymptotic analysis contains a nonlinear integral equation for the unknown coefficient. The features of its solution are discussed. Numerical experiments demonstrate the possibility of solving problems of such class using the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, approaches to the numerical recovering of the initial condition in the inverse problem for a nonlinear singularly perturbed reaction–diffusion–advection equation are considered. The feature of the formulation of the inverse problem is the use of additional information about the value of the solution of the equation at the known position of a reaction front, measured experimentally with a delay relative to the initial moment of time. In this case, for the numerical solution of the inverse problem, the gradient method of minimizing the cost functional is applied. In the case when only the position of the reaction front is known, the method of deep machine learning is applied. Numerical experiments demonstrated the possibility of solving such kinds of considered inverse problems. [ABSTRACT FROM AUTHOR]