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Numerical Analysis and Applications

2020 year, number 4

New accuracy estimates for methods for localizing discontinuity lines of a noisy function

A.L. Ageev, T.V. Antonova
Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia
Keywords: ill-posed problems, regularization method, discontinuity lines, global localization, discretization, separability threshold

Abstract >>
We consider the ill-posed problem of localizing (finding the position) the discontinuity lines of a function of two variables, provided that the function of two variables is smooth outside of the discontinuity lines, and at each point on the line has a discontinuity of the first kind. There is a uniform grid with the step τ . It is assumed that we know the averages on the square τ x τ of the perturbed function at each node of the grid. The perturbed function approximates the exact one the in space L 2( mathbb R 2). The perturbation level δ is known. Earlier, the authors investigated (obtained accuracy estimates) the global discrete regularizing algorithms for approximating a set of discontinuity lines of a noisy function. However, stringent smoothness conditions were superimposed on the discontinuity line. The main result of this paper is the improvement of localizing the accuracy estimation methods, which allows replacing the smoothness requirement with a weaker Lipschitz condition. Also, the conditions of separability are formulated in a more general form, as compared to previous studies. In particular, it is established that the proposed algorithm make it possible to obtain the localization accuracy of the order O( δ ). Also, estimates of other important parameters characterizing the localization algorithm are given.

Construction of reachable sets of controlled systems with second order of accuracy with respect to time step

A.A. Ershov1,2
1Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia
2Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg, Russia
Keywords: modified Euler method, second-order Runge-Kutta method, control system, reachable set, switching of control

Abstract >>
The paper investigates the pixel method for constructing reachable sets of a dynamic controlled system. Sufficient conditions for a control system have been obtained under which the explicit second order Runge-Kutta method (a modified Euler method) provides the second order of accuracy with respect to a time step in constructing reachable sets, even if discontinuous functions are in the class of admissible controls.

An efficient algorithm for stochastic ensemble smoothing

E.G. Klimova
Federal Research Center for Information and Computational Technologies SB RAS, Novosibirsk, Russia
Keywords: data assimilation, ensemble Kalman filter, ensemble smoother

Abstract >>
The state of the environment using a mathematical model and observational data The state of the environment using a mathematical model and observational data using a data assimilation procedure is assessed. The Kalman ensemble filter is one of the widespread data assimilation algorithms at present. An important component of the data assimilation procedure is the assessment not only of the predicted values, but also of the parameters that are not described by the model. A single improvement procedure from observational data in the Kalman ensemble filter may not provide a required accuracy. In this regard, the ensemble smoothing algorithm, in which data from a certain time interval are used to estimate values at a given time, is becoming increasingly popular. This paper considers a generalization of the previously proposed algorithm, which is a version of the Kalman stochastic ensemble filter. The generalized algorithm is an ensemble smoothing algorithm, in which smoothing is performed for the average value of a sample and then the ensemble of perturbations is transformed. The transformation matrix proposed in the paper is used to estimate both the predicted value and the parameter. An important advantage of the algorithm is its locality, which makes it possible to estimate a parameter in a given domain. The paper provides a rationale for the applicability of this algorithm to the implementation of ensemble smoothing. Test calculations were performed with the proposed numerical algorithm with a 1-dimensional model of transport and diffusion of passive impurity. The algorithm proposed is effective and can be used to assess the state of the environment.

Mathematical modeling and forecasting of COVID-19 in Moscow and Novosibirsk region

O.I. Krivorotko1,2,3, S.I. Kabanikhin1,2,3, N.Yu. Zyatkov1, A.Yu. Prikhodko1,2,3, N.M. Prokhoshin2,3, M.A. Shishlenin1,2,3
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia
2Mathematical Center in Akademgorodok, Novosibirsk, Russia
3Novosibirsk State University, Novosibirsk, Russia
Keywords: mathematical modeling, pandemic, COVID-19, SIER-HCD, SIER-D, scenarios, inverse problem, identifiability, optimization, differential evolution, annealing simulation, genetic algorithm, Moscow, Novosibirsk region

Abstract >>
We investigate the inverse problems of finding unknown parameters of the SEIR-HCD and SEIR-D mathematical models of the spread of COVID-19 coronavirus infection based on additional information about the number of detected cases, mortality, self-isolation coefficient and tests performed for the city of Moscow and the Novosibirsk region since 23.03.2020. In the SEIR-HCD model, the population is divided into seven, and in SEIR-D - into five groups with similar characteristics and with transition probabilities depending on a specific region. An analysis of the identifiability of the SEIR-HCD mathematical model was made, which revealed the least sensitive unknown parameters as related to additional information. The task of determining parameters is reduced to the minimization of objective functionals, which are solved by stochastic methods (simulated annealing, differential evolution, genetic algorithm). Prognostic scenarios for the disease development in Moscow and in the Novosibirsk region were developed and the applicability of the developed models was analyzed.

