New Algorithm to Solve Mixed Integer Quadratically Constrained Quadratic Programming Problems Using Piecewise Linear Approximation

Techniques and methods of linear optimization underwent a significant improvement in the 20th century which led to the development of reliable mixed integer linear programming (MILP) solvers. It would be useful if these solvers could handle mixed integer nonlinear programming (MINLP) problems. Piecewise linear approximation (PLA) is one of most popular methods used to transform nonlinear problems into linear ones. This paper will introduce PLA with brief a background and literature review, followed by describing our contribution before presenting the results of computational experiments and our findings. The goals of this paper are (a) improving PLA models by using nonuniform domain partitioning, and (b) proposing an idea of applying PLA partially on MINLP problems, making them easier to handle. The computational experiments were done using quadratically constrained quadratic programming (QCQP) and MIQCQP and they showed that problems under PLA with nonuniform partition resulted in more accurate solutions and required less time compared to PLA with uniform partition.

A two-stage method for real-time baseline drift compensation in gas sensors

Baseline drift caused by slowly changing environment and other instability factors affects significantly the performance of gas sensors, resulting in reduced accuracy of gas classification and quantification of the electronic nose. In this work, a two-stage method is proposed for real-time sensor baseline drift compensation based on estimation theory and piecewise linear approximation. In the first stage, the linear information from the baseline before exposure is extracted for prediction. The second stage continuously predicts changing linear parameters during exposure by combining temperature change information and time series information, and then the baseline drift is compensated by subtracting the predicted baseline from the real sensor response. The proposed method is compared to three efficient algorithms and the experiments are conducted towards two simulated datasets and two surface acoustic wave sensor datasets. The experimental results prove the effectiveness of the proposed algorithm. Moreover, the proposed method can recover the true response signal under different ambient temperatures in real-time, which can guide the future design of low-power and low-cost rapid detection systems.

Approximations of Fuzzy Numbers by Using r-s Piecewise Linear Fuzzy Numbers Based on Weighted Metric

Using simple fuzzy numbers to approximate general fuzzy numbers is an important research aspect of fuzzy number theory and application. The existing results in this field are basically based on the unweighted metric to establish the best approximation method for solving general fuzzy numbers. In order to obtain more objective and reasonable best approximation, in this paper, we use the weighted distance as the evaluation standard to establish a method to solve the best approximation of general fuzzy numbers. Firstly, the conceptions of I-nearest r-s piecewise linear approximation (in short, PLA) and the II-nearest r-s piecewise linear approximation (in short, PLA) are introduced for a general fuzzy number. Then, most importantly, taking weighted metric as a criterion, we obtain a group of formulas to get the I-nearest r-s PLA and the II-nearest r-s PLA. Finally, we also present specific examples to show the effectiveness and usability of the methods proposed in this paper.

Simulating Nonlinear Dynamics of a 3D Crystal Lattice of Metals

Oscillations of crystal lattices determine important material properties such as thermal conductivity, heat capacity, thermal expansion, and many others; therefore, their study is an urgent and important problem. Along with experimental studies of the nonlinear dynamics of a crystal lattice, effective computer simulation techniques such as ab initio simulation and the molecular dynamics method are widely used. Mathematical simulation is less commonly used since the calculation error there can reach 10 %. Herewith, it is the least computationally intensive. This paper describes the process and results of mathematical simulation of the nonlinear dynamics of a 3D crystal lattice of metals using the Lennard-Jones potential in the MatLab software package, which is well-proven for solving technical computing problems. The following main results have been obtained: 3D distribution of atoms over the computational cell has been plotted, proving the possibility of displacement to up to five interatomic distances; the frequency response has been evaluated using the Welch method with a relative RMS error not exceeding 30 %; a graphical dependence between the model and the reference cohesive energy data for a metal HCP cell has been obtained with an error of slightly more than 3 %; an optimal model for piecewise-linear approximation has been calculated, and its 3D interpolation built. All studies performed show good applicability of mathematical simulation to the problems of studying dynamic processes in crystal physics.

Investigation of the dielectric fatigue on the example of lead titanate films PbTiO3

The phenomenon of dielectric fatigue of active dielectrics, which consists in a decrease in the residual polarization depending on the number of switching cycles, is researched. A model of the dependence of the residual polarization of ferroelectric materials on the number of switching cycles is proposed. The model is based on piecewise - linear approximation of the results of measurements of the hysteresis loops of thin films PbTiO3 at a temperature T = 470 (°C), the electric field strength E = 100 (kV/cm). The developed model was used in the development of a technique for studying dielectric fatigue, depending on different modes of material switching.

