Relative Error Reduction Research Articles

Integrating spoken instructions into flight trajectory prediction to optimize automation in air traffic control

The booming air transportation industry inevitably burdens air traffic controllers’ workload, causing unexpected human factor-related incidents. Current air traffic control systems fail to consider spoken instructions for traffic prediction, bringing significant challenges in detecting human errors during real-time traffic operations. Here, we present an automation paradigm integrating controlling intent into the information processing loop through the spoken instruction-aware flight trajectory prediction framework. A 3-stage progressive multi-modal learning paradigm is proposed to address the modality gap between the trajectory and spoken instructions, as well as minimize the data requirements. Experiments on a real-world dataset show the proposed framework achieves flight trajectory prediction with high predictability and timeliness, obtaining over 20% relative reduction in mean deviation error. Moreover, the generalizability of the proposed framework is also confirmed by various model architectures. The proposed framework can formulate full-automated information processing in real-world air traffic applications, supporting human error detection and enhancing aviation safety.

Optimal wideband digital fractional-order differentiators using gradient based optimizer

In this paper, we propose a novel optimization approach to designing wideband infinite impulse response (IIR) digital fractional order differentiators (DFODs) with improved accuracy at low frequency bands. In the new method, the objective function is formulated as an optimization problem with two tuning parameters to control the error distribution over frequencies. The gradient based optimizer (GBO) is effectively employed on the proposed objective function. A wide range of design examples are presented to illustrate the effectiveness of the proposed approach. The proposed approximations are compared to those of recent literature in terms magnitude, phase, and group delay errors. The result reveal that our method can attain approximations have a favorable low frequency performance (with about 60% of relative magnitude error reduction) and maintain a comparable accuracy at most of the Nyquist band to those of the existing ones. Thus, our approximations can be attractive for low frequency applications.

Fast and accurate approximation algorithms for computing floating point square root

AbstractThe square root is one of the most used functions in many different engineering and scientific applications. We propose new methods for calculating the square root function that are based on the Newton–Raphson method with Heron iteration. A modification of Heron’s formula combined with an improved selection of the magic constants enables a significant reduction of the maximum relative error (MRE). Simple modifications to the Newton–Raphson formula and the magic number method enable implementation on platforms with limited hardware resources, such as microcontrollers and FPGAs, with variable accuracy. Implementations of new approximation algorithms in the C programming language were carefully tested and evaluated against their software and hardware counterparts on the most popular platforms, e.g., CPUs from Intel, AMD and ARM, GPU from Nvidia and IPU from Graphcore. The proposed numerical algorithms are shown to be superior in terms of computational time, the number of clock cycles, accuracy, MRE, and root mean square deviation.

Residual fatigue life prediction based on a novel improved Manson-Halford model considering loading interaction effect

The classical nonlinear fatigue cumulative damage model Manson-Halford is widely used in fatigue life analysis and prediction, but this model lacks consideration for the effects of load interaction. At present, some studies have proposed new correction models to overcome the shortcomings of classical models. However, the problem with these models is that the parameters are difficult to determine, resulting in a cumbersome application process or the correction process is based only on a single parameter. Insufficient correction leads to large fluctuations in life prediction errors. To overcome the above problems, this article proposes a new correction method and establishes a new improved model. Unlike existing models, the newly improved model is based solely on S-N curve parameters and does not require additional material parameters. It fully considers the relationship between adjacent loads and the difference in fatigue life under different stress states, and dynamically modifies the classical model based on larger influence weights. By conducting multi-level fatigue tests on 300M steel, a new improved model was used to accurately predict its remaining fatigue life. Compared with the classical model, the prediction error was reduced by 11.75 %. Utilized fatigue test data from various other materials to predict the fatigue life of the improved model under different levels of stress loading. The results indicate that in the vast majority of cases, the improved model has the highest prediction accuracy among all compared models, with a minimum relative prediction error of only 3.79 % and small fluctuations in prediction error. Compared with the classical model, the maximum reduction in relative prediction error after model improvement reached 28.73 %, indicating the effectiveness and accuracy of the new improved model.

