Home – Home – Jornals – Numerical Analysis and Applications 2016 number 4

This study deals with the application of the solution method to recover the initial tsunami waveform in a tsunami source area by inverting the remote water-level measurements for a real event. The inverse problem in question is regarded as the so-called ill-posed problem and it is regularized by means of the least square inversion using the truncated Singular Value Decomposition method. The method presented allows one to control the instability of the numerical solution and to obtain an acceptable result in spite of the ill-posedness of the problem. Moreover, it is possible to make a preliminary prediction of the quality of the inversion with a given set of observational stations and to estimate further changes in the inversion result after modifying the monitoring system.

In this paper we propose an iterative method for solving the equation, where a mapping \Upsilon acts in metric spaces, is covering in the first argument and Lipschitzian in the second one. Each subsequent element of a sequence of iterations is defined by the previous one as a solution to the equation, where can be an arbitrary point sufficiently close to. The conditions for convergence and error estimates have been obtained. The method proposed is an iterative development of the Arutyunov method for finding coincidence points of mappings. In order to determine it is proposed to perform one step using the Newton-Kantorovich method or the practical implementation of the method in linear normed spaces. The obtained method of solving the equation of the form coincides with the iterative method proposed by A.I. Zinchenko, M.A. Krasnosel'skii, I.A. Kusakin.

The notion of a special quality for approximate solutions to ill-posed inverse problems is introduced. A posteriori estimates of the quality are studied for different regularizing algorithms (RA). Examples of typical quality functionals are provided, which arise in solving linear and nonlinear inverse problems. The techniques and the numerical algorithm for calculating a posteriori quality estimates for approximate solutions of general nonlinear inverse problems are developed. The new notions of optimal and extra-optimal quality of a regularizing algorithm are introduced. The theory of regularizing algorithms with optimal and extra-optimal quality is presented, which includes an investigation of optimal properties for estimation functions of the quality. Examples of regularizing algorithms with extra-optimal quality of solutions are given, as well as examples of regularizing algorithms without such property. The results of numerical experiments illustrate a posteriori quality estimation.

This paper considers the solution of the two-dimensional wave equation with the use of the Laguerre transform. The optimal parameters of finite difference schemes for this equation have been obtained. Numerical values of these optimal parameters are specified. The finite difference schemes of second order with optimal parameters give the accuracy of the solution to the equations close to the accuracy of the solution by the scheme of fourth order. It is shown that using the Laguerre decomposition, the number of optimal parameters in comparison with the Fourier decomposition can be reduced. This reduction leads to simplification of finite difference schemes and to reduction of the number of computations, i.e. the efficiency of the algorithm.

Data assimilation problem for non-stationary model is considered as a sequence of the linked inverse problems which reconstruct, taking into account the different sets of measurement data, the space-time structure of the state functions. Data assimilation is carried out together with the identification of additional unknown function, which we interpret as a function of model uncertainty. The variational principle serves as a basis for constructing algorithms. Different versions of the algorithms are presented and analyzed. Based on the discrepancy principle, a computationally efficient algorithm for data assimilation in a locally one-dimensional case is constructed. The theoretical estimation of its performance is obtained. This algorithm is one of the core components of the data assimilation system in the frames of splitting scheme for the non-stationary three-dimensional transport and transformation models of atmospheric chemistry.

A hypersingular integral on the interval of integration with the weight function is considered. We prove the spectral ratios for hypersingular integrals on [-1, 1]. The quadrature formulas for certain integrals with the weight function are constructed. The estimation error is presented.

On non-matching grids discrete analogue conjugate-operator models of heat conduction, keeping the structure of the original model are constructed. Numerical experiments show that the difference scheme converges with second order of accuracy for the case of discontinuous parameters of the medium in the Fourier law and non-uniform grids.

It is shown that due to the periodic changes in metacentric heights of a pontoon on the astir surface of liquids in the sump of an open coal mine, the pontoon is capable to produce parametric pitching, both in the longitudinal and in the transverse directions. The equation, describing parametric pitching, is transformed to the Mathieu equation, whose factors depend both on the own frequencies and the pontoon floatability features on «calm water», and on the frequency of fluctuation of a liquid, which, in turn, is defined by the sump size. The installed regularities between parameters, characterizing parametric pitching in the longitudinal and transverse directions, and areas of its instability are revealed.

We give a complete description of the set of matrix pairs such that T is a real Toeplitz matrix, H is a real Hankel matrix, and TH = HT.