Small Relative Error Research Articles

Prediction of viscoelastic and printability properties on binder-free TiO2-based ceramic pastes by DIW through a machine learning approach

Ceramic 3D printing has become an increasingly popular manufacturing technique due to its potential to create complex geometries with high precision. However, predicting the printability of ceramic pastes remains a challenge, as it depends on various rheological properties. In this study, we propose a feed-forward deep neural network model that predicts the printability of ceramic pastes based on two suggested criteria, a shear-thinning ability and a gel point greater than 1×103 Pa. The model is trained on a dataset built from rheological and viscoelastic characterizations of pastes, and validated on a separate test set. Our results show that the proposed learning model achieves small relative error in predicting the gel point of the ceramic pastes, with a mean value of 8.99181 and a standard deviation of 1.812864. This model has the potential to improve the efficiency and quality of ceramic 3D printing by enabling rapid and accurate predictions of paste printability.

An improved analytical method for rarefied aerothermodynamics on a complex geometry in free-molecular flows

A numerical method for high altitude aerothermodynamic analysis code for Very Low Earth Orbit (VLEO) satellite and spacecraft, named as HACS, is presented in this article. HACS is a computational tool that addresses the challenges by integrating the free-molecular gas kinetic model with novel particle sampling method and robust particle tracing algorithm to predict the shadow region on surfaces. To ensure the reliability and accuracy of the HACS, a rigorous verification and validation process has been systematically conducted by comparing the aerothermodynamic properties of forces, moments, and heat flux with the direct simulation Monte-Carlo (DSMC) results. The HACS is applied to the realistic geometries of VLEO satellite, the Apollo re-entry module, and spacecraft including fin and wing shapes. Results demonstrate that HACS provides the aerothermodynamic properties within the time frame of 60 seconds for the complex geometries and the accuracy of the HACS is guaranteed above 120 km altitude. The maximum relative error of the aerothermodynamic properties is less than 5% compared to the DSMC results at the lowest altitude of about 120 km, and has smaller relative errors as the altitude increases.

Research on Hybrid Approach for Maximum Power Point Tracking of Photovoltaic Systems under Various Operating Conditions

Based on the characteristics of the whale optimization algorithm (WOA) and perturbation observation (P&O) method, this paper proposes a novel hybrid approach called the improved chaotic whale optimization combined with perturb and observe (ICWOA-P&O) method for maximum power point tracking (MPPT) control to solve the challenge of low efficiency in photovoltaic (PV) power generation under local shadows. First, the ICWOA is used for a global search to quickly locate the position of the maximum power point (MPP). Then, the P&O method is used for a fine-grained local search to quickly track the position of the global maximum power point (GMPP) with low oscillation. To ensure accuracy, the tracking performance of the ICWOA-P&O method is comprehensively compared with the WOA-P&O, WOA, and PSO models under four conditions: uniform irradiance, static local shading, dynamic shading, and sudden changes in irradiance and temperature. The simulation results verify that under the above four conditions, the ICWOA-P&O method can track the MPP continuously and stably and greatly improves the convergence time and accuracy. Compared with the other three methods, the ICWOA-P&O method can effectively obtain the fastest tracking speed (less than 0.1 s), the highest tracking accuracy (more than 99.97%), the smallest relative error (less than 0.03%), and the smallest oscillation fluctuation. Finally, this study integrated the ICWOA-P&O algorithm into the designed MPPT controller hardware and established a practical PV experimental platform based on the ICWOA-P&O control algorithm for practical tests.