Research on high-precision angular measurement based on machine learning and optical vortex interference technology

Abstract The paper innovatively constructs a regression prediction model based on the Stacking ensemble learning algorithm by utilizing the distortion degree of vortex optical interference patterns, achieving high-precision measurement of small angles. It constructs a regression prediction model based on the Stacking ensemble learning algorithm. Initially, in the spiral optical conjugate interference system, minute variations in the optical axis yield corresponding interference patterns, within an angle range of 0.0006° to 0.3°. The angle formed between the centroids of the upper two petals in the deformed interference patterns and the center is extracted as a feature for feature extraction. A dataset is established and randomly divided into training, validation, and testing sets in a 6:2:2 ratio. Subsequently, four models—support vector regression, particle swarm optimization back propagation, Gaussian process regression, and the stacking ensemble algorithm—are optimized for hyperparameters, trained, and evaluated based on coefficients of determination, root mean square error, and mean absolute error to compare their predictive performance. Through multiple rounds of training and prediction on randomly partitioned datasets, it is evident that the ensemble model exhibits a reduction in relative error compared to single learners, demonstrating that the Stacking-based ensemble algorithm can combine the strengths of base learners, showcasing superior predictive performance and enhanced stability. Moreover, the Stacking ensemble model achieves a measurement precision of 0.0006°, with a relative error maintained within 0.6%, indicating the feasibility of achieving high-precision measurement of tiny angles in the optical axis using machine learning and spiral optical conjugate interference systems.

A novel prediction model of total drag coefficient of an MPV vehicle based on LSA optimization

Abstract Environmental degradation and energy scarcity are caused by the ongoing rise in the number of automobiles. The most efficient way to lower air resistance while driving is to boost vehicle energy efficiency and reduce energy consumption. In this case, a specific MPV model serves as the study subject in this work, and we use CFD simulation to estimate the drag coefficient along with the selection of important factors. Furthermore, to increase prediction accuracy, a lightning search algorithm (LSA)-based BP-neural-network prediction model is created and developed. The results show that the average relative error reduction of 2.83% in the drag resistance model optimized by the lightning search method represents a significant improvement in accuracy. This phenomenon can be explained by the LSA’s ability to resolve the BP neural network’s initial weight and threshold uncertainty as well as the issue of local minima being quickly reached. The high prediction accuracy drag prediction model, obtained herein, is useful for creating and producing automobile shapes with low drag coefficients.

Calibrating the global network of gravitational wave observatories via laser power calibration at NIST and PTB

Current gravitational wave (GW) observatories rely on photon calibrators that use laser radiation pressure to generate displacement fiducials used to calibrate detector output signals. Reducing calibration uncertainty enables optimal extraction of astrophysical information such as source distance and sky position from detected signals. For the ongoing O4 observing run that started on 24 May 2023, the global GW detector network is employing a new calibration scheme with transfer standards calibrated at both the National Institute of Standards and Technology (NIST) and the Physikalisch-Technische Bundesanstalt (PTB). These transfer standards will circulate between the observatories and the metrology institutes to provide laser power calibration traceable to the International System of Units (SI) and enable assessment and reduction of relative calibration errors for the observatory network. The Laser Interferometer Gravitational-Wave Observatory (LIGO) project and the Virgo project are currently participating in the new calibration scheme. The Large-scale Cryogenic Gravitational-wave Telescope project (KAGRA) is expected to join in 2024, with the LIGO Aundha Observatory in India joining later. Before implementing this new scheme, a NIST-PTB bilateral comparison was conducted. It validated the scale representation by both laboratories, with a degree of equivalence of −0.2% and an associated expanded uncertainty of 0.32% (k = 2) which is significantly lower than previous studies. We describe the transfer of power sensor calibration, including detailed uncertainty estimates, from the transfer standards calibrated by NIST and PTB to the sensors operating continuously at the interferometer end stations. Finally, we discuss the ongoing calibration of Pcal-induced displacement fiducials for the O4 observing run. Achieved combined standard uncertainty levels as low as 0.3% facilitate calibrating the interferometer output signals with sub-percent accuracy.

Super-resolution of digital rock images with hybrid attention multi-branch neural network