Development of a metaheuristic programming method for the nonlinear models synthesis

O.G. Monakhov, E.A. Monakhova
Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia
Keywords: metaheuristic programming method, genetic algorithm, genetic programming, grammatical evolution, Cartesian Genetic Programming, nonlinear models, bioinspired algorithms

Abstract >>
The solution of the problem of building nonlinear models (mathematical expressions, functions, algorithms, programs) based on an experimental data set, a set of variables, a set of basic functions and operations is considered. A metaheuristic programming method for the evolutionary synthesis of nonlinear models has been developed that has a representation of a chromosome in the form of a vector of real numbers and allows the use of various bioinspired (nature-inspired) optimization algorithms in the search for models. The effectiveness of the proposed algorithm is estimated using ten bioinspired algorithms and compared with a standard algorithm of genetic programming, grammatical evolution and Cartesian Genetic Programming. The experiments have shown a significant advantage of this approach as compared with the above algorithms both with respect to time for the solution search (greater than by an order of magnitude in most cases), and the probability of finding a given function (a model) (in many cases at a twofold rate).

On monotonicity of CABARET scheme approximating the multidimensional scalar conservation law

V.V. Ostapenko, T.V. Protopopova
М.А. Lavrentiev's Institute of Hydrodynamics, Siberian Branch of Russian Academy of Sciences, Novosibirsk, Russia
Keywords: CABARET difference scheme, multidimensional scalar conservation law, monotonicity property

Abstract >>
The monotonicity of the two-layer with respect to time CABARET scheme approximating the multidimensional scalar conservation law is analyzed. There is proposed a modification of this scheme. This modification of the CABARET scheme retains the monotonicity of the one-dimensional difference solutions in the linear approximation, and, as a result, it provides an increased smoothness during the calculation of the multidimensional discontinuous solutions. The results of test calculations are given. They illustrate the advantages of the modified scheme.

Solving the (1+n)-dimensional fractional Burgers equation by natural decomposition method

M. Cherif1,2, D. Ziane1, A.K. Alomari3, K. Belghaba1
1Laboratory of mathematics and its applications, University of Oran1 Ahmed Ben Bella, Oran, Algeria
2Oran's Hight School of Electrical and Energetics Engineering, Oran, Algeria
3Department of Mathematics, Science Yarmouk University, Irbid, Jordan
Keywords: Adomian decomposition method, natural transform, (1+n)-dimensional Burgers equation, Caputo fractional derivative

Abstract >>
In this paper, we extend the natural transform combined with the Adomian decomposition method for solving nonlinear partial differential equations with time-fractional derivatives. We apply the proposed method to obtain approximate analytical solutions of the (1+ n )-dimensional fractional Burgers equation. Some illustrative examples are given, which reveal that this is a very efficient and accurate analytical method for solving nonlinear fractional partial differential equations.

Sensitivity of functionals of the solution of a variational data assimilation problem with simultaneous reconstruction of heat fluxes and the initial state for the sea thermodynamics model

V.P. Shutyaev, E.I. Parmuzin
Marchuk Institute of Computational Mathematics of the Russian Academy of Sciences, Moscow, Russia
Keywords: variational data assimilation, optimal control, adjoint equations, covariance matrices, sensitivity of functionals, sea thermodynamics model

Abstract >>
For the mathematical model of the sea thermodynamics, developed at the Institute of Numerical Mathematics of the Russian Academy of Sciences, the problem of variational data assimilation is considered, aimed at simultaneous reconstruction of the sea surface heat flux and the initial state of the model. The sensitivity of functionals with respect to observational data in the considered problem of variational assimilation is studied, and the results of numerical experiments for the model of the Baltic Sea dynamics are presented.