Polynomial Basis Research Articles

Positivity-preserving discontinuous Galerkin scheme for linear hyperbolic equations with characteristics-informed augmentation

This paper presents a positivity-preserving discontinuous Galerkin (DG) scheme for the linear hyperbolic problem with variable coefficients on structured Cartesian domains. The standard DG spaces are augmented with either polynomial or non-polynomial basis functions. The primary purpose of these augmented basis functions is to ensure that the cell average from the unmodulated DG scheme remains positive. We explicitly obtain suitable basis functions by inspecting the method of characteristics on an auxiliary problem. A key result is proved which demonstrates that the unmodulated augmented DG scheme will retain a positive cell average, provided that the inflow, source term, and variable coefficients are positive. A simple scaling limiter can then be leveraged to produce a high-order conservative positivity-preserving DG scheme. Numerical experiments demonstrate the scheme is able to retain high-order accuracy as well as robustness for variable coefficients. To improve efficiency, an inexact augmented basis function can be obtained rather than a analytic non-polynomial solution to the auxiliary problem from the method of characteristics.

Meshfree RBF–FD methods for numerical simulation of PDE problems

The meshfree RBF–FD method is becoming an increasingly popular discretization method for challenging PDE problems due to its flexibility with respect to geometry and its ease of implementation. Using a combination of polyharmonic spline basis functions and polynomial basis functions has proven to be particularly successful with a convergence rate depending on the polynomial degree and good behavior at boundaries due to the contribution of the spline part. A challenge that has been observed is that derivative boundary conditions can give rise to large errors. A known remedy for this problem is to introduce one or more layers of ghost points, which improves the boundary approximations. In a recent paper, Tominec, Larsson, Heryudono (2021), it was shown that using oversampling is another effective way to handle this problem. Oversampling also allows for theoretical analysis of the method, which can then be viewed as a discretization of a continuous least-squares projection. By looking deeper into the error estimates we gain an understanding about how to best implement the method.

Representation in Fréchet Spaces of Hyperbolic Theta and Integral Operator Bases for Polynomials

Representation in Fréchet Spaces of Hyperbolic Theta and Integral Operator Bases for Polynomials

Comparison between RBF and FDM for solving nonlinear time-dependent PDEs using Newton’s Method

In this paper, the improved localized method of approximated particular solutions (ILMAPS) using polyharmonic splines with constant polynomial basis is used to approximate solutions of nonlinear time-dependent PDEs. The discretization process is done through a simple implicit time stepping and a collocation technique on a set of points in local domains of influence. Resulted system of nonlinear algebraic equations is solved by Newton’s method. Numerical experiments suggest that the proposed method yields superior accuracy than other RBFs as well as FDM for solving nonlinear PDEs.

On conjugacy of additive actions in the affine Cremona group

An additive action on an irreducible algebraic variety X is an effective action with an open orbit of the vector group . Any two additive actions on X are conjugate by a birational automorphism of X. We prove that, if X is the projective space, the conjugating element can be chosen in the affine Cremona group and it is given by so-called basic polynomials of the corresponding local algebra.

A numerical investigation of a well-known nonlinear Newell-Whitehead-Segel equation using the rank polynomial of the star graph

Mathematical models of pattern formation are indispensable tools in various fields, from developmental biology to ecology, providing insights into complex phenomena and contributing to our understanding of the natural world. These patterns have been extensively studied using reaction-diffusion and NewellWhiteheadSegel models. This article intended to find an approximate numerical solution to the NewellWhiteheadSegel equation. The appearance of stripe patterns in two-dimensional systems is explained in nonlinear systems using the NewellWhiteheadSegel equation. Based on the function basis of rank polynomials of star graphs and the well-posed operational matrices, the rank polynomial collocation method is constructed. The alleged rank polynomial collocation method created a system of nonlinear algebraic equations from the nonlinear NewellWhiteheadSegel equation. The nonlinear NewellWhiteheadSegel equation solution is approximated by solving the resulting system via Newton’s Raphson method. Numerical instances are provided to illustrate the validity and effectiveness of the technique. Verification of accuracy is accomplished by calculating error norms. The obtained numerical results show a reasonable degree of consistency with the findings reported in the current literature. The scheme’s primary benefit is the algorithm’s ease of implementation.