High-resolution digital rock images are vital in studying microstructure and fluid flow characterization of oil and gas reservoirs. Computed tomography (CT) is currently the primary tool to acquire digital rock images because of its non-destructiveness and three-dimensional imaging capability. However, the restriction of imaging capability of CT sets a compromise between image resolution and field of view (FOV). The Hybrid Attention Multi-branch Super-Resolution Network (HAMSR) proposed in this study aims to remedy this deficiency. HAMSR employs a limited discrete distribution to capture all possible high-order information and generate reliable high-resolution images. The network is built to prioritize the recovery of pores, cracks, and textures in digital rock images by introducing channel and spatial attention mechanisms. The structure together with modules of HAMSR have been designed and optimized to yield superior performance with fewer parameters and less training time. HAMSR and several contemporary advanced models are trained on the DeepRock-SR 2D dataset which contains lots of sandstone and carbonate images. It is evident that HAMSR-generated images have minimal noise, clearer edges, and more micropores. Quantitative calculations show that compared to traditional bicubic interpolation, HAMSR acquires a 2.669 dB and 1.376 dB increase in peak signal-to-noise ratio (PSNR) respectively on the test sets (27%–38% reduction in relative error). Furthermore, the high-resolution (HR) images reconstructed by HAMSR are generally consistent with real HR images concerning statistical characteristics such as porosity, pore size distribution, shape factor distribution, and two-point correlation function. This work provides a new super-resolution model to generate high-quality digital rock images for further analysis.

Addressing the Accuracy-Cost Trade-off in Material Property Prediction Using a Teacher-Student Strategy.

Deep learning has catalyzed a transformative shift in material discovery, offering a key advantage over traditional experimental and theoretical methods by significantly reducing associated costs. Models adept at predicting properties from chemical compositions alone do not require structural information. However, this cost-efficient approach compromises model precision, particularly in Chemical Composition-based Property Prediction Models (CPMs), which are notably less accurate than Structure-based Property Prediction Models (SPMs). Addressing this challenge, our study introduces a novel Teacher-Student (TS) strategy, where a pretrained SPM serves as an instructive 'teacher' to enhance the CPM's precision. This TS strategy successfully harmonizes low-cost exploration with high accuracy, achieving a significant 47.1% reduction in relative error in scenarios involving 100 data entries. We also evaluate the effectiveness of the proposed strategy by employing perovskites as a case study. This method represents a significant advancement in the exploration and identification of valuable materials, leveraging CPM's potential while overcoming its precision limitations.

Shear wave anisotropy response characteristics of laminated shale and its correction methods

Laminated shale exhibits strong anisotropic characteristics in horizontal wells, significantly affecting the application of shear wave travel time data. Using equivalent medium theory and the elastic wave equation, we constructed a physical model for laminated shale, simulating the impact of lamination angle and lamination density on shear wave velocity and shear wave anisotropy coefficient. We established the relationship between the shear wave anisotropy coefficient and the cosine of the lamination angle. The simulation results indicate that shear wave velocity decreases with an increase in lamination angle and density, while the shear wave anisotropy coefficient gradually decreases with an increase in lamination angle and initially increases and then decreases with an increase in lamination density. Moreover, it exhibits a power exponential relationship with the cosine of the lamination angle. After correction, the minimum horizontal principal stress and fracture pressure calculated from shear wave data show an average relative error reduction of 6.72% and 7.77%, respectively, compared to measured data. This demonstrates that the correction method effectively eliminates the impact of laminated shale anisotropy on shear waves.

Element Aggregation for Estimation of High-Dimensional Covariance Matrices

This study addresses the challenge of estimating high-dimensional covariance matrices in financial markets, where traditional sparsity assumptions often fail due to the interdependence of stock returns across sectors. We present an innovative element-aggregation method that aggregates matrix entries to estimate covariance matrices. This method is designed to be applicable to both sparse and non-sparse matrices, transcending the limitations of sparsity-based approaches. The computational simplicity of the method’s implementation ensures low complexity, making it a practical tool for real-world applications. Theoretical analysis then confirms the method’s consistency and effectiveness with its convergence rate in specific scenarios. Additionally, numerical experiments validate the method’s superior algorithmic performance compared to conventional methods, as well as the reduction in relative estimation errors. Furthermore, empirical studies in financial portfolio optimization demonstrate the method’s significant risk management benefits, particularly its ability to effectively mitigate portfolio risk even with limited sample sizes.