Correlation between the Mechanical Properties and Laser-Induced Breakdown Spectroscopy for Human Teeth

The present study investigates the correlation between mechanical properties and laser-induced breakdown spectroscopy (LIBS) for different tooth samples. A set of Fourteen samples collected from the Al-Kut Specialized Dental Centre in Wasit Governorate was analysed. Tooth samples were divided into groups depending on age and gender (men, women). The optical emission of the ionic-to- atomic intensity ratio of the Ca spectral line has been correlated with Leeb hardness test (LHT) values, compressive strength, and Young's modulus measured by standard tools. The ratio ionic line to atomic line CaII / CaI between the ionic calcium line at 396.84 nm and the atomic line at 558.87 nm for teeth is obtained using LIBS technique employing Nd: YAG laser with a wavelength of 1064 nm, an energy of 200 Mj, and a pulse duration of 10 ns. A linear relationship was observed between compressive strength, Young's modulus and hardness, and the concentrations of the elements (Ca). The small relative errors for hardness, compressive strength, and Young's modulus ranged from (1.44-5.57) %, (1.94-6.07) %, and (0.243-4.01) %, respectively. These findings validate applying LIBS to check out some important mechanical properties.

Comprehensive assessment of the accuracy of heart rate determination with the personal wearable device

BACKGROUND: Increased resting heart rate (HR) is an independent risk factor for overall mortality, sudden cardiac death, and death from cardiovascular diseases (CVD) and a worsening factor for CVD patients’ prognosis. Moreover, HR is easy to measure and monitor independently using various devices. The accuracy of personal wearable devices used in assessing HR should be investigated. AIM: To evaluate the accuracy of HR determination with the personal wearable device (PWD). MATERIALS AND METHODS: An open, observational, single-center study was performed. Participants underwent physical examination, electrocardiography, HR recording using PWD, and electrocardiography Holter monitoring (HMECG). For each participant, the study duration was 14 days. The reliability of HR measured in everyday life was assessed using the PWD in comparison with HMECG as the “gold” standard. For statistical analysis, Microsoft Excel 2016 and STATISTICA 10.0 software were used. To assess the reliability of HR measured in everyday life using PWD in relation to HMECG, the hypothesis of the equality of average HR values measured by both methods was tested, a correlation analysis of time series of HR values was conducted, and Bland–Altman plots were used to visualize differences in HR values. RESULTS: The study involved 45 healthy individuals (22 men, 49.24±6.47 years). HR data were obtained over a 24-hour period, including the subjects' routine working day. The results demonstrated good convergence between data on HR using PWD and HMECG. The relative error in determining the HR of PWD did not exceed 3.2%. The smallest relative error was recorded at night and early morning hours (0.3÷1.1%). During the daytime it was slightly higher — from 2.0 to 3.2%. A significantly high positive correlation was obtained between HR recorded using both devices, both at night (MCorrP from 0.75 to 0.85; p 0.001) and during daytime (MCorrP from 0.77 to 0.85; p 0.001). CONCLUSION: The accuracy comparison of 24-hour HR measurements by PWD showed that they were mostly within the acceptable error range (less than 3.2%). Future studies should include HR PWD assessment in patients with various diseases.

Prediction of Geometric Dimensions of Deposited Layer Produced Using Laser-Arc Hybrid Additive Manufacturing.

Laser-arc hybrid additive manufacturing (LAHAM) holds substantial potential in industrial applications, yet ensuring dimensional accuracy remains a major challenge. Accurate prediction and effective control of the geometrical dimensions of the deposited layers are crucial for achieving this accuracy. The width and height of the deposited layers, key indicators of geometric dimensions, directly affect the forming precision. This study conducted experiments and in-depth analysis to investigate the influence of various process parameters on these dimensions and proposed a predictive model for accurate forecasting. It was found that the width of the deposited layers was positively correlated with laser power and arc current and negatively correlated with scanning speed, while the height was negatively correlated with laser power and scanning speed and positively with arc current. Quantitative analysis using the Taguchi method revealed that the arc current had the most significant impact on the dimensions of the deposited layers, followed by scanning speed, with laser power having the least effect. A predictive model based on extreme gradient boosting (XGBoost) was developed and optimized using particle swarm optimization (PSO) for tuning the number of leaf nodes, learning rate, and regularization coefficients, resulting in the PSO-XGBoost model. Compared to models enhanced with PSO-optimized support vector regression (SVR) and XGBoost, the PSO-XGBoost model exhibited higher accuracy, the smallest relative error, and performed better in terms of Mean Relative Error (MRE), Mean Square Error (MSE), and Coefficient of Determination R2 metrics. The high predictive accuracy and minimal error variability of the PSO-XGBoost model demonstrate its effectiveness in capturing the complex nonlinear relationships between process parameters and layer dimensions. This study provides valuable insights for controlling the geometric dimensions of the deposited layers in LAHAM.