CHANGE OF ENERGY CHARACTERISTICS OF LITHOSPHERE ELEMENTS DURING THEIR DEFORMATION

Purpose. The purpose of this study is to establish analytical patterns for predicting changes in stress and energy density spent on the destruction of rocks according to experimental studies. To solve this purpose in the article were set the following scientific problems: 1) analytical description of the dependence of the stress σij on the main deformation εij; 2) establishment of calculation parameters that are included in the analytical patterns; 3) analytical description and study of fracture energy density curves. Methodology. In the course of analytical and experimental researches of full diagrams of deformation of rocks the mathematical model of dependence of the stress on the deformation is developed. Physico-mechanical processes of all characteristic sections of the complete deformation diagram were also analyzed and described. Analysis of the resulting curve showed that the rock mass and elements of the lithosphere are not perfectly elastic or plastic objects. Along with the elastic ones, plastic ones are always present to one degree or another. The integration of the obtained analytical expression σ11 = f(ε11) allowed to establish the volumetric energy density spent on the destruction of the rock sample under the action of external load. The maximum activation energy for the considered rock is 0.67 MJ/m3. A comparison of the experimental and calculated values of the energy dependence u(ε1) shows a coincidence over almost the entire range of deformation changes (ε11 = 0..0.04). Findings. The study of rock samples at hard stress allowed to obtain a complete deformation characteristics of the rock. The curve that surrounds the deformation cycles (1) combines pre-boundary, boundary, extremal modes of deformation and destruction of rocks. Equation (4) allows us to establish that the destruction can occur at different values of energy density U(ε). Originality. An analytical description of the energy diagram of deformation and a complete diagram of stress change in the form of a single dependence, which takes into account the boundary and extremal areas, was developed in the work. In contrast to the method of piecewise linear approximation, this approach corresponds to the physics of the process and reduces errors in calculations. Practical implications. Theoretical and experimental analysis of the obtained energy fracture diagrams and complete stress change diagrams in rocks allows to estimate the bearing capacity of a rock mass or other solid body. This allows you to predict critical values of stresses and external loads to prevent failure in a timely manner.

Odd Mathieu Functions Application to Synthesize a Multi-element Radiator Flat-topped Radiation Pattern

The paper studies synthesizing capabilities of a flat-topped radiation pattern when using the expansion of the target radiation pattern into a series in terms of odd Mathieu functions. As parameters for comparing the target and synthesized radiation patterns, we used a main-lobe width at a level of -1 dB and an irregularity of the top of the main-lobe of the radiation pattern. The sector-shaped radiation pattern has been synthesized for linear radiators of various lengths. The convergence of the coefficients of the Mathieu series in the synthesis of the sector-shaped radiation pattern has been estimated. It is shown that the use of piecewise-linear approximation of the target radiation pattern in the synthesis using a series expansion into odd Mathieu functions allows us to improve the quality of the radiation pattern formed.The task that involved finding the amplitude-phase distribution for a linear emitter with a length of 3λ, 4λ and 5λ (λ is operation wavelength) for a target radiation pattern was solved. The target amplitude distribution has the following electrical characteristics: the main-lobe width is 37.5° at a level of -1 dB and the side lobe level (SLL) is -20 dB. The synthesis procedure was performed for two cases. In the first case, the target radiation pattern is represented by a piecewise constant function with a given width. In the second case, the target pattern was specified using piecewise linear approximation of the top and slopes of the main lobe.Comparison of the radiation patterns obtained shows that in the first case, the main-lobe width of the radiation pattern at a level of -1 dB is 34°, the SLL varies from -15.6 to -17 dB, and the irregularity of the main-lobe top of the radiation pattern lies within 0.9 ... 1.2 dB. In the second case, the main-lobe width of the antenna radiation pattern at a level of -1 dB is 36.5°, the SLL is -17.5 dB, and the irregularity of the main-lobe top is 0.4 dB at most. When used, the considered under consideration enables us to obtain both the synthesized patterns for linear radiators of various lengths, and the corresponding amplitude-phase distributions and coefficients of the Mathieu series. An estimate of the convergence of the Mathieu series shows that the use of linear approximation of the target radiation pattern in some cases allows up to 2.7-fold increase in acceleration of the convergence of the Mathieu series. The accuracy of reproducing the sector-shaped pattern by the synthesis method using the expansion into odd Mathieu functions gives good results when synthesizing the amplitude-phase distribution for the linear radiators with an electric length of 5λ or more.

Mutual relationships between SARS-CoV-2 test numbers, fatality and morbidity rates

Background The number of SARS-CoV-2 tests conversely to other factors, such as age of population or comorbidities, influencing SARS-CoV-2 morbidity and fatality rates, can be increased or decreased by decision makers depending on the development of the pandemic, operational capacity, and financial restraints. The key objective of this study is to identify and describe, within the probabilistic approach, the relationships between SARS-CoV-2 test numbers and the mortality and morbidity rates. Methods The study is based on a statistical analysis of 1058 monthly observations relating to 107 countries, from six different continents, in an 11-month period from March 2020 to January 2021. The variable utilised can be defined as the number of tests performed in a given country in 1 month, to the number of cases reported in a prior month and morbidities and mortalities per 1 million population. The probabilities of different mortality and morbidity rates for different test numbers were determined by moving percentiles and fitted by the power law and by the three-segment piecewise-linear approximation based on Theil Sen trend lines. Results We have identified that for a given probability the dependence of mortality and morbidity rates on SARS-CoV-2 test rates follows a power law and it is well approximated by the three Theil Sen trend lines in the three test rate ranges. In all these ranges Spearman rho and Kendall tau-b rank correlation coefficients of test numbers and morbidity with fatality rates have values between − 0.5 and − 0.12 with p-values below 0.002. Conclusions According to the ABC classification: the most important, moderately important, and relatively unimportant ranges of test numbers for managing and control have been indicated based on the value of the Theil Sen trend line slope in the three SARS-CoV-2 test rate ranges identified. Recommendations for SARS-CoV-2 testing strategy are provided.

Using the inverse distribution function method and the modified superposition method in the NMPUD computer system

This paper presents a computer system for modelling one-dimensional random variables NMPUD, developed in the laboratory of mathematical modelling of Lyceum No. 130 in Novosibirsk. The results of the numerical experiments and the considerations justifying the practicability for using in the NMPUD system: the elementary densities constructed by the technology of sequential (inserted) substitutions, the densities representing weighted sums of elementary densities (which can be simulated using the modified discrete superposition method), the algorithms for a piecewise linear approximation of unknown densities using a given sample, the algorithms of the modified superposition method for computational modelling of random variables with piecewise linear densities, are also presented.