Dense monocular depth estimation for stereoscopic vision based on pyramid transformer and multi-scale feature fusion

Stereoscopic display technology plays a significant role in industries, such as film, television and autonomous driving. The accuracy of depth estimation is crucial for achieving high-quality and realistic stereoscopic display effects. In addressing the inherent challenges of applying Transformers to depth estimation, the Stereoscopic Pyramid Transformer-Depth (SPT-Depth) is introduced. This method utilizes stepwise downsampling to acquire both shallow and deep semantic information, which are subsequently fused. The training process is divided into fine and coarse convergence stages, employing distinct training strategies and hyperparameters, resulting in a substantial reduction in both training and validation losses. In the training strategy, a shift and scale-invariant mean square error function is employed to compensate for the lack of translational invariance in the Transformers. Additionally, an edge-smoothing function is applied to reduce noise in the depth map, enhancing the model's robustness. The SPT-Depth achieves a global receptive field while effectively reducing time complexity. In comparison with the baseline method, with the New York University Depth V2 (NYU Depth V2) dataset, there is a 10% reduction in Absolute Relative Error (Abs Rel) and a 36% decrease in Root Mean Square Error (RMSE). When compared with the state-of-the-art methods, there is a 17% reduction in RMSE.

Enhanced POLYMER atmospheric correction algorithm for water-leaving radiance retrievals from hyperspectral/multispectral remote sensing data in inland and coastal waters.

Accurate retrieval of the water-leaving radiance from hyperspectral/multispectral remote sensing data in optically complex inland and coastal waters remains a challenge due to the excessive concentrations of phytoplankton and suspended sediments as well as the inaccurate estimation and extrapolation of aerosol radiance over the visible wavelengths. In recent years, reasonably accurate methods were established to estimate the enhanced contribution of suspended sediments in the near-infrared (NIR) and shortwave infrared (SWIR) bands to enable atmospheric correction in coastal waters, but solutions to derive the dominant phytoplankton contribution in the NIR and SWIR bands are less generalizable and subject to large uncertainties in the remotely-derived water color products. These issues are not only associated with the standard atmospheric correction algorithm in the SeaDAS processing system but with the non-traditional algorithms such as POLYMER (POLYnomial-based approach established for the atmospheric correction of MERIS data). This study attempts to enhance the POLYMER algorithm to enable atmospheric correction of hyperspectral and multispectral remote sensing data over a wide range of inland and ocean waters. The original POLYMER algorithm is less suitable owing to its complete reliance on a polynomial approach to model the atmospheric reflectance as a function of the wavelength and retrieve the water-leaving reflectance using two semi-analytical models (MM01 and PR05). The polynomial functions calculate the bulk atmospheric contribution instead of using an explicit method to estimate aerosol radiance separately, resulting the erroneous water color products in inland and coastal waters. The modified POLYMER algorithm (mPOLYMER) employs more realistic approaches to estimate aerosol contributions with a combination of UV and Visible-NIR bands and enables accurate retrievals of water-leaving radiance from both hyperspectral and multispectral remote sensing data. To assess the relative performance and wider applicability of mPOLYMER, the original and enhanced algorithms were tested on a variety of HICO, MSI and MODIS-Aqua data and the retrieved Lwn products were compared with AERONET-OC and OOIL-regional in-situ data. Expectedly, the mPOLYMER algorithm greatly improved the accuracy of Lwn (in terms of magnitude and spectral shape) when applied to MODIS-Aqua and HICO data in highly turbid productive waters (with higher concentrations of phytoplankton or with dense algal blooms) in Muttukadu Lagoon, Lake Erie, Yangtze River Estuary, Baltic Sea and Arabian Sea. In contrast, the original POLYMER algorithm overestimated Lwn in the visible and NIR bands and produced unphysical negative Lwn or distorted Lwn spectra in turbid productive waters. The mPOLYMER yielded a relative mean error reduction of more than 50% (i.e., from 79% to 34%) in Lwn for a large number of matchup data. The improved accuracy and data quality is because the mPOLYMER algorithm's funio and coefficients sufficiently accounted for the enhanced backscattering contribution of phytoplankton and suspended sediments in optically complex waters.

BERT-Based Arabic Diacritization: A state-of-the-art approach for improving text accuracy and pronunciation

In order to accurately represent the meaning and pronunciation of Arabic words and sentences, the presence of diacritics plays a crucial role. Over the years, researchers have dedicated significant efforts to enhancing automated diacritization systems. This paper introduces a novel approach for Arabic diacritization utilizing Bidirectional Encoder representations from Transformers (BERT) models. To evaluate the effectiveness of the proposed approach, two publicly available datasets, namely the Arabic Diacritization (AD) dataset and the Tashkeela Processed (TP) dataset, were employed. The performance of the models was assessed using various error metrics, including Diacritic Error Rate (DER) and Word Error Rate (WER). The findings demonstrate the superior performance of BERT in the diacritization process, surpassing all models employed in other diacritization systems. On the AD dataset, the proposed system achieved state-of-the-art (SOTA) syntactic DER and WER of 1.14% and 3.34%, respectively. For morphological diacritization, the best results yielded a DER of 0.92% and a WER of 1.91%. These outcomes reflect a remarkable relative error reduction of over 30% compared to previous research. Additionally, on the TP dataset, the BERT models exhibited a substantial decrease in DER, reducing the benchmark from 4.0% to 1.11%. Furthermore, this study introduces a real-time diacritization system called SUKOUN, which offers diacritized text through a user-friendly website. A comparison with existing automatic diacritization tools, using six example texts, reveals the superior prediction accuracy and preservation of input format provided by SUKOUN.