Automatic Segmentation of Type A Aortic Dissection on Computed Tomography Images Using Deep Learning Approach.

Type A aortic dissection (TAAD) is a life-threatening aortic disease. The tear involves the ascending aorta and progresses into the separation of the layers of the aortic wall and the occurrence of a false lumen. Accurate segmentation of TAAD could provide assistance for disease assessment and guidance for clinical treatment. This study applied nnU-Net, a state-of-the-art biomedical segmentation network architecture, to segment contrast-enhanced CT images and quantify the morphological features for TAAD. CT datasets were acquired from 24 patients with TAAD. Manual segmentation and annotation of the CT images was used as the ground-truth. Two-dimensional (2D) nnU-Net and three-dimensional (3D) nnU-Net architectures with Dice- and cross entropy-based loss functions were utilized to segment the true lumen (TL), false lumen (FL), and intimal flap on the images. Four-fold cross validation was performed to evaluate the performance of the two nnU-Net architectures. Six metrics, including accuracy, precision, recall, Intersection of Union, Dice similarity coefficient (DSC), and Hausdorff distance, were calculated to evaluate the performance of the 2D and 3D nnU-Net algorithms in TAAD datasets. Aortic morphological features from both 2D and 3D nnU-Net algorithms were quantified based on the segmented results and compared. Overall, 3D nnU-Net architectures had better performance in TAAD CT datasets, with TL and FL segmentation accuracy up to 99.9%. The DSCs of TLs and FLs based on the 3D nnU-Net were 88.42% and 87.10%. For the aortic TL and FL diameters, the FL area calculated from the segmentation results of the 3D nnU-Net architecture had smaller relative errors (3.89-6.80%), compared to the 2D nnU-Net architecture (relative errors: 4.35-9.48%). The nnU-Net architectures may serve as a basis for automatic segmentation and quantification of TAAD, which could aid in rapid diagnosis, surgical planning, and subsequent biomechanical simulation of the aorta.

Comparative study on calculation methods for compressive strength of low strength aggregate concrete

The compressive strength of concrete with low strength aggregate volume fractions of 10%, 20%, and 30% was calculated using both the graphical analysis method and mesomechanics method. In the calculation method based on graphical analysis, three-dimensional random spherical aggregate models of concrete with different volume contents of low strength aggregate were established and sliced. The results show that the graphical analysis method can effectively calculate the compressive strength of concrete with different volume contents of low strength aggregate. In the graphical analysis method, the relative errors of the calculated compressive strength of concrete with low strength aggregate volume fractions of 10%, 20%, and 30% were 4.84%, 4.84%, and 6.43%, respectively. Three-phase concrete models composed of mortar, aggregate, and interfacial transition zone were analyzed through the method of mesomechanics. The calculation results of the mesomechanics method show that the compressive strength of concrete was controlled by low strength aggregate, and the calculated compressive strength of concrete decreased with the increase in low strength aggregate volume content. In the mesomechanics method, the relative errors of the calculated compressive strength of concrete with low strength aggregate volume fractions of 10%, 20%, and 30% were 1.36%, 1.74%, and 3.7%, respectively. It can be found that the calculation results of the mesomechanics method are closer to the test values and have smaller relative errors, which indicate that the calculation method of mesomechanics theory is superior to the method of graphical analysis.