Accurate neural-network-based fitting of full-dimensional N2 –Ar and N2–CH4 two-body potential energy surfaces aimed at spectral simulations

We describe the development of machine-learned (ML) potentials for flexible, weakly interacting monomers. A recently suggested permutationally invariant polynomial neural network (PIP-NN) approach is utilised to represent the full-dimensional two-body component of the molecular pair energy. To ensure the asymptotic zero-interaction limit, a tailored subset of the full invariant polynomial basis set is utilised, and their variables are modified to achieve a better fit of the correct asymptotic behaviour at a long range. This new technique is used to build full-dimensional potentials for the two-body N 2 –Ar and N 2 –CH 4 interactions by fitting databases of ab initio energies calculated at the coupled-cluster level of theory. The second virial coefficient, fully accounting for molecular flexibility, is then calculated within the classical framework using the obtained PIP-NN potential surfaces. A trajectory-based simulation of the N 2 –Ar collision-induced absorption is conducted, covering both the far- and mid-infrared ranges.

A comparative analysis of feature selection models for spatial analysis of floods using hybrid metaheuristic and machine learning models.

The research aims to propose a feature selection model for hydraulic analysis as such a model has not been proposed previously. For this purpose, hybrids of three metaheuristic algorithms, particle swarm optimization (PSO), ant colony optimization (ACO), and genetic algorithm (GA) with two machine learning models which are support vector machine (SVM) and K-nearest neighbor (KNN) are employed. The dataset considered was hydraulic having an association with flood and possessed topographic, geo-environmental, and human-induced variables. The dataset considered had multicollinearity heteroscedasticity and autocorrelation problems. The metaheuristic algorithms were evaluated by varying the number of population size. Among them, PSO performed better by providing an appropriate number of features with a lower number of iterations. We have analyzed the performance of SVM with different kernels; linear, radial basis function (RBF), sigmoid, and polynomial, as the original SVM is designed only for linear datasets but the hydraulic dataset possesses non-linear characteristics as well. The performance of different kernels in terms of their accuracies is evaluated and recorded. This study showed that RBF performed the best and sigmoid showed the least accuracy for GA, PSO, and ACO algorithms. The performance of KNN is evaluated in terms of accuracies by varying the K-values. It was found that KNN shows low accuracy with a small K-value which then attained a maximum level by increasing K-values, and it finally started decreasing, explicitly, by further enhancing K-values. While comparing the performance of hybrids of GA, PSO, and ACO with SVM and KNN, it was analyzed that KNN performed better with these meta-heuristics with PSO-KNN which performed the best among the baseline models. Thus, the study proposes that PSO-KNN can be utilized as a feature selection technique to obtain optimal data subsets for hydraulic modeling and analysis.

A framework for performance enhancement of classifiers in detection of prostate cancer from microarray gene

Prostate cancer is a major world health problem for men. This shows how important early detection and accurate diagnosis are for better treatment and patient outcomes. This study compares different ways to find Prostate Cancer (PCa) and label tumors as normal or abnormal, with the goal of speeding up current work in microarray gene data analysis. The study looks at how well several feature extraction methods work with three feature selection strategies: Harmonic Search (HS), Firefly Algorithm (FA), and Elephant Herding Optimization (EHO). The techniques tested are Expectation Maximization (EM), Nonlinear Regression (NLR), K-means, Principal Component Analysis (PCA), and Discrete Cosine Transform (DCT). Eight classifiers are used for the task of classification. These are Random Forest, Decision Tree, Adaboost, XGBoost, and Support Vector Machine (SVM) with linear, polynomial, and radial basis function kernels. This study looks at how well these classifiers work with and without feature selection methods. It finds that the SVM with radial basis function kernel, using DCT for feature extraction and EHO for feature selection, does the best of all of them, with an accuracy of 94.8 % and an error rate of 5.15 %.

