In this paper, the relationship between representation formulas for unique regularized solutions of inverse source, as well as backward problems with final overdetermination for evolution equations and singular value decomposition (SVD) of corresponding input–output operators, is analysed. For each considered inverse problem this representation formula is derived via the solution of appropriate adjoint problems and the parameter of regularization. The Lagrange multiplier method is then used to show that each regularized inverse problem is equivalent to a coupled system of two (direct and adjoint) parabolic (or hyperbolic) problems, and the Lagrange multiplier is the weak solution of the corresponding adjoint problem. In the constant coefficient linear parabolic and hyperbolic equations cases, it is proved that the coupled problem is equivalent to the normal equation corresponding to each inverse/backward problem. As a result, an equivalence of the representation formula for unique regularized solution and the SVD of the input–output operator is obtained. This approach, in particular, allows one to construct regularizing filters for a considered class of inverse problems. Some numerical examples are presented to illustrate the usefulness and effectiveness of the proposed approach.

The direct task of the subsurface exploration of a homogeneous medium with surface relief by the resistivity method is analyzed. To calculate the resistivity field for such a medium, the method of integral equations was successfully applied for the first time. The corresponding integral equation for the density of secondary current sources on the surface of the medium was established. The method of computational grid construction, adapted to the characteristics of the surface relief, was developed for the numerical solution of the integral equation. This method enables the calculation of the resistivity field of a point source on a surface that is not smooth and allows for steep ledges. Numerical examples of the calculation of resistivity fields and apparent resistivity are shown. The anomalies of apparent resistivity arising from the deviation of the surface shape from a flat medium were quantitatively established as model examples. Calculations of apparent resistivity for the direct current sounding method were carried out using modifications of the electrical tomography approach.

This study explores an application of the quasi-solution and the Fourier methods used to solve a mixed initial-boundary problem involving Laplace’s equation in a rectangle. The problem is reduced to a system consisting of mutually dependent direct and adjoint problems. The explicit formula for the quasi-solution is obtained using the Fourier method. Computational experiments are performed for different types of synthetic data, and admissible parameter ranges are established.