Research of fresh/salt water in Upper-middle Pleistocene (qp\(_{2–3}\)) in Bac Lieu Province

This study identified the relationship between Total Dissolved Solids (TDS) and Electrical Conductivity (EC) with the TDS = 0.6628*EC = 0.1706 correlation, in which the correlation coefficient is r = 0.984 and the Standard Error (±%) is 2.78%. The TDS content is determined by small relative error and a high correlation coefficient of the conductivity, which applies to calculate the TDS content of other groundwater samples in Bac Lieu Province. Also, the temperature difference in the electrical conductivity measurement affects the TDS content; thus, the study has determined that the temperature compensated constant between the electrical conductivity on the field and those at 25oC is 0.002953. According to the data and the analysis of the TDS content, the fresh-salt water distribution map of the qp2–3 aquifers has been established, and its boundary has been more detailed and standardized than in the previous study. In particular, TDS values range from 0.437 g/L to 2.0 g/L. The areas that contain brackish and salt water are distributed discontinuously, forming the saline zones in Loc Ninh, Phuoc Long communes in Hong Dan district; Phuoc Long, Hung Phu communes in Phuoc Long district; Vinh Hau, Vinh Binh communes in Hoa Binh district, and Thuan Hoa commune in Bac Lieu city. The saltwater area of the qp2–3 aquifers is about 398 km2, accounting for 16% of the study’s total area. The results fresh-salt water distributed mapping provide the environmental management institution with a comprehensive view of the distribution of fresh-salt water to propose effective groundwater exploitation policies.

Reference solutions for compressible single-phase flows in heated and cooled ducts

Analytical/quasi-analytical solutions are proposed for a steady, compressible, single-phase flow in a rectilinear duct subjected to heating followed by cooling. The flow is driven by the pressure ratio between an upstream tank and a variable outlet pressure. The article proposes a methodology to determine the full flow behavior, as a function of pressure ratio and heat-flux distribution. Following an analogy done with the study of compressible flows in nozzles, a behavioral classification of non-adiabatic compressible flows is proposed through the definition of critical pressure ratios. It is demonstrated that a critical pressure ratio distinguishes subsonic and supersonic outlet regimes and that there cannot be a steady shock wave in such configuration. The behavior of this critical pressure ratio is studied for limit cases of heat flux, delineating physical boundaries. An abacus is also proposed for a given couple of heating and cooling powers, as both values are needed to characterize the flow. Results are studied for parameters such as pressure ratio and outlet heat power. A short validation of a numerical simulation tool is provided, yielding excellent results and very small relative errors.

In silico modified UV spectrophotometric approaches to resolve overlapped spectra for quality control of rosuvastatin and teneligliptin formulation

Abstract A binary blend of rosuvastatin (ROS) and teneligliptin (TEN) used for the management of cardiovascular complications require a simple, analytical process for the quality assurance of this formulation. UV absorption spectra of ROS and TEN showed overlapping spectra. Hence, the overlapped spectra of ROS and TEN were separated by ratio difference, ratio first derivative; constant extraction coupled with exponentiation with division spectrum, and induced dual-wavelength methods. The proposed methods were authenticated by following the international council for harmonization criteria. A good linear relationship was demonstrated by all four methods, in 2–15 and 2–30 µg/mL for ROS and TEN, respectively. The high percentage retrieval of 98.96–100.22 and 98.72–99.73% for ROS and TEN, respectively, with small relative error, assured the correctness of the techniques. The validated techniques were employed for concurrent evaluation of ROS and TEN from binary formulation and laboratory-prepared mixture. The standard addition process verified the reliability of the projected procedures. The developed methods showed same accuracy and precision when compared to the HPLC methods along with safer solvent. Finally, the environmental sustainability of the presented UV spectroscopic procedures was found to be better than the reported HPLC method. Hence, eco-friendly, simple, and accurate mathematically processed UV spectroscopic procedures can be employed for simultaneous quantification of ROS and TEN for routine quality control study.