Nonlinear dynamical response of sinusoidal impulsive actuated piezoelectric/porous sandwich nanoharvesters via GM-based meshfree collocation formulations

In this paper a new effective computational approach based upon a meshless collocation formulation of the Gurtin-Murdoch (GM) continuum elasticity is described. The newly developed computational model for the third-order shear flexible elastic plates is then employed to analyze the nonlinear dynamical response of piezoelectric/porous sandwich nanoharvesters subjected to a sinusoidal impulsive actuation. The material properties relevant to the porous passive core having uniform or two graded through thickness porosity dispersions are estimated using the approach of Gaussian random field. The weak form of model conception is discretized and solved numerically via employing an incorporation of the polynomial and radial basis functions having the capability to remove any feasible singularity as well as taking more precise approximation into account within the GM-based meshfree collocation formulations. It is deduced that by contemplating the surface stress tensor via the established GM-based model, the achieved voltage from the sinusoidal impulsive actuated sandwich nanoharvesters reduces, especially for those possessing lower thickness. Also, it is deduced that for the fully simply supported impulsive actuated sandwich nanoharvester, by changing the porosity decoration of distribution from the uniform decoration to FGO and FGX graded ones, the significance of the role of surface stress tensor in the achieved voltage reduces from 12.45% to 12.28%, and enhances from 12.45% to 12.64%, respectively. For the fully clamped impulsive actuated sandwich nanoharvester, by changing the porosity decoration of distribution from the uniform decoration to FGO and FGX graded ones, the significance of the role of surface stress tensor in the achieved voltage reduces from 15.01% to 14.68%, and enhances from 15.01% to 15.37%, respectively.

Optimizations of the Accelerated Axial Polynomial Method of Characteristics of the APOLLO3® Deterministic Neutron Transport Code

For some years now, the TDT (two- and three-dimensional transport) solver of the APOLLO3® deterministic neutron transport code has been able to perform lattice calculations on three-dimensional extruded and unstructured geometries. A polynomial expansion of the angular flux has been implemented to better describe the flux gradient axially to reduce the number of computational meshes required to reach a given accuracy. Then the polynomial approximation was extended to macroscopic cross sections to perform evolution calculations. Besides these transport schemes, synthetic acceleration has also been implemented, relying on double PN approximations of the angular flux on the boundaries of the spatial regions. The solver has already introduced several techniques to reduce the transport and memory footprint; for example, for the storage of the surfaces crossed by a trajectory or the classification of chords. In this paper, new optimizations are presented. One deals with how monomials of the polynomial basis are integrated along trajectories. Another one concerns the computation of the source term of the transmission equation in the case of polynomial cross sections. The last optimization exploits the fact that, along horizontal trajectories, the flux and the cross sections are constant to speed up the sweep algorithm. Calculations on 5 × 5 and 7 × 7 pressurized water reactor assemblies were performed to assess the gains of these recently developed strategies. The results show good improvements both in computing time and in memory footprint reductions.

Digital Self-Interference Canceler with Joint Channel Estimator for Simultaneous Transmit and Receive System.

Simultaneous transmit and receive wireless communications have been highlighted for their potential to double the spectral efficiency. However, it is necessary to mitigate self-interference (SI). Considering both the SI channel and remote transmission (RT) channel need to be estimated before equalizing the received signal, we propose two adaptive algorithms for linear and nonlinear self-interference cancellation (SIC), based on a multi-layered joint channel estimator structure. The proposed algorithms estimate the RT channel while performing SIC, and the multi-layered structure ensures improved performance across various interference-to-signal ratios. The M-estimate function enhances the robustness of the algorithm, allowing it to converge even when affected by impulsive noise. For nonlinear SIC, this paper introduces an adaptive algorithm based on generalized Hammerstein polynomial basis functions. The simulation results indicate that this approach achieves a better convergence speed and normalized mean squared difference compared to existing SIC methods, leading to a lower system bit error rate.

A Spectrally Accurate Step-by-Step Method for the Numerical Solution of Fractional Differential Equations

In this paper we consider the numerical solution of fractional differential equations. In particular, we study a step-by-step procedure, defined over a graded mesh, which is based on a truncated expansion of the vector field along the orthonormal Jacobi polynomial basis. Under mild hypotheses, the proposed procedure is capable of getting spectral accuracy. A few numerical examples are reported to confirm the theoretical findings.

