Stepwise Approximation Research Articles

Automated control of the immune response to the causants of infectious disease

The problems that arose as a result of the rapid spread of the new coronavirus infection COVID-19 and the inability of health care institutions to provide the necessary medical care to a large number of patients in a timely manner showed the need to develop new approaches to diagnosis and the formation of an individual treatment program. Important in this direction is the development and implementation of specialized decision support systems based on the integration of expert systems for the analysis and interpretation of medical data and mathematical modeling tools. In order to form effective treatment programs, it is necessary to ensure the availability of reliable tools for predicting the dynamics of the disease in the conditions of external medical influences, various protection mechanisms, spatial effects, mixed infections etc. The purpose of this work is to generalize the basic model of viral infection to predict the complex controlled influence of adsorption and immune therapy on the development of the disease, taking into account diffusion disturbance and temperature response of the body, which is an important element of specialized decision-making systems for the formation of effective treatment programs. To find the solution of the original model singularly perturbed problem with delays, the procedure of stepwise numerical asymptotic approximation of the solutions of the corresponding specially constructed sequence of problems without delay has been modernized. Based on the obtained results of computer modeling, it was confirmed that the introduction of adsorbents during periods of intensive growth of the number of viral elements makes it possible to sufficiently effectively reduce the rate of their reproduction, and, therefore, provides an additional way of controlling the number of antigens in the body and preventing their supercritical growth. It is emphasized that taking into account the features of the action of adsorption drugs is important for making optimized decisions regarding the use of adsorption therapy

Heat and mass transfer in the processes of drying thin natural leathers pasted on smooth surfaces

The results of the study of drying thin natural leather pasted on a smooth surface are presented. The results of solving the differential equation of non-stationary heat conduction with constant thermophysical coefficients are used to determine the average temperature during the period of decreasing drying rate. The calculated values of temperatures for moisture-proof and moisture-free skin surfaces are given. Methods for simplifying solutions of nonlinear equations with the aim of linearizing these equations for approximating solutions with variable transport coefficients are considered. The use of the method of piecewise stepwise approximation of the transfer coefficients with constant values of these coefficients over short time intervals showed that for low-intensity drying processes, the solutions of the linear heat equation are implemented quite accurately, confirming the regularities obtained empirically. The calculation of the duration of drying according to the method of B. S. Sazhin is given. The results of the processing of experimental curves for drying leathers pasted are presented. The reliability of the obtained equations is verified and the experiment is compared with the calculated values using the formulas. The obtained approximate analytical solutions with reliable values of the transfer coefficients, confirmed by experimentally established regularities, are of practical importance. Together with experimental methods, analytical methods make it possible to establish optimal drying regimes and more accurately generalize experimental data. 

The logit dynamic in supermodular games with a continuum of strategies: A deterministic approximation approach

We consider large population supermodular games with pairwise interaction and a continuous strategy set. Our objective is to establish convergence of the logit dynamic in such games to logit equilibria. For this purpose, we apply the deterministic approximation approach, which interprets a deterministic dynamic as an approximation of a stochastic process. We first establish the closeness of this dynamic with a step–wise approximation. We then show that the logit stochastic process is close to the step–wise logit dynamic in a discrete approximation of the original game. Combining the two results, we obtain our deterministic approximation result. We then apply this result to supermodular games. Over finite but sufficiently long time horizons, the logit stochastic process converges to logit equilibria in a discrete approximation of the supermodular game. By the deterministic approximation approach, so does the logit dynamic in the continuum supermodular game.

Mesh-free least-squares reverse-time migration

AbstractThe reflectivity of the subsurface can be precisely determined using least-squares reverse-time migration (LSRTM). As LSRTM necessitates solving the wave equation, the numerical solution method of the wavefield directly determines the quality of the migration image. The conventional LSRTM method usually uses the finite difference method based on a regular grid to calculate the wavefield. Due to the stepwise approximation of an irregular surface with a regular grid, scattering noise may occur in the propagation of the wavefield, which affects the quality of the image. In addition, the conventional LSRTM cannot effectively handle the models with rugged topography. The finite difference method generated by radial basis functions (FD-RBF) is a mesh-free method and can construct interpolation functions to solve the wave equations numerically according to arbitrarily distributed spatial coordinate points. Therefore, we use the FD-RBF method to develop a mesh-free LSRTM approach to eliminate the influence of the inherent limitation of a regular grid on the imaging. Numerical examples show that the mesh-free LSRTM method can better represent the curved or steep interface within the model and is also suitable for models with rugged topography. The LSRTM method can provide higher-quality images and effectively reduce the memory required for calculations.

