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Numerical Analysis and Applications

2016 year, number 4

Recovering a tsunami source and designing an observational system based on the r-solution method

T. A. Voronina
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva 6, Novosibirsk, 630090, Russia
Keywords: цунами, численное моделирование, обратная некорректная задача, регуляризация, сингулярное разложение, решения, tsunami numerical modeling, ill-posed inverse problem, regularization, singular value decomposition and solution

Abstract >>
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.

On iterative methods for solving equations with covering mappings

T. V. Zhukovskaia1, E. S. Zhukovskiy2,3
1Tambov State Technical University, 392000, Russian Federation, Tambov, Sovetskaya st., 106, Russia
2Tambov State University named after G.R. Derzhavin, 392000, Russian Federation, Tambov, Internatsional'naya st., 33
3Peoples' Friendship University of Russia, 117198, Russian Federation, Moscow, Mikluho-Maklaya st., 6
Keywords: итерационные методы решения уравнений, накрывающие отображения метрических пространств, приближенное решение, iterative methods for solving equations, covering mappings in metric spaces, approximate solution

Abstract >>
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.

Regularizing algorithms with optimal and extra-optimal quality

A.S. Leonov
National Research Nuclear University, 115409, Moscow, Kashirskoe shosse, 31
Keywords: некорректные задачи, регуляризующие алгоритмы, качество приближенного решения, апостериорная оценка качества, РА с экстраоптимальным качеством, ill-posed problems, regularizing algorithms, quality of approximate solution, a posteriori quality estimates, RA with extra-optimal quality

Abstract >>
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.

Optimal finite difference schemes for the wave equation

A. F. Mastryukov
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva 6, Novosibirsk, 630090, Russia
Keywords: волновое уравнение, электромагнитные волны, конечно-разностный метод, оптимальный, точность, метод Лагерра, система линейных уравнений, итерации, wave equation, electromagnetic wave, finite-difference, optimal, accuracy, Laguerre method, linear system of equations

Abstract >>
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.

Sequential data assimilation algorithms in air quality monitoring models based on weak-constraint variational principle

A. V. Penenko, V. V. Penenko, E. A. Tsvetova
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Lavrentieva, 6, Novosibirsk, 630090, Russia
Keywords: усвоение данных, вариационный принцип, слабые ограничения, прямые и обратные задачи, модель как регуляризатор, последовательные алгоритмы, data assimilation, variational principle, weak-constraint, direct and inverse problems, model as regularizer, sequential algorithms

Abstract >>
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.

Quadrature interpolation type formulas for hypersingular integrals in the interval of integration

L. Yu. Plieva1,2
1Southern Mathematical Institute of the Vladikavkaz Scientific Center of the Russian Academy of Sciences, str. Vatutina, 53, Vladikavkaz, Republic North Ossetia-Alania, 362027, Russia
2North Ossetian State University named after K.L. Hetagurov, str. Vatutina, 44, Vladikavkaz, Republic North Ossetia-Alania, 362025, Russia
Keywords: гиперсингулярный интеграл, квадратурная формула, оценка погрешности, hypersingular integral, quadrature formula, the estimation error

Abstract >>
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.

A difference scheme for a conjugate-operator model of the heat conduction problem on non-matching grids

S. B. Sorokin1,2
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva, 6, Novosibirsk, 630090, Russia
2Novosibirsk State University, Pirogova st., 2, Novosibirsk, 630090, Russia
Keywords: задача теплопроводности, математическая модель, дискретный аналог, несогласованная сетка, сходимость, разностная схема, problem of heat conductivity, mathematical model, discrete analog, non-matching grid, convergence, difference scheme

Abstract >>
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.

Analysis of the pontoon fluctuations with a seasonally changing parameter of stability on the astir surface of finite water depth

S. V. Cherdantsev1, N. V. Cherdantsev2
1Kuzbass State Technical University, Vesennaya str., 28, Kemerovo, 650026, Russia
2Institute of Coal, Siberian Branch of Russian Academy of Sciences, Leningradskii pr., 10, Kemerovo, 650065, Russia
Keywords: зумпф угольного разреза, понтон, потенциал скоростей, частота волн, ватерлиния, метацентрические высоты, присоединенные массы жидкости, параметрическая качка понтона, уравнение Матье, диаграмма Айнса-Стретта, sump of an open coal mine, pontoon, potential of the velocities, frequency of the waves, waterline, metacentric heights, added masses of liquid, parametric pitching pontoon, Mathieu equation, Inc-Strutt stability diagram

Abstract >>
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.

Classification of real pairs of commuting Toeplitz and Hankel matrices

V. N. Chugunov1, Kh. D. Ikramov2
1Institute of Numerical Mathematics, Russian Academy of Sciences, Gubkina st., 8, Moscow, 119991, Russia
2Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics, Leninskie Gory, Moscow, 119899, Russia
Keywords: теплицева матрица, ганкелева матрица, циркулянт, косой циркулянт, перестановочность, Toeplitz matrix, Hankel matrix, circulant, skew-circulant, commutativity

Abstract >>
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.