Short-term PV power prediction based on VMD-CNN-IPSO-LSSVM hybrid model

Abstract This article discusses the significance and obstacles of short-term power prediction in photovoltaic systems and introduces a hybrid model for photovoltaic short-term power prediction technology based on variational mode decomposition (VMD), convolutional neural network (CNN), improved particle swarm optimization (IPSO) and least squares support vector machine (LSSVM). In the initial stage, the photovoltaic generation signal is decomposed into multiple Intrinsic mode functions (IMFs) using VMD to enhance the extraction of signal time–frequency characteristics. Subsequently, CNN is utilized for feature learning and extraction of each IMF, modeling the nonlinear and non-stationary features. Following this, the IPSO-LSSVM optimization algorithm is employed to establish and optimize multiple LSSVM models, predicting power fluctuations at different time scales. Finally, the predictions from each model are synthesized to obtain the final photovoltaic short-term power forecast. Through validation with actual photovoltaic generation data, this hybrid model demonstrates high accuracy and reliability in short-term power prediction, showing an average relative error and root mean square error reduction of 15.23 and 53.60%, respectively, compared to a certain comparative model. This proposed method based on VMD-CNN-IPSO-LSSVM hybrid model for photovoltaic power prediction holds promising prospects and practical value in the operation and scheduling of photovoltaic generation systems.

Research on micro/nano scale 3D reconstruction based on scanning electron microscope

Scanning electron microscopy (SEM) has an important application in the petroleum field, which is often used to analyze the microstructure of reservoir rocks, etc. Most of these analyses are based on two-dimensional images. In fact, SEM can carry out micro-nano scale three-dimensional measurement, and three-dimensional models can provide more accurate information than two-dimensional images. Among the commonly used SEM 3D reconstruction methods, parallax depth mapping is the most commonly used method. Multiple SEM images can be obtained by continuously tilting the sample table at a certain Angle, and multiple point clouds can be generated according to the parallax depth mapping method, and a more complete point clouds recovery can be achieved by combining the point clouds registration. However, the root mean square error of the point clouds generated by this method is relatively large and unstable after participating in point clouds registration. Therefore, this paper proposes a new method for generating point clouds. Firstly, the sample stage is rotated by a certain angle to obtain two SEM images. This operation makes the rotation matrix a known quantity. Then, based on the imaging model, an equation system is constructed to estimate the unknown translation parameters, and finally, triangulation is used to obtain the point clouds. The method proposed in this paper was tested on a publicly available 3D SEM image set, and the results showed that compared to the disparity depth mapping method, the point clouds generated by our method showed a significant reduction in root mean square error and relative rotation error in point clouds registration.

AI-based denoising of head impact kinematics measurements with convolutional neural network for traumatic brain injury prediction.

Wearable devices are developed to measure head impact kinematics but are intrinsically noisy because of the imperfect interface with human bodies. This study aimed to improve the head impact kinematics measurements obtained from instrumented mouthguards using deep learning to enhance traumatic brain injury (TBI) risk monitoring. We developed one-dimensional convolutional neural network (1D-CNN) models to denoise mouthguard kinematics measurements for tri-axial linear acceleration and tri-axial angular velocity from 163 laboratory dummy head impacts. The performance of the denoising models was evaluated on three levels: kinematics, brain injury criteria, and tissue-level strain and strain rate. Additionally, we performed a blind test on an on-field dataset of 118 college football impacts and a test on 413 post-mortem human subject (PMHS) impacts. On the dummy head impacts, the denoised kinematics showed better correlation with reference kinematics, with relative reductions of 36% for pointwise root mean squared error and 56% for peak absolute error. Absolute errors in six brain injury criteria were reduced by a mean of 82%. For maximum principal strain and maximum principal strain rate, the mean error reduction was 35% and 69%, respectively. On the PMHS impacts, similar denoising effects were observed and the peak kinematics after denoising were more accurate (relative error reduction for 10% noisiest impacts was 75.6%). The 1D-CNN denoising models effectively reduced errors in mouthguard-derived kinematics measurements on dummy and PMHS impacts. This study provides a novel approach for denoising head kinematics measurements in dummy and PMHS impacts, which can be further validated on more real-human kinematics data before real-world applications.