Моделирование кинетики нагревания подложки при распылении сэндвич-мишени медь-титан

This study focuses on the special aspects of the heat flow formation during the sputtering of a sandwich target with an internal copper plate and an external titanium one in argon. The kinetics of the heat flow generated by the external plate was determined by the numerical solution of the two-dimensional uniform Fourier’s equation with a set of boundary and zero initial conditions. The Heat Transfer Module of the COMSOL Multiphysics software was used for numerical calculations. In calculations, the grid element size was chosen to be 3.64 mm, which ensures a calculation error of ≤ 5 %. Solving the Fourier’s equation in the time range of 0…500 s with a step of 1 s resulted in a set of the surface temperature distributions of the external plate. When studying heat flows, the external plate was presented as a set of concentric radiation sources with a constant surface temperature. The central source had the shape of a disk, all the others were ring-shaped. The geometric dimensions of the sources and their temperatures were determined by solving the problem of stepwise approximation of the external plate’s surface temperature distribution with a relative error of 10 %. This article provides calculation results for power of 1,000 W and total relative area of the cutouts of 0.25. The comparison of calculated and experimental data is presented.

Experimental Study of the Suppression Effect of N2/CO2 On the LPG Explosion Behavior in a Half-Open Duct with Large Aspect Ratios

ABSTRACT This work aims to investigate the effect of N2/CO2 on the explosion behavior of premixed liquefied petroleum gas (LPG)-air in a half-open duct. Experiments were performed in a duct (L/D = 47.5) with the ignition end closed and the other end open. Firstly, the optimal explosion concentration of LPG was determined using the stepwise approximation method, secondly, the effects of different volume fractions (VFs) of N2/CO2 on the pressure dynamic process, flame propagation velocity, and explosion severity of LPG-air explosion were analyzed, and finally the inerting effects of N2 and CO2 were compared using the control variable method. The results show that the explosion pressure and flame propagation velocity reach a maximum when the VF of LPG is 4.8% (without inert gas), and two pressure peaks (the first pressure peak (P 1 ) and the second pressure peak (P 2 )) are observed. When the VF of LPG is 4.8%, the maximum pressure and the average flame propagation velocity in the tube decrease and the explosion intensity inside and outside the duct was effectively mitigated as the VF of N2/CO2 increases, and the explosion is completely suppressed as the VF of inert gas reached 30%. P 2 is significantly suppressed, and the maximum explosion pressure (P max ) occurs at when the VF of N2/CO2 is above 15%. The suppression effects of CO2 are more remarkable at a lower VF (below 18%), those of N2 are more significant at a higher VF (above 18%), and P 1 also follows this rule. For a given volume fraction of inert gas, the suppression effect of N2/CO2 on the degree of explosion severity becomes more significant when approaching the optimal LPG explosion concentration (4.8%).

Application of a spectral method to simulate quasi-three-dimensional underwater acoustic fields