Kernelized Normalizing Constant Estimation: Bridging Bayesian Quadrature and Bayesian Optimization

In this paper, we study the problem of estimating the normalizing constant through queries to the black-box function f, which is the integration of the exponential function of f scaled by a problem parameter lambda. We assume f belongs to a reproducing kernel Hilbert space (RKHS), and show that to estimate the normalizing constant within a small relative error, the level of difficulty depends on the value of lambda: When lambda approaches zero, the problem is similar to Bayesian quadrature (BQ), while when lambda approaches infinity, the problem is similar to Bayesian optimization (BO). More generally, the problem varies between BQ and BO. We find that this pattern holds true even when the function evaluations are noisy, bringing new aspects to this topic. Our findings are supported by both algorithm-independent lower bounds and algorithmic upper bounds, as well as simulation studies conducted on a variety of benchmark functions.

An efficient online successive reanalysis method for dynamic topology optimization

In this study, an efficient reanalysis strategy for dynamic topology optimization is proposed. Compared with other related studies, an online successive dynamic reanalysis method and a Proper Orthogonal Decomposition (POD)-based approximate dynamic displacement strategy are integrated. In the dynamic reanalysis, the storage of the stiffness matrix decomposition can be avoided, and the reduced basis vectors should be successively updated according to the structural state in each iteration. Thus, the bottleneck of the combined approximation method for large-scale dynamic topology optimization can be overcome. Sequentially, POD is applied to obtain the approximate dynamic displacement, in which the Proper Orthogonal Mode (POM) of the displacement field is applied to obtain the approximate equivalent static loads of the Equivalent Static Load (ESL) method. Compared to the exact equivalent static loads at all the time intervals, the number of equivalent static loads is significantly reduced. Finally, the 2D and 3D test results indicate that the proposed method has remarkable speed-up effect under the premise of small relative error, support the strength of the proposed strategy.

Evaluation of droplet evaporation characteristics and its influencing factors in high-temperature gaseous environment

Droplet evaporation is widely present in nature and industrial production, involving heat and mass transfer between the droplet and the surrounding gas environment. This study investigates the evaporation of individual spherical droplets in a high-temperature gas environment under both stationary and forced convection conditions, using theoretical and numerical approaches. The results show that the established evaporation model can effectively reflect the temperature change during droplet evaporation. When the droplet evaporates in a gaseous environment, the equilibrium temperature of the droplet is influenced by environmental parameters such as temperature and humidity. Compared to environmental temperature, humidity has a more significant impact on the equilibrium temperature of the droplet, making it the dominant factor affecting the evaporation temperature. The evaporation rate of the droplet is influenced by environmental humidity. When the air humidity is below 0.5, the variable-rate evaporation aligns closely with the numerical simulation results (droplet lifetime). When the air humidity is above 0.5, the simulation results are slightly higher but still have a small relative error. As the environmental humidity increases, the evaporation rate of the droplet is suppressed, leading to an increase in its lifetime. Establishing an appropriate droplet evaporation model not only deepens our understanding of the internal mechanisms of droplet evaporation but also provides practical guidance for enhancing heat and mass transfer in droplets.

Inter-comparison study of wind measurement between the three-lidar-based virtual tower and four lidars using VAD techniques

ABSTRACT The accurate three-dimensional wind field obtained from a Doppler lidar not only helps to comprehend the refined structure of complex airflow but also provides important and valuable solutions for many fields. However, the underlying homogeneity assumption of the typical wind retrieval methods, such as Doppler Beam Swinging (DBS) and Velocity Azimuth Display (VAD) based on a single-lidar, will introduce the measurement uncertainty in complex terrain. In this paper, a new design of a wind measurement campaign involving seven lidars was carried out, which contained the three-lidar-based Virtual Tower (VT) using a time-space synchronization technique and four single-lidars with different elevation angles. This study investigates the performance of VT and VAD measurements under various conditions and evaluates the sensitivity of wind measurement uncertainty of VAD to the horizontal spatial- and probe volume-average effects associated with elevation angles of the laser beam. The inter-comparison results between VT and four VADs show consistent trends with small relative errors under neutral atmospheric conditions with weak wind velocity. Under convective or high Turbulence Intensity (TI) conditions, the relative errors between VT and VAD become larger and more fluctuant. Moreover, it is found that the measurement uncertainty of VAD increases at a given elevation angle with the increasing measurement heights, which is caused by the horizontal homogeneity associated with the conical scanning area. Additionally, the simulated and measured results of four VADs indicate that a larger elevation angle corresponds to a lower measurement uncertainty for a given height.