TriKSV-LG: a robust approach to disease prediction in healthcare systems using AI and Levy Gazelle optimization

A seamless connection between the Internet and people is provided by the Internet of Things (IoT). Furthermore, lives are enhanced using the integration of the cloud layer. In the healthcare domain, a reactive healthcare strategy is turned into a proactive one using predictive analysis. The challenges faced by existing techniques are inaccurate prediction and a time-consuming process. This paper introduces an Artificial Intelligence (AI) and IoT-based disease prediction method, the TriKernel Support Vector-based Levy Gazelle (TriKSV-LG) Algorithm, which aims to improve accuracy, and reduce the time of predicting diseases (kidney and heart) in healthcare systems. The IoT sensors collect information about patients’ health conditions, and the AI employs the information in disease prediction. TriKSV utilizes multiple kernel functions, including linear, polynomial, and radial basis functions, to classify features more effectively. By learning from different representations of the data, TriKSV better handles variations and complexities within the dataset, leading to more robust disease prediction models. The Levy Flight strategy with Gazelle optimization algorithm tunes the hyperparameters and balances the exploration and exploitation for optimal hyperparameter configurations in predicting chronic kidney disease (CKD) and heart disease (HD). Furthermore, TriKSV's incorporation of multiple kernel functions, combined with the Gazelle optimization strategy, helps mitigate overfitting by providing a more comprehensive search space for optimal hyperparameter selection. The proposed TriKSV-LG method is applied to two different datasets, namely the CKD dataset and the HD dataset, and evaluated using performance measures such as AUC-ROC, specificity, F1-score, recall, precision, and accuracy. The results demonstrate that the proposed TriKSV-LG method achieved an accuracy of 98.56% in predicting kidney disease using the CKD dataset and 98.11% accuracy in predicting HD using the HD dataset.

An Intuitively Derived Decoupling and Calibration Model to the Multiaxis Force Sensor Using Polynomials Basis

An Intuitively Derived Decoupling and Calibration Model to the Multiaxis Force Sensor Using Polynomials Basis

Spatial Functional Data analysis: Irregular spacing and Bernstein polynomials

Spatial Functional Data (SFD) analysis is an emerging statistical framework that combines Functional Data Analysis (FDA) and spatial dependency modeling. Unlike traditional statistical methods, which treat data as scalar values or vectors, SFD considers data as continuous functions, allowing for a more comprehensive understanding of their behavior and variability. This approach is well-suited for analyzing data collected over time, space, or any other continuous domain. SFD has found applications in various fields, including economics, finance, medicine, environmental science, and engineering. This study proposes new functional Gaussian models incorporating spatial dependence structures, focusing on irregularly spaced data and reflecting spatially correlated curves. The model is based on Bernstein polynomial (BP) basis functions and utilizes a Bayesian approach for estimating unknown quantities and parameters. The paper explores the advantages and limitations of the BP model in capturing complex shapes and patterns while ensuring numerical stability. The main contributions of this work include the development of an innovative model designed for SFD using BP, the presence of a random effect to address associations between irregularly spaced observations, and a comprehensive simulation study to evaluate models’ performance under various scenarios. The work also presents one real application of Temperature in Mexico City, showcasing practical illustrations of the proposed model.

Prediction-Based Power Consumption Monitoring of Industrial Equipment Using Interpretable Data-Driven Models