The calculation of a three-dimensional underwater acoustic field has always been a key problem in computational ocean acoustics. Traditionally, this solution is usually obtained by directly solving the acoustic Helmholtz equation using a finite difference or finite element algorithm. Solving the three-dimensional Helmholtz equation directly is computationally expensive. For quasi-three-dimensional problems, the Helmholtz equation can be processed by the integral transformation approach, which can greatly reduce the computational cost. In this paper, a numerical algorithm for a quasi-three-dimensional sound field that combines an integral transformation technique, stepwise coupled modes and a spectral method is designed. The quasi-three-dimensional problem is transformed into a two-dimensional problem using an integral transformation strategy. A stepwise approximation is then used to discretize the range dependence of the two-dimensional problem; this approximation is essentially a physical discretization that further reduces the range-dependent two-dimensional problem to a one-dimensional problem. Finally, the Chebyshev–Tau spectral method is employed to accurately solve the one-dimensional problem. We provide the corresponding numerical program SPEC3D for the proposed algorithm and describe several representative numerical examples. In the numerical experiments, the consistency between SPEC3D and the analytical solution/high-precision finite difference program COACH verifies the reliability and capability of the proposed algorithm. A comparison of running times illustrates that the algorithm proposed in this paper is significantly faster than the full three-dimensional algorithm in terms of computational speed.

A novel algorithm to solve for an underwater line source sound field based on coupled modes and a spectral method

A high-precision numerical sound field is the basis of underwater target detection, positioning and communication. A line source in a plane is a common type of sound source in computational ocean acoustics. The exciting waveguide in a range-dependent ocean environment is often structurally complicated; however, traditional algorithms often assume that the waveguide has a simple seabed boundary and that the line source is located at a horizontal range of 0 m, although this ideal situation is rarely encountered in the actual ocean. In this paper, a novel algorithm is designed that can solve for the sound field excited by a line source at any position in a range-dependent ocean environment. The proposed algorithm uses the classic stepwise approximation approach to address the range dependence of the environment and uses the Chebyshev–Tau spectral method to solve for the horizontal wavenumbers and modes of approximately range-independent segments. Once the modal information of these flat segments has been obtained, a global matrix is constructed to solve for the coupling coefficients of all segments, and finally, the complete sound field is synthesized. Numerical experiments using a robust numerical program developed based on this algorithm verify the correctness and usability of our novel algorithm and software. Furthermore, a detailed analysis and test of the computational cost of this algorithm show that it is efficient.

АПРОКСИМАЦІЯ ОБВІДНИХ КВАЗІСИНУСОЇДАЛЬНИХ ЦИФРО-АНАЛОГОВИХ СИГНАЛІВ

The principle of setting the instantaneous values of signals with smoothed envelopes using first-order basis splines, which is optimal from the point of view of equipment costs and the computing power of a microprocessor is proposed. A computer model of a multiphase generator has been developed with the help of which the principles of stepwise and linear approximation of the signal envelope have been investigated. References 11, figures 5, table 1.

Approximation of SDE solutions using local asymptotic expansions

We develop an asymptotic expansion for small time of the density of the solution of a non-degenerate system of stochastic differential equations with smooth coefficients, and apply this to the stepwise approximation of solutions. The asymptotic expansion, which takes the form of a multivariate Edgeworth-type expansion, is obtained from the Kolmogorov forward equation using some standard PDE results. To generate one step of the approximation to the solution, we use a Cornish–Fisher type expansion derived from the Edgeworth expansion. To interpret the approximation generated in this way as a strong approximation we use couplings between the (normal) random variables used and the Brownian path driving the SDE. These couplings are constructed using techniques from optimal transport and Vaserstein metrics. The type of approximation so obtained may be regarded as intermediate between a conventional strong approximation and a weak approximation. In principle approximations of any order can be obtained, though for higher orders the algebra becomes very heavy. In order 1/2 the method gives the usual Euler approximation; in order 1 it gives a variant of the Milstein method, but which needs only normal variables to be generated. However the method is somewhat limited by the non-degeneracy requirement.

Properties of Harmonic and Composite Half-Waves, Determination of the Uniform Time Sampling Interval of Digital Signal Processors