Research on the Application of Gray Prediction and BP Neural Network in English-Chinese Translation Quality Assessment

Abstract Gray prediction model and BP neural network are increasingly used in many fields as important prediction and analysis tools. The article first elaborates on the related algorithm of the gray prediction method. It constructs the gray prediction model, then constructs the BP neural network, and builds the gray BP neural network translation quality assessment model through the modeling of the simulation of the artificial neural network, and finally takes eight postgraduate students of translation majors of Beijing Foreign Studies University as the subjects, and the eight students translate two articles with similar word counts and difficulties, and score the translation quality of the subjects, and reasonably evaluate and scientifically predict these translations in order to validate the grey translation quality of subjects. The quality of translation is scored, and these translations are reasonably evaluated and scientifically predicted to verify the role and effect of the gray BP neural network in the quality of English-Chinese translation. The study concludes that the results based on the gray prediction model are that the prediction results are in the error range of 0%–7% compared with the original data, and the overall error is small. The prediction results of the compensatory fuzzy neural network prediction model for all 8 subjects before and after translation were more accurate than those predicted by the traditional fuzzy neural network method, with a good fit to the actual monitoring data and a small relative error. Only one subject out of the 8 spent more time on Chapter B than on Chapter A. Chapter B was more time-consuming than Chapter A. The time spent on Chapter A was more than the time spent on Chapter A. Chapter B was more time-consuming than Chapter A. Chapter B was more time-consuming than Chapter A. One subject spent equal time on both articles. The remaining 6 subjects did not spend as much time on chapter B as they did on chapter A.

ПРОГНОЗУВАННЯ ФІНАНСОВИХ ПОКАЗНИКІВ СУБ’ЄКТА ПІДПРИЄМНИЦЬКОЇ ДІЯЛЬНОСТІ: ВИБІР ОПТИМАЛЬНОГО МЕТОДУ

The use of economic and mathematical methods for solving financial problems is an integral part of effective management of enterprise resources and making sound economic decisions. This approach is becoming increasingly important in today’s world, as the dynamism of the financial environment and the variety of factors influencing the economy require complex analysis and precise calculations. The use of mathematical tools also allows to reduce the risks of making wrong decisions and improve the quality of forecasts. The article considers the necessity of forecasting financial indicators of a business entity as a process of predicting future financial results and the state of the enterprise for effective management and strategic decision-making, which gives an opportunity to ensure better resource planning, timely response to changes and reduce financial risks, contributing to the business sustainability and success. An analytical straight-line alignment of the dynamics of the amount of net revenue from sales was carried out on the basis of the financial statements of a real enterprise; a straight line equation characterising the amount of revenue from sales was derived and its forecast value for 2024 was calculated. The equation of the second-order parabola, which characterizes the trend in the value of sales revenue, was also derived and its forecast value for 2024 was found. For each method, the value of the standard square deviation of the actual levels of the dynamic series from the adjusted ones, i.e. the relative error of calculations, was calculated, and the optimal forecasting option was chosen – the one with the smaller relative error. It was found that the straight line more accurately reflects the main trend of a series of dynamics, which characterises the amount of revenue from sales of a business entity. On the basis of this conclusion and the obtained forecast data for 2024, the lower and upper limits of the projected value of sales revenue were calculated. It is concluded that forecasting of the financial performance of an enterprise allows to find optimal options for further development and planning of business activities and is a promising and relevant direction of predicting trends in the development of an enterprise, contributes to effective management and making sound strategic decisions.

