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Research presented in this paper was initiated by the publication [A. D. Agaltsov and R. G. Novikov, Examples of solving the inverse scattering problem and the equations of the Veselov–Novikov hierarchy from the scattering data of point potentials, Russian Math. Surveys74 (2019), 3, 373–386] and is based on its results. Two sets of the complex monopole scattering coefficients are distinguished among the possible values of these coefficients for nonabsorbing inhomogeneities. These sets differ in phases of the scattering coefficients. In order to analyze the features and possibilities of reconstructing the inhomogeneities of both sets, on the one hand, the inverse problem is solved for each given value of the monopole scattering coefficient using the Novikov functional algorithm. On the other hand, the scatterer is selected in the form of a homogeneous cylinder with the monopole scattering coefficient that coincides with the given one. The results obtained for the monopole inhomogeneity and for the corresponding cylindrical scatterer are compared in the coordinate and spatial-spectral spaces. The physical reasons for the similarities and differences in these results are discussed when the amplitude of the scattering coefficient changes, as well as when passing from one set to another. [ABSTRACT FROM AUTHOR]

This is a review paper of the role of Carleman estimates in the theory of multidimensional coefficient inverse problems since the first inception of this idea in 1981. [ABSTRACT FROM AUTHOR]

The article analyzes the development of the variational Tikhonov regularization method for ill-posed problems. A. N. Tikhonov published papers on the development of efficient solution methods for a broad class of ill-posed problems in 1963. He was also credited for coming up with the generalization of the notion of approximate solution. Definitions of the regularizing algorithm and regularized solution are provided.

In this paper we show that an additional a priori information about a sourcewise representation of the solution of nonlinear ill-posed problem can be used for constructing regularizing algorithms with a posteriori error estimates. We describe these algorithms. Algorithms for the case when operators are linear can be obtained from these general algorithms. [ABSTRACT FROM AUTHOR]

An algorithm for the simultaneous determination of two coefficients in an inverse problem for equations of the parabolic type is studied. The model under investigation has been proposed by H. de Vriend and M. Capobianco to describe the long-time coastal profile evolution. The iterative inversion procedure is based on the minimization of a suitable cost functional. Results of numerical tests are shown to illustrate the performance of the algorithm. The real measured data of the coastal profile evolution have been numerically processed to validate the aforementioned diffusion type model. The paper is targetly oriented to attract attention of professionals, who study Inverse Problems, to such new area of applications as Coastal Engineering. [ABSTRACT FROM AUTHOR]

In this paper, an algorithm for solving a 3D inverse kinematic problem of seismics for continuous-inhomogeneous media with zones of considerable horizontal velocity gradients is proposed. This method is a further development of the "nonclassical equation" method proposed earlier for 2D media. [ABSTRACT FROM AUTHOR]

Modeling of large lakes or oceans for environmental studies or deformable blood vessels for mini-invasive surgery involve dynamical free boundaries governed by their interaction with their immediate environment. The object of this paper is to further Shape Analysis techniques in the formulation and analysis of dynamical free boundary problems defined over a time-varying domain (noncylindrical problems). The generic example of a potential incompressible fluid flow with a pressure condition (Bernouilli type) on the dynamical free boundary is presented along with a discrete time algorithm to follow the evolution of the time-space differential equation for the normal component of the velocity field on the time-varying free boundary. These ideas and constructions can be extended to more complex and realistic models. [ABSTRACT FROM AUTHOR]