When building autonomous real-time systems (RTS), it is necessary to solve the problem of optimal multitasking loading of a number of parallel functioning digital signal processors. One of the reserves for achieving the desired result is the implementation of samples from the sensor signals of information about the magnitude of the signal most rarely in time. In this case, it is necessary to provide a linear or stepwise approximation of the signal by samples with an acceptable reconstruction error. One of the system tasks of these processors is filtering signals or limiting the spectrum to the cutoff frequency. A distinctive feature of the approach proposed in the article is the fulfillment of the condition: if the measurement of this frequency is difficult (for example, in the electromechanical means of the RTS), then for such signals it is proposed to match the maximum values of the harmonic half-wave parameters: approximation error, speed and acceleration. The study opens up the prospect of applying new approaches to sampling the time of signals in the amplitude-time domain and determining the equivalent cutoff frequency of the signal spectrum for such signals. In this article, the dependences of the value of the unit of system time for input-output of data on the degree of agreement between the maximum values of the signal parameters are obtained. A mathematical model of the extreme behavior of a signal between two adjacent samples is given in the form of a harmonic half-wave. The study is also extended to convex composite harmonic functions, according to which the signal can deviate from the results of a linear or stepwise approximation of the signal for these samples. The comparison of the models by the value of the relative time sampling intervals, depending on the degree of matching of the maximum parameters of the harmonic half-wave, is carried out. When comparing, in addition to these maximum parameters, the relationship of the maximum signal speed with the error of approximating the samples by steps and the relationship of the maximum acceleration of the signal with the maximum error of the linear approximation was taken into account. The results make it possible to determine the duration of the intervals of uniform sampling of the signal time based on the results of the inspection of the control object, substantiate a significant increase in the sampling interval of time or a similar increase in the number of tasks to be solved per unit of system time.

Calculating the Wave Force on Partially Immersed Large-Scale Horizontal Cylinders

Large-scale interceptors constitute the main structure of offshore self-driven floating marine litter collection devices, and the structural stability of such interceptors under the action of waves directly influences the overall safety of the device. When the ratio of the diameter of a horizontal cylinder in such interceptors to the incident wavelength is larger than 0.25, the wave force can be calculated by using the diffraction theory, by considering the problem as that of the interaction between the waves and a partially immersed large-scale horizontal cylinder. In this study, an analytical approach to calculate the wave force on a partially immersed large-scale horizontal cylinder was formulated by using the stepwise approximation method. Physical model tests were conducted to investigate the effects of different factors (wave height, period, and immersion depth) on the wave force on a large-scale horizontal cylinder under conditions involving short-period waves. The results show that both horizontal and vertical wave forces on the cylinder increase as the wave height (immersion depth) increases in most cases. The vertical wave force decreases with the decrease of the period. For the horizontal wave force, it increases with the decrease of the period when the wavelength is larger than the diameter of the cylinder and decreases with the decrease of the period when the wavelength is smaller than the diameter of the cylinder.

Heat and mass transfer in the process of heat treatment and drying of natural leather with the convective method of energy supply

In the work, the authors investigated the possibility of using the results of analytical solutions of the linear differential equations of unsteady heat conduction with constant heat transfer coefficients to calculate the temperature of the material during heat treatment of leathers. Heat treatment of natural leathers as heat-sensitive materials is carried out under mild temperature conditions and high air moisture contents, the temperature does not undergo significant changes, and the heat transfer coefficients change almost linearly. When using analytical solutions, the authors made the assumptions that for small temperature gradients over the cross section of a thin body, the thermal transfer of matter can be neglected and for values of the heat and mass transfer Biot criteria less than unity, the main factor, limiting heat and mass transfer, is the interaction of the evaporation surface of the body with the environment; so, in solving the differential heat equation we can restrict ourselves to one first member of an infinite series. In this case, a piecewise stepwise approximation of all thermophysical characteristics with constant values of these coefficients at the calculated time intervals was applied, which made it possible to take into account the change in the transfer coefficients throughout the entire heat treatment process. Processing of experimental data showed that in low-intensity processes with reliable values of the transfer coefficients, it is possible to use the results of solutions of differential equations of unsteady heat conduction in heat transfer calculations. The results of the study of heat transfer during drying of leather confirm the laws of temperature change established experimentally. Together with experimental studies of drying processes, analytical studies are of great practical importance in the development of new methods for calculating heat and mass transfer in wet bodies.