Efficient energy optimization and scheduling in industrial factories depend on accurate, reasonable, and real-time monitoring of equipment power consumption. However, the power prediction of industrial equipment requires a large number of process data, which could be inevitably contaminated by some imperfect data, due to the harsh environments. Monitoring or predicting equipment power consumption is usually not feasible using data-driven black-box models with imperfect data. This paper proposes a prediction-based approach for power consumption monitoring using an interpretable data-driven model. First, a data preprocessing method is used to remove outliers and fill in missing values. Then, a Volterra polynomial basis function (VPBF) model is built to predict equipment power consumption. This model decomposes power values into a series of basis functions consisting of input parameters. Moreover, to compensate for data dropouts during the power consumption monitoring process, a networked predictive monitoring system is also proposed. Finally, this paper presents two case studies based on actual production equipment in an industrial manufacturing factory. The results demonstrate that the proposed approach can achieve satisfactory monitoring accuracy and adaptation ability. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This paper is motivated by the problem of power consumption monitoring of industrial equipment in harsh production environments. A conventional solution is to predict the power consumption using data-driven black-box models, with limited feasibility and interpretability. This paper proposes a new power consumption monitoring approach, utilizing an interpretable data-driven model and a networked predictive method. This approach accurately reveals the transparent relationships between the output and input parameters. The power prediction result is consequently interpretable and more reasonable. Meanwhile, this approach actively compensates for data dropouts in the network, which can help operators real-time grasp the power consumption of equipment. Furthermore, this approach can be integrated into energy optimization systems as a basis of optimization and decision-making. Practical applications in an aluminum manufacturing company located in Guangdong Province demonstrate that this approach is feasible and applicable. Future research is to expand this approach further for energy optimization and scheduling.

A novel dynamic risk assessment method for hazardous chemical warehouses based on improved SVM and mathematical methodologies

Effective dynamic risk assessment is crucial for identifying process hazards and preventing accidents. The rapid development of modern technology urgently requires the development of new dynamic risk assessment methods to address current societal needs, especially in situations where a large amount of real-time data is available. Consequently, this manuscript presents an improved support vector machine model (SVM), integrated mathematical modeling, to facilitate dynamic risk assessment for hazardous chemical warehouses (HCWs). Firstly, an indicator framework and a quantitative risk factor table are crafted. Then a mathematical model using knowledge graph and DEMATEL-variable weight theory (VWT) is established, which combines accident information with VWT to optimize the weight calculation and provides reliable prior knowledge for the improved SVM model. Ultimately, the improved SVM model is crafted by the establishment of an improved hybrid kernel of polynomial and radial basis function kernel, fine-tuning its hyperparameters with particle swarm optimization, and utilizing sensitivity analysis to simplify the model. The model is formulated to accommodate the intricate non-linear relationships within indicators and dynamic risk, thereby establishing dynamic risk assessment rules. Through a case study, the improved SVM model demonstrates higher precision in dynamic risk prediction compared to original models. The results substantiate that the model can provide innovative insight and methodology for the precise evaluation of dynamic risk in HCWs.

WaveNet: Tackling Non-stationary Graph Signals via Graph Spectral Wavelets

In the existing spectral GNNs, polynomial-based methods occupy the mainstream in designing a filter through the Laplacian matrix. However, polynomial combinations factored by the Laplacian matrix naturally have limitations in message passing (e.g., over-smoothing). Furthermore, most existing spectral GNNs are based on polynomial bases, which struggle to capture the high-frequency parts of the graph spectral signal. Additionally, we also find that even increasing the polynomial order does not change this situation, which means polynomial-based models have a natural deficiency when facing high-frequency signals. To tackle these problems, we propose WaveNet, which aims to effectively capture the high-frequency part of the graph spectral signal from the perspective of wavelet bases through reconstructing the message propagation matrix. We utilize Multi-Resolution Analysis (MRA) to model this question, and our proposed method can reconstruct arbitrary filters theoretically. We also conduct node classification experiments on real-world graph benchmarks and achieve superior performance on most datasets. Our code is available at https://github.com/Bufordyang/WaveNet

Computing compact finite difference formulas under radial basis functions with enhanced applicability

In this work, we use the combination of radial basis functions (RBFs) and polynomials to derive compact finite difference (FD) formulas. We use RBFs and integrated RBFs (IRBFs), to obtain two sets of formulas: RBF-Hermite FD (RBF-HFD) and IRBF weight formulas. The analytical coefficients and truncation errors for these formulations are computed for the first derivative, second derivative, and 2D Laplacian operator. Our presented approach expands the usability of the weights by introducing a general framework for deriving weighting formulas. Finally, the computational results underscore the effectiveness of the introduced approach.