Abstract: In this paper, we introduce a notion of controllability of a nonparametric statistical test and compare the powers of a new controllable nonparametric statistical test and the Wilcoxon— Mann—Whitney test for cases with samples from exponential distributions. PubDate: 2019-07-01

Abstract: Stochastic differential equations (SDEs) with first integrals are considered. Exact solutions of such SDEs belong to smooth manifolds with probability 1. However, numerical solutions do not belong to the manifolds, but belong to their neighborhoods due to numerical errors. The main goal of this paper is to construct modified numerical methods for solving SDEs that have first integrals. In this study, exact solutions for three SDE systems with first integrals are obtained, and the modification of numerical methods is tested on these systems. PubDate: 2019-07-01

Abstract: A hierarchy of minimal mathematical models of the dynamics of the p53-Mdm2- microRNA system has been developed. The models are based on differential equations with a time delay, describing complex interaction mechanisms in the signal system of the p53 protein. We consider two types of interaction of p53 with microRNAs: a positive direct connection and a positive feedback. The feedback of microRNA-p53 is due to a negative effect of the microRNA on the Mdm2 protein, which is a negative regulator of p53. To approximate the direct positive effect of p53 on the microRNAs, a linear function or a representation of the Goldbeter-Koshland type is used. A comparison of numerical solutions with medical and biological data of a number of specific p53-dependent microRNAs is made, which proves that the models and the numerical analysis results are adequate. Special attention is given to analysis of a positive feedback of p53 and microRNAs. The minimal models allow us to consider the most general regularities of the p53-dependent microRNAs. Within the framework of these mathematical models it is shown that it is possible to neglect the Mdm2-miRNA connection for at least some of the most studied microRNAs associated with a direct positive connection with p53. However, those of the microRNAs that are an important negative regulator of Mdm2, can have the most significant impact on the entire p53-Mdm2-microRNA system. Conditions are obtained to manifest the regulatory function of microRNAs with respect to p53. The results of the numerical experiments indicate that such microRNAs can be used as a factor of an anticancer therapy. PubDate: 2019-07-01

Abstract: The paper presents a description of a method for simulation of the motion of bodies in viscous incompressible fluid with the use of a technique of computation on overset grids (“chimera” technique). Equations describing the flow of viscous incompressible fluid are approximated by the finite volume method on an arbitrary unstructured grid. Their iterative solution is implemented using the algorithm SIMPLE. This paper describes the basic equations in the case of moving grid. The features of implementation of grid boundary conditions that are set in the course of construction of interpolation pattern are described. A method for overcoming numerical instability when a solid body model is used is demonstrated. The specificity of taking into account the forces of gravitation in the presence of multiphase media is described. The results of solving the problem of motion of cylinder in fluid, fall of sphere into fluid, and flooding of a ship's model are presented. PubDate: 2019-07-01

Abstract: It is known that dual representation of problems (through the main function and its adjoint in the Lagrange sense) makes it possible to formulate an effective perturbation theory on which the successive approximation method in the inverse problem theory relies. Let us suppose that according to preliminary predictions, a solution to the inverse problem (for example, the structure of medium of interest) belongs to a certain set A. Then selecting a suitable (trial, reference) element a0 as an unperturbed one and applying the perturbation theory, one can approximately describe the behavior of a solution to the forward problem in this domain and find a subset A0 that matches the measurement data best. However, as the accuracy requirements increase, the domain of applicability of the first approximation A0 is rapidly narrowing, and its expansion via addition of higher terms of the expansion complicates the solving procedure. For this reason, a number of works have searched for unperturbed approaches, including the method of variational interpolation (VI method). In this method, not one but several reference problems a1, a2,...,an are selected, from which a linear superposition of the principal function and the adjoint one is constructed, followed by determination of coefficients from the condition of stationarity of the form of the desired functional representation. This paper demonstrates application of the VI method to solving inverse problems of cosmic rays astrophysics in the simplest statement. PubDate: 2019-07-01

Abstract: This paper presents multi-parameter asymptotic description of the stress field near the tip of a central crack in a linear-elastic plate under: (1) normal tensile stress; (2) transverse shear; (3) mixed mode deformation in the full range of mixed modes of loading, from the opening mode fracture to antiplane shear. A multi-parameter expansion of the stress tensor components including higher order terms has been constructed. All the scale (amplitude) factors—coefficients of the complete Williams asymptotic expansion—have been determined as functions of the crack length and parameters of loading. The expansion constructed and formulas obtained for the expansion coefficients can be used for keeping any preassigned number of terms in asymptotic representations of mechanical fields at a crack tip in a plate. The number of components to keep at different distances from the tip of defect was subjected to analysis. The angles of crack propagation under conditions of mixed-mode loading were calculated using a multi-parameter expansion of stress field by means of (1) the maximum tangential stress criterion and (2) the criterion of minimum elastic strain energy density. PubDate: 2019-07-01