Estimation of Water Consumption of Haloxylon ammodendron Sand-Fixing Forest in Minqin Oasis-Desert Ecotone of China Based on Leaf Index, China

In order to accurately estimate the water consumption of a Minqin desert Haloxylon ammodendron sand fixation forest, we used thermal equilibrium stem flow measurement to monitor the stem flow dynamics of Minqin desert H. ammodendron (5, 10, 15, and 20 years old, respectively, denoted as H5, H10, H15, and H20). In this study, we chose the growth index of assimilated branches (expressed as Ab in the following text) as the scalar quantity for expansion, and then estimated the water consumption of the H. ammodendron sand fixation forest by fitting the correlation between the water consumption of single H. ammodendron plants and the surface area and dry weight of single Ab. The results showed that the growth of assimilated H. ammodendron branches and the plant’s monthly water consumption were synchronized, and that the growth of Ab and water consumption were significantly higher during the fast-growth period of H. ammodendron (July) compared to during the leaf-expanding (May) and leaf-falling periods (October). After correlation analysis, it was determined that there was a linear function between the surface area and dry weight of Ab and water consumption in H. ammodendron. The simulated water consumption values (kg) of H. ammodendron sand fixation forests were greatly affected by stand density and branching configuration, and the simulated values of H. ammodendron assimilated branch growth (cm) were slightly smaller than the measured values for forests of different ages, while the simulated values of assimilated branch surface area (cm2) had a much smaller relative error (8.92%) on average. Therefore, it is feasible to use the surface area and dry weight of Ab to calculate the water consumption of H. ammodendron sand fixation forests. This can enable the reliable estimation of the water consumption of H. ammodendron forests and provide a reference for desert vegetation sand fixation afforestation and its ecological water management.

Relative error-based time-limited [formula omitted] model order reduction via oblique projection

In time-limited model order reduction, a reduced-order approximation of the original high-order model is obtained that accurately approximates the original model within the desired limited time interval. Accuracy outside that time interval is not that important. The error incurred when a reduced-order model is used as a surrogate for the original model can be quantified in absolute or relative terms to access the performance of the model reduction algorithm. The relative error is generally more meaningful than an absolute error because if the original and reduced systems’ responses are of small magnitude, the absolute error is small in magnitude as well. However, this does not necessarily mean that the reduced model is accurate. The relative error in such scenarios is useful and meaningful as it quantifies percentage error irrespective of the magnitude of the system’s response. In this paper, the necessary conditions for a local optimum of the time-limited H2 norm of the relative error system are derived. Inspired by these conditions, an oblique projection algorithm is proposed that ensures small H2-norm relative error within the desired time interval. Unlike the existing relative error-based model reduction algorithms, the proposed algorithm does not require solutions of large-scale Lyapunov and Riccati equations. The proposed algorithm is compared with time-limited balanced truncation, time-limited balanced stochastic truncation, and time-limited iterative Rational Krylov algorithm. Numerical results confirm the superiority of the proposed algorithm over these existing algorithms.

Апроксимація дистрибутивної функції відбивної здатності поверхні поліном третього степеня

In the article a bidirectional reflectance distribution function based on a polynomial of the third degree is developed. The main disadvantages of Schlick, Phong, Blinn reflectance models are analyzed. The approximation of Blinn-Phong model by a cubic polynomial is proposed to improve the productivity of three-dimensional image formation. Formulas for the coefficients of the approximation cubic polynomial are calculated. The disadvantages of using the proposed cubic polynomial for reproducing the glare’s attenuation zone are considered. A function for high-precision reproduction of this zone is proposed, formulas of its coefficients are calculated. A combined function, combining a cubic polynomial for reproducing the glare’s epicenter zone and a function for reproducing the attenuation zone, is proposed. The plot of the combined function is built. It is shown that the combined function reproduces the glare’s epicenter and attenuation zones with small relative and absolute errors. The developed reflectance model provides a highly-productive formation of three-dimensional scenes in computer graphics systems. The proposed distribution function of surface reflectivity can be used in computer graphics systems.