Basis Function Method Research Articles

A Function-Based Behavioral Modeling Method for Air Combat Simulation

A Function-Based Behavioral Modeling Method for Air Combat Simulation

Hybrid Haar wavelet and meshfree methods for hyperbolic double interface problems: Numerical implementations and comparative performance analysis

This paper introduces a variety of approaches for solving 2D and 3D hyperbolic double interface problems. The methods are based on the Haar wavelet method, multiquadric radial basis function method, and integrated multiquadric radial basis function method. Temporal derivatives are handled using the second central difference and the Houbolt method. Various numerical approaches based on these methods are developed, and their implementations are discussed in complete detail. The paper evaluates and compares the performances of these approaches using both linear and nonlinear 2D and 3D double interface hyperbolic problems. Error analysis, conducted using the L-infinity norm, and efficiency assessments measured through CPU times contribute to a comprehensive understanding of the applicability and comparative effectiveness of the proposed methods. This study provides valuable insights for researchers and practitioners dealing with the challenges posed by interface problems in general.

Causal inference with hidden mediators

Summary Proximal causal inference was recently proposed as a framework to identify causal effects from observational data in the presence of hidden confounders for which proxies are available. In this paper, we extend the proximal causal inference approach to settings where identification of causal effects hinges upon a set of mediators that are not observed, yet error prone proxies of the hidden mediators are measured. Specifically, (i) we establish causal hidden mediation analysis, which extends classical causal mediation analysis methods for identifying natural direct and indirect effects under no unmeasured confounding to a setting where the mediator of interest is hidden, but proxies of it are available; (ii) we establish a hidden front-door criterion, criterion to allow for hidden mediators for which proxies are available; (iii) we show that the identification of a certain causal effect called the population intervention indirect effect remains possible with hidden mediators in settings where challenges in (i) and (ii) might co-exist. We view (i)–(iii) as important steps towards the practical application of front-door criteria and mediation analysis as mediators are almost always measured with error and, thus, the most one can hope for in practice is that the measurements are at best proxies of mediating mechanisms. We propose identification approaches for the parameters of interest in our considered models. For the estimation aspect, we propose an influence function-based estimation method and provide an analysis for the robustness of the estimators.

Using the Probability Density Function-Based Channel-Combination Bloch-Siegert Method Realizes Permittivity Imaging at 3T.

Magnetic resonance electrical properties tomography (MR EPT) can retrieve permittivity from the B1+ magnitude. However, the accuracy of the permittivity measurement using MR EPT is still not ideal due to the low signal-to-noise ratio (SNR) of B1+ magnitude. In this study, the probability density function (PDF)-based channel-combination Bloch-Siegert (BSS) method was firstly introduced to MR EPT for improving the accuracy of the permittivity measurement. MRI experiments were performed using a 3T scanner with an eight-channel receiver coil. The homogeneous water phantom was scanned for assessing the spatial distribution of B1+ magnitude obtained from the PDF-based channel-combination BSS method. Gadolinium (Gd) phantom and rats were scanned for assessing the feasibility of the PDF-based channel-combination BSS method in MR EPT. The Helmholtz-based EPT reconstruction algorithm was selected. For quantitative comparison, the permittivity measured by the open-ended coaxial probe method was considered as the ground-truth value. The accuracy of the permittivity measurement was estimated by the relative error between the reconstructed value and the ground-truth value. The reconstructed relative permittivity of Gd phantom was 52.413, while that of rat leg muscle was 54.053. The ground-truth values of relative permittivity of Gd phantom and rat leg muscle were 78.86 and 49.04, respectively. The relative error of average permittivity was 33.53% for Gd and 10.22% for rat leg muscle. The results indicated the high accuracy of the permittivity measurement using the PDF-based channel-combination BSS method in MR EPT. This improvement may promote the clinical application of MR EPT technology, such as in the early diagnosis of cancers.