Enhancing the hydrologic system differential response method for flood forecasting correction

The differential response of hydrologic system (HSDR) can represent the relation between errors of model outputs and influencing variables, which has been proposed for error correction in flood forecasting. In this study, the HSDR method with stepwise approximation is established to update areal mean rainfall in whole basin (HSDR_AMP) and spatial distributed rainfall in each sub-basin (HSDR_SDP), respectively. The regularized least square algorithm is employed to calculate the solutions. The performance of the proposed method on improving the predictions by the Xinanjiang (XAJ) model is demonstrated in a synthetic case with different rainfall errors and in real basins with different flow characteristics. The results show that 1) the HSDR can improve flood predictions by updating rainfall; 2) the improved HSDR with stepwise approximation outperforms that without iteration; 3) the HSDR_SDP converges faster than the HSDR_AMP; 4) the HSDR_SDP with enough observations used for updating can perform better due to the consideration of rainfall spatial distribution; 5) when increasing lead time the HSDR performance generally deteriorates slowly than the autoregressive (AR) technique. To some extent, the HSDR for error updating can utilize physical mechanism of hydrological process represented by hydrological models. With clear concept, simple structure and stable performance, the enhanced method is worthy of further applications for real-time flood forecasting and ultimately extend the understanding of error feedback in hydrologic process.

Spatial runoff updating based on the hydrologic system differential response for flood forecasting

Abstract Runoff controls water volume in the rainfall–runoff process and plays a dominant role in flood forecasting. This study proposes a spatial runoff updating approach based on the hydrologic system differential response (HSDR). The first-order partial derivative is employed to express the hydrologic response to runoff change. A stepwise approximation for the HSDR is suggested to reduce the effect of linearization of the nonlinear hydrologic system. The regularized least square algorithm is used to calculate the estimated errors of runoff. The HSDRs for spatial distributed runoff (SDR) updating and areal mean runoff (AMR) updating are examined to correct the predictions of the Xinanjiang model in two basins in China. The case results show that the HSDR for runoff updating can improve flood predictions; the HSDR with stepwise approximation outperforms that without it; the HSDR_SDR performs better than the HSDR_AMR; and with increasing lead time, the HSDR method exhibits more stable performance than the autoregressive (AR) technique on streamflow correction. The proposed HSDR_SDR method can decompose the information of residuals between observations and calculations to update spatial runoff through the response matrix for each sub-basin. With simple structure and stable performance, the HSDR_SDR is convenient and effective for real-time flood forecasting.

Accounting Method of Filling Cells for the Solution of Hydrodynamics Problems with a Complex Geometry of the Computational Domain

This article is devoted to the development and application of the filling cells method for the solution of hydrodynamics problems with a complicated geometry of the computational domain, in particular, a liquid domain, to increase the smoothness and accuracy of the finite-difference solution. The spatial-two-dimensional flow problem of a viscous fluid between two coaxial semicylinders and the spatial-three-dimensional problem of wave propagation in the coastal zone demonstrate the possibilities of the proposed method. The rectangular grids are used to solve these problems, taking into account the filling of cells. The approximation of problems are used to split schemes in time for physical processes and the approximation in spatial variables is made using the balance method, taking into account the filling of cells and without it. An analytical solution describing the Taylor-Couette flow is used as the reference to assess the accuracy of the numerical solution of the first problem. The simulation is performed on a a sequence of condensing computational grids with the following dimensions: 11 × 21, 21 × 41, 41 × 81, and 81 × 161 nodes in the case of using the method and without using it. In the case of the direct use of rectangular grids (stepwise approximation of boundaries), the relative error of the calculations reaches 70%; under the same conditions, the use of the proposed method allows us to reduce the error to 6%. It is shown that splitting up the rectangular grid by factors of 2 to 8 in each of the spatial directions does not lead to the same increase in the accuracy of the numerical solutions obtained taking into account the filling of the cells.