Abstract: We present some approaches to solving a problem of shallow water oscillations in a parabolic basin (including an extra case of a horizontal plane). Some requirements on the form of the solutions and effects of Earth’s rotation and bottom friction are made. The resulting solutions are obtained by solving ODE systems. The corresponding free surfaces are first- or second-order ones. Some conditions of finiteness and localization of the flow are analyzed. The solutions are used to verify the numerical algorithm of the large-particle method. The efficiency of the method is discussed in tests on wave run-up on shore structures. PubDate: 2019-07-01

Abstract: The presence of the nonlocal term in the nonlocal problems destroys the sparsity of the Jacobian matrices when solving the problem numerically using finite elementmethod and Newton–Raphson method. As a consequence, computations consume more time and space in contrast to local problems. To overcome this difficulty, this paper is devoted to the analysis of a linearized theta-Galerkin finite element method for the time-dependent nonlocal problem with nonlinearity of Kirchhoff type. Hereby, we focus on time discretization based on θ-time stepping scheme with θ ∈ [½, 1). Some error estimates are derived for the standard Crank–Nicolson (θ = ½), the shifted Crank–Nicolson (θ = ½ + δ, where δ is the time-step) and the general case (θ ≠ ½ + kδ, where k = 0, 1). Finally, numerical simulations that validate the theoretical findings are exhibited. PubDate: 2019-07-01

Abstract: A mathematical model of sea thermodynamics, developed at the Institute of Numerical Mathematics of the Russian Academy of Sciences, is considered. A problem of variational assimilation of sea surface temperature data is studied with allowance for observational error covariance matrices. Based on variational assimilation of satellite observation data, an inverse problem of restoring the heat flux on the sea surface is solved. The sensitivity of some functionals to observation data in the variational assimilation problem is investigated, and the results of numerical experiments with the model applied to Baltic Sea dynamics are presented. PubDate: 2019-04-01

Abstract: In this paper, we present a two-grid scheme for a semilinear parabolic integro-differential equation using a new mixed finite element method. The gradient for the method belongs to the square integrable space instead of the classical H(div; Ω) space. The velocity and the pressure are approximated by the P02–P1 pair which satisfies the inf-sup condition. Firstly, we solve an original nonlinear problem on the coarse grid in our two-grid scheme. Then, to linearize the discretized equations, we use Newton iteration on the fine grid twice. It is shown that the algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy h = O(H6 lnH 2). As a result, solving such a large class of nonlinear equations will not be much more difficult than the solution of one linearized equation. Finally, a numerical experiment is provided to verify theoretical results of the two-grid method. PubDate: 2019-04-01

Abstract: An adaptive analog of Nesterov’s method for variational inequalities with a strongly monotone operator is proposed. The main idea of the method is an adaptive choice of constants in the maximized concave functionals at each iteration. In this case there is no need in specifying exact values of the constants, since this method makes it possible to find suitable constants at each iteration. Some estimates for the parameters determining the quality of the solution to the variational inequality are obtained as functions of the number of iterations. PubDate: 2019-04-01

Abstract: Randomized Monte Carlo algorithms are constructed by a combination of a basic probabilistic model and its random parameters to investigate parametric distributions of linear functionals. An optimization of the algorithms with a statistical kernel estimator for the probability density is presented. A randomized projection algorithm for estimating a nonlinear functional distribution is formulated and applied to the investigation of the criticality fluctuations of a particle multiplication process in a random medium. PubDate: 2019-04-01

Abstract: We consider two related discrete optimization problems of searching for a subset in a finite set of points in Euclidean space. Both problems are induced by versions of a fundamental problem in data analysis, namely, that of selecting a subset of similar elements in a set of objects. In each problem, given an input set and a positive real number, it is required to find a cluster (i.e., a subset) of the largest size under constraints on a quadratic clusterization function. The points in the input set, which are outside the sought-for subset, are treated as a second (complementary) cluster. In the first problem, the function under the constraint is the sum over both clusters of the intracluster sums of the squared distances between the elements of the clusters and their centers. The center of the first (i.e., the sought-for) cluster is unknown and determined as a centroid, while the center of the second one is fixed at a given point in Euclidean space (without loss of generality, at the origin of coordinates). In the second problem, the function under the constraint is the sum over both clusters of the weighted intracluster sums of the squared distances between the elements of the clusters and their centers. As in the first problem, the center of the first cluster is unknown and determined as a centroid, while the center of the second one is fixed at the origin of coordinates. In this paper, we show that both problems are strongly NP-hard. Also, we present exact algorithms for the problems in which the input points have integer components. If the space dimension is bounded by some constant, the algorithms are pseudopolynomial. PubDate: 2019-04-01