Efficiently solving large-scale electrostatic lightning problems by integrating characteristic basis functions into boundary element method

The evaluation of the electric field near objects of complex geometries is critical in the pre-bridging regime of lightning strikes in high-voltage transmission systems, which is subject to an open domain problem and can be computationally expensive. In this paper, the characteristic basis function method (CBFM) is incorporated into the conventional boundary element method (BEM) to efficiently handle large-scale electrostatic problems. Within the CBFM, low-level basis functions have been replaced by high-level macro characteristic basis functions, resulting in a significant reduction in the number of degrees of freedom. This reduction in unknowns allows iteration-free direct solvers to approximate the matrices, further reducing the computational complexity of the problem. Numerical examples of the surface charge distribution of an actual 500 kV tower-transmission-line system and a three MW wind turbine in the lightning environment demonstrate the accuracy of the proposed CBFM, while requiring considerably much fewer computational resources compared with conventional BEM. Furthermore, CBFM is highly parallelizable, implying the feasibility of handling large electrostatic lightning problems with complex geometries.

Influence of Electronic Correlations on Electron-Phonon Interactions of Molecular Systems with the GW and Coupled Cluster Methods.

Electron-phonon interactions are of great importance to a variety of physical phenomena, and their accurate description is an important goal for first-principles calculations. Isolated examples of materials and molecular systems have emerged where electron-phonon coupling is enhanced over density functional theory (DFT) when using the Green's-function-based ab initio GW method, which provides a more accurate description of electronic correlations. It is, however, unclear how general this enhancement is and how employing high-end quantum chemistry methods, which further improve the description of electronic correlations, might further alter electron-phonon interactions over GW or DFT. Here, we address these questions by computing the renormalization of the highest occupied molecular orbital energies of Thiel's set of organic molecules by harmonic vibrations using DFT, GW, and equation-of-motion coupled-cluster calculations. We find that, depending on the amount of exact exchange included in the DFT starting point, GW can increase the magnitude of the electron-phonon coupling across Thiel's set of molecules by an average factor of 1.1-1.8 compared to the underlying DFT, while equation-of-motion coupled-cluster leads to an increase of 1.4-2. The electron-phonon coupling predicted with the ab initio GW method is generally in much closer agreement to coupled cluster values compared to DFT, establishing GW as a promising route for accurately computing electron-phonon phenomena in molecules and beyond at a much lower computational cost than higher-end quantum chemistry techniques.

Method of Radial Basis Functions for Solving the Partial Integro-Differential Equation of Diffusion with Nonlocal Effects

Method of Radial Basis Functions for Solving the Partial Integro-Differential Equation of Diffusion with Nonlocal Effects

A Hybrid Algorithm for Multiple Disease Prediction: Radial Basis Function and Logistic Regression

Disease prediction is an important aspect of modern medicine, which aims to diagnose disease early and provide appropriate treatment to patients. This research uses a hybrid approach that combines the RBF (Radial Basis Function) kernel algorithm with logistic regression to predict various diseases in medical datasets. This method is intended to improve prediction performance by exploiting the advantages of each algorithm. This research uses a dataset containing medical information about several diseases collected from the Kaggle dataset. First, the RBF kernel is applied to transform the data features into a more informative, non-linear representation. Then, the logistic regression model is used to make predictions based on the features that have been processed by the RBF kernel. In this research, the hybrid RBF (Radial Basis Function) method was proven to be superior in predicting multiple diseases. This method shows the highest accuracy of 0.9460, as well as excellent precision, recall, and F1-score values of 0.8680, 0.8097, and 0.8294, respectively. The advantage of the hybrid RBF method lies in its ability to capture complex patterns in data that other methods often cannot identify, as well as its ability to handle non-linear decision boundaries, which are a common characteristic in medical datasets. Keywords: Disease prediction, Hybrid approach, RBF kernel algorithm, Logistic regression, medical datasets

Multi-Agent System Based Cooperative Control for Speed Convergence of Virtually Coupled Train Formation.

This paper investigates the problem of spacing control between adjacent trains in train formation and proposes a distributed train-formation speed-convergence cooperative-control algorithm based on barrier Lyapunov function. Considering practical limitations such as communication distance and bandwidth constraints during operation, not all trains can directly communicate with the leader and obtain the expected trajectory it sends, making it difficult to maintain formation consistency as per the predetermined ideal state. Furthermore, to address the challenge of unknown external disturbances encountered by trains during operation, this paper designs a distributed observer deployed on each train in the formation. This observer can estimate and dynamically compensate for unknown reference trajectories and disturbances solely based on the states of adjacent trains. Additionally, to ensure that the spacing between adjacent trains remains within a predefined range, a safety hard constraint, this paper encodes the spacing hard constraint using barrier Lyapunov function. By integrating nonlinear adaptive control theory to handle model parameter uncertainties, a barrier Lyapunov function-based adaptive control method is proposed, which enables all trains to track the reference trajectory while ensuring that the spacing between them remains within the preset interval, therefore guaranteeing the asymptotic stability of the closed-loop system. Finally, a practical example using data from the Guangzhou Metro Line 22, specifically the route from Shiguang Road Station to Chentougang Station over three stations and two sections, is utilized to validate the effectiveness and robustness of the proposed algorithm.

