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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 behavior of the multiplicative acceleration of Selmer's algorithm is widely unknown and no general result on convergence has been detected yet. Solely for its 2-dimensional, periodic expansions, there exist some results on convergence and approximation due to Fritz Schweiger. In this paper we show that periodic expansions of any dimension do in fact converge and that the coordinates of the limit points are rational functions of the largest eigenvalue of the periodicity matrix. [ABSTRACT FROM AUTHOR]

In elliptic curve cryptosystems, scalar multiplications performed on the curves have much effect on the efficiency of the schemes, and many efficient methods have been proposed. In particular, recoding methods of the scalars play an important role in the performance of the algorithm used. For integer radices, the non-adjacent form (NAF) and its generalizations (e.g., the generalized non-adjacent form (GNAF) and the radix- r non-adjacent form ( rNAF)) have been proposed for minimizing the non-zero densities in the representations of the scalars. On the other hand, for subfield elliptic curves, the Frobenius expansions of the scalars can be used for improving efficiency. Unfortunately, there are only a few methods apply the techniques of NAF or its analogue to the Frobenius expansion, namely τ-adic NAF techniques on Koblitz curves and hyperelliptic Koblitz curves. In this paper, we try to combine these techniques, namely recoding methods for reducing non-zero density and the Frobenius expansion, and propose two new efficient recoding methods of scalars on more general family of subfield elliptic curves in odd characteristic. We also prove that the non-zero densities for the new methods are same as those for the original GNAF and rNAF. We estimate scalar multiplication costs on the above subfield elliptic curves in terms of elliptic curve operations and finite field operations for several previous methods and the proposed methods. In addition, we implement scalar multiplication on an subfield elliptic curve belonging to the above family, for the previous methods and a proposed method. As a result, our estimation and implementation show that the speed of the proposed methods improve between 8% and 50% over that for the Frobenius expansion method. [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 propose and justify at the differential level an iteration method for solving the Stokes problem in a rectangular domain with the rate of convergence independent of the ratio of the sides. The numerical implementation of the algorithm is constructed for the cavity test problem and computational results are given. [ABSTRACT FROM AUTHOR]

In this paper, we consider algorithms to pack rectangles into a strip. As the main result we present an algorithm that packs rectangles online and for which the ratio of expected wasted area to expected occupied area tends to zero as the number of rectangles increases. The research was supported by the Russian Foundation for Basic Research, grants 05–01–00798 and 04–01–00359. [ABSTRACT FROM AUTHOR]

In this paper we present two recent results for matrices arising in the mixed finite element method on triangular meshes. The first result is an algebraic proof for the discrete LBB condition. The second one is a new algorithm for the construction of a two-level spectrally equivalent preconditioner for the condensed matrix. [ABSTRACT FROM AUTHOR]

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]

The distribution of jobs in a system with m identical parallel processors which minimises the load of the maximally loaded processor is an NP-hard problem. Many approximate algorithms are developed for this problem, but for the version of the problem where the jobs arrive ad must be treated on-line there is no algorithm possessing the guaranteed estimate which is less than 1 + 1 / ... for m ≥ 4 and tends to 1.837 as m → ∞. In this paper, we consider the version of the problem where jobs arrive one by one and must be treated on-line under the additional condition that the total duration of the jobs is known. For this version of the problem we suggest an algorithm with the guaranteed estimate equal to 5/3. [ABSTRACT FROM AUTHOR]