Structure of a Planar Brush of Grafted Polyelectrolyte Stars

For planar brushes made of grafted polyelectrolyte stars, an analytical theory based on a stepwise approximation of the brush density profile is developed. Particular attention is paid to the effect of formation of a layered structure with the division of stars into two populations and the influence of arm charge and salt concentration on this phenomenon. It is shown that an increase in the number of stars with an extremely stretched arm by which the star is grafted to the surface is facilitated by an increase in the grafting density, the number of arms, or the degree of ionization of the arms. In the latter case, the average brush density decreases, while in the former two cases it increases. It is demonstrated that the theory based on the three-step approximation of the density profile describes well the results of self-consistent field modeling and the best agreement is achieved at a high degree of ionization, a large number of arms, and a high grafting density.

NUMERICAL REALIZATION OF THE ALGORITHM FOR SOLVING THE TASKS OF HYDROPHYSICS IN THE AREA WITH COMPLEX GEOMETRY

The work is devoted to the development and numerical implementation of an algorithm for solving the problem of modeling the process of hydrophysics, describing the transport of polluting biogenic substances in a complex shape. The velocity field of the water flow, calculated according to the hydrodynamic model of the Sea of Azov, is used in the model of transport of polluting nutrients as input data. To test the accuracy of the numerical solution of the hydrodynamic model problem, a test problem of viscous fluid flow between two coaxial semicylinders is used. To solve the interrelated problems of hydrophysics, including a model of hydrodynamic processes and a model of transport of polluting nutrients, rectangular grids are used, taking into account the “fullness” of the cells. The approximation of the problem in time is made on the basis of splitting schemes for physical processes. To assess the accuracy of the numerical solution of hydrodynamic problems, an analytical solution describing the Couette–Taylor flow is used as a reference. The simulation was carried out on a sequence of thickening computational grids with dimensions: 11×21, 21×41, 41×81 and 81×161 nodes in the cases of a smooth and stepped border. To increase the smoothness of the solution, grids were used that take into account the “fullness” of the cells. In the case of a stepwise approximation of the interface between two media, the calculation error reaches 70 % of the error solution of the problem. When using grids that take into account the “fullness” of cells, the error in the numerical solution of model problems of hydrodynamics, caused by the approximation of the boundary, does not exceed 6 % of the error solution of the problem. The example shows that the fragmentation of the grid by 8 times for each of the spatial directions does not lead to an increase in the accuracy that solutions obtained on the grids, which take into account the “fullness” of the cells, possess.

METHOD OF FORECASTING THE TERMS OF SETTLING OF STRUCTURES ON SOILS WITH SILT LAYERS

The method of forecasting the size and timing of sedimentation of structures on soils with silt layers, based on the results offield observations, has been developed. It is suggested to use a set of exponential dependencies with a constant component.The algorithm and software for calculation of the envelope by experimental data by a stepwise approximation are developed.The process of consolidation of soils is considered as a combination of simultaneous and independent flow of phases of primary filtration consolidation and secondary consolidation of creep. According to the results of data processing of observations by direct iterative calculations by finding the minimum nonconnectedness by the method of least squares.

Virtual reality-based measurement of ocular deviation in strabismus

Background and objectiveStrabismus is an eye movement disorder in which shows the abnormal ocular deviation. Cover tests have mainly been used in the clinical diagnosis of strabismus for treatment. However, the whole process depends on the doctor's level of experience, which could be subjected to several factors. In this study, an automated technique for measurement of ocular deviation using a virtual reality (VR) device is developed. MethodsA VR display system in which the screens that have the fixation target are changed alternately between on and off stages is used to simulate the normal strabismus diagnosis steps. Patients watch special-designed 3D scenes, and their eye motions are recorded by two infrared (IR) cameras. An image-processing-based pupil tracking technique is then applied to track their eye movement. After recording eye motion, two strategies for strabismus angle estimation are implemented: direct measurement and stepwise approximation. The direct measurement converts the eye movement to a strabismus angle after considering the eyeball diameter, while the stepwise approximation measures the ocular deviation through the feedback calibration process. ResultsExperiments are carried out with various strabismus patients. The results are compared to those of their doctors’ measurement, which shows good agreement. ConclusionsThe results clearly indicate that these techniques could identify ocular deviation with high accuracy and efficiency. The proposed system can be applied in small space and has high tolerance for the unexpected head movements compared with other camera-based system.