A mesh-free framework for high-order simulations of viscoelastic flows in complex geometries

The accurate and stable simulation of viscoelastic flows remains a significant computational challenge, exacerbated for flows in non-trivial and practical geometries. Here we present a new high-order meshless approach with variable resolution for the solution of viscoelastic flows across a range of Weissenberg numbers. Based on the Local Anisotropic Basis Function Method (LABFM) of King et al. (2020), highly accurate viscoelastic flow solutions are found using Oldroyd B and PPT models for a range of two dimensional problems — including Kolmogorov flow, planar Poiseulle flow, and flow in a representative porous media geometry. Convergence rates up to 9th order are shown. Three treatments for the conformation tensor evolution are investigated for use in this new high-order meshless context (direct integration, Cholesky decomposition, and log-conformation), with log-conformation providing consistently stable solutions across test cases, and direct integration yielding better accuracy for simpler unidirectional flows. The final test considers symmetry breaking in the porous media flow at moderate Weissenberg number, as a precursor to a future study of fully 3D high-fidelity simulations of elastic flow instabilities in complex geometries. The results herein demonstrate the potential of a viscoelastic flow solver that is both high-order (for accuracy) and meshless (for straightforward discretisation of non-trivial geometries including variable resolution). In the near-term, extension of this approach to three dimensional solutions promises to yield important insights into a range of viscoelastic flow problems, and especially the fundamental challenge of understanding elastic instabilities in practical settings.

Introducing the evaluation condition number: A novel assessment of conditioning in radial basis function methods

Background and Objective:RBF methods are crucial for reconstructing functions from data sites, like solving partial differential equations. However, ill-conditioning is common, and the standard condition number may not fully address it as it focuses solely on the Kernel matrix. Methods:The concept of the “largest manageable scale” is introduced, representing the maximum scale where errors are minimized without amplification. To determine this scale, we propose a cheaply available “evaluation condition number” (EVN) that evaluates the conditioning of the linear system arising from kernel methods, as well as “reliable digits” that indicate how many digits in the function evaluation are accurate. The results are implemented across various changes in scale, types of RBFs, and sizes of kernel matrices. Results and Conclusions:The evaluation condition number enhances solution quality estimation in kernel method linear systems over standard methods. We establish the connection between three conditioning criteria: the standard, effective, and our proposed evaluation condition numbers. Furthermore, equipped with the EVN criterion, we demonstrate the advantage of normalization techniques, such as truncated SVD over full SVD and MATLAB backslash. Our results confirm the proposed approach’s effectiveness in maintaining stability and accuracy despite rank loss. We provide the code implementation at: https://doi.org/10.24433/CO.4444506.v2.

A utility-based three-way group decision consensus model with overlapping subgroups

This paper introduces a relative utility function-based consensus method for three-way group decision (TWGD) in a social network scenario where overlapping subgroups and non-overlapping subgroups co-exist. Firstly, we construct a three-way decision (TWD) model rooted in utility theory, which proposes a novel objective calculation method for the relative utility functions based on evaluation values. The relative utility matrices of decision makers (DMs) are derived on this basis. Secondly, a utility and trust-driven overlapping clustering algorithm is designed to maximize the overall trust level and the difference between subgroup clustering centers, aiming to divide the group into several manageable subgroups. In addition, we identify contrarian DMs, triggering a delegation-exit mechanism, and determine the weights of subgroups and DMs. Thirdly, a three-way group consensus decision model is formulated to provide targeted adjustment strategies for different types of DMs. To improve group consensus, we address the influence of key DMs on the opinions of overlapping subgroups with a minimum adjustment consensus model. Finally, we demonstrate