Abstract: An automated system for identification of conditions for conduction of homogeneous and heterogeneous reactions includes mathematical modeling of a chemical reaction, determination of criteria for conditions of its optimization and variable parameters, setting and solving a multiobjective optimization problem and an optimal control problem, and development of efficient algorithms for a computing experiment. Modeling and optimization of homogeneous and heterogeneous catalytic reactions are done. Reaction conditions optimal for achievement of given criteria are determined. PubDate: 2019-04-01

Abstract: In this article, a weighted finite difference scheme is proposed for solving a class of parameterized singularly perturbed problems (SPPs). Depending upon the choice of the weight parameter, the scheme is automatically transformed fromthe backward Euler scheme to amonotone hybrid scheme. Three kinds of nonuniform grids are considered, especially the standard Shishkin mesh, the Bakhavalov–Shishkinmesh and the adaptive grid. Themethods are shown to be uniformly convergent with respect to the perturbation parameter for all three types of meshes. The rate of convergence is of first order for the backward Euler scheme and second order for themonotone hybrid scheme. Furthermore, the proposed method is extended to a parameterized problem with mixed type boundary conditions and is shown to be uniformly convergent. Numerical experiments are carried out to show the efficiency of the proposed schemes, which indicate that the estimates are optimal. PubDate: 2019-04-01

Abstract: A method of obtaining analytical solutions to thermal conduction problems is considered. It is based on a separation of thermal conduction into two stages of its time evolution using additional boundary conditions and additional sought-for functions in the heat balance integral method. Solving the partial differential equations is thus reduced to the integration of two ordinary differential equations for some additional sought-for functions. The first stage is characterized by fast convergence of an analytical solution to the exact one. For the second stage, an exact analytical solution is obtained. The additional boundary conditions for both stages are such that their fulfillment by the sought-for solution is equivalent to the fulfillment of the original equation at the boundary points and at the temperature perturbation front. It is shown that if the equation is valid at the boundary points, it is also valid inside the domain. PubDate: 2019-04-01

Abstract: A widespread approach to solving the Cauchy problem for Laplace’s equation is to reduce it to an inverse problem. As a rule, an iterative procedure is used to solve this problem. In this study, an efficient direct method for numerically solving the inverse problem in rectangular domains is described. The main idea is to expand the desired solution with respect to a basis consisting of eigenfunctions of a difference analogue of Laplace’s operator. PubDate: 2019-01-01

Abstract: The Kalman filter is currently one of the most popular approaches to solving the data assimilation problem. A major line of the application of the Kalman filter to data assimilation is the ensemble approach. In this paper, a version of the stochastic ensemble Kalman filter is considered. In this algorithm, an ensemble of analysis errors is obtained by transforming an ensemble of forecast errors. The analysis step is made only for a mean value. Thus, the ensemble π-algorithm combines the advantages of stochastic filters and the efficiency and locality of square root filters. A numerical method of implementing the ensemble π-algorithm is proposed, and the validity of this method is proved. This algorithm is used for a test problem in a three-dimensional domain. The results of numerical experiments with model data for estimating the efficiency of the algorithm are presented. A comparative analysis of the behavior of the root-mean-square errors of the ensemble π-algorithm and the Kalman ensemble filter by means of numerical experiments with a one-dimensional Lorentz model is performed. PubDate: 2019-01-01

Abstract: Algorithms for solving an inverse source problem for production–destruction systems of nonlinear ordinary differential equations with measurement data in the form of time series are presented. A sensitivity operator and its discrete analogue are constructed on the basis of adjoint equations. This operator relates perturbations of the sought-for parameters of the model to those of the measured values. The operator generates a family of quasi-linear operator equations linking the required unknown parameters and the data of the inverse problem. A Newton–Kantorovich method with right-hand side r-pseudo-inverse matrices is used to solve the equations. The algorithm is applied to solving an inverse source problem for an atmospheric pollution transformation model. PubDate: 2019-01-01

Abstract: In this paper, we obtain conditions of existence and uniqueness of periodic solutions for a nonlinear fourth-order differential equation utilizing explicit Green’s function and fixed point index theorem combining with operator spectral theorem. We discuss an iteration method for constant coefficient nonlinear differential equations and establish the theorem on the existence of positive solutions for a fourth-order boundary value problem with a variable parameter. Finally, we give an example to illustrate our results. PubDate: 2019-01-01