Lapse rate-adjusted bias correction for CMIP6 GCM precipitation data: An application to the Monsoon Asia Region.

Bias correction (BC) of General Circulation Models (GCMs) variables is a common practice when it is being used for climate impact assessment studies at regional scales. The present study proposes a bias correction method (LR-Reg) that first adjusts the original GCM precipitation for local lapse rate corrections and later bias corrects the lapse rate-adjusted GCMs precipitation data with linear regression coefficients. We evaluated LR-Reg BC method in comparison to Linear Scaling (LS) and Quantile Mapping (QMap) BC methods, and NASA's downscaled NEX data for Monsoon Asia region. This study used Coupled Model Intercomparison Project Phase 6 (CMIP6)-based MIROC6 GCM precipitation with historical and projected shared socio-economic pathways (SSP) scenarios (SSP245 and SSP585) datasets. The BC comparison results show that the relative percentage reduction in mean absolute error (MAE) values of LR-Reg over LS-BC was up to 10-30% while this relative reduction in MAE values of LR-Reg was 30-50% over QMap-BC and 75-100% over NASA's NEX-data. The future projected precipitation over Monsoon Asia during dry season shows more decreased precipitation by up to 100% mostly in the south Asia while during wet season shows more increased precipitation by up to 50% mostly in the northeastern China and in the Himalayan belts with respect to the baseline condition (1970-2005). The results on the average precipitation per 0.25 degree increase in latitude analysis shows that the maximums of average monsoon precipitation during baseline period occur at 0 and 25 degree latitudes while the projected monsoon precipitation during both SSP scenarios occurs at 10 and 20 degree latitudes which clearly shows an inward shift in the latitude axis for the projected precipitation in the Monsoon Asia.

Research on Missing Value Imputation to Improve the Validity of Air Quality Data Evaluation on the Qinghai-Tibetan Plateau

In the Qinghai-Tibet Plateau region, operational deficiencies and limited maintenance capacities often impair automatic air quality monitoring stations. This results in frequent data omissions, compromising the reliability of environmental assessment data. Therefore, an effective data imputation method is required to address the gaps in observational records. Utilizing a Sequence-to-Sequence framework, we introduce a model termed Bidirectional Recurrent Imputation for Time Series-Attention-based Long Short-Term Memory (BRITS-ALSTM). The encoder of BRITS-ALSTM applies BRITS to integrate single-station historical characteristics with multi-station correlation features. Concurrently, the decoder employs LSTM within an attention mechanism to capitalize on previously observed data, thereby generating hourly imputations for missing air quality data values. The model was trained using six types of air quality data from 16 stations across Qinghai Province. Through localized testing and parameter optimization, BRITS-ALSTM achieved a reduction in mean relative error (MRE) by 74.88% compared to the baseline mean-filling approach. Additionally, ablation studies demonstrated an improvement in the coefficient of determination R-squared (R2) from 0.67 to 0.76, outperforming the standalone BRITS. Consequently, BRITS-ALSTM enhances the accuracy of air quality data evaluations in the Tibetan Plateau and offers an efficacious strategy for data imputation in elevated terrains.

New “Walk on Equations” Monte Carlo Algorithm for Linear Systems

A novel version of Monte Carlo algorithm for solving systems of linear algebraic equations is presented and studied. The algorithm is similar to the “Walk on Equations” Monte Carlo method recently developed by Ivan Dimov, Sylvain Maire and Jean Michel Sellier. It is done a comparison with the Gauss-Seidel method for matrices up to size of 212. The algorithm could be drastically improved by choosing appropriate values for the relaxation parameters, which in turn leads to dramatic reduction in time and lower relative errors for a given number of iterations. What is more, a sequential Monte Carlo method of John Halton based on an iterative use of the control variate method has been applied. Some of the most important numerical applications are the large system, coming from a finite element approximation of problems, describing a beam structure in constructive mechanics, and the block-diagonal matrices, which come from discretization of models in a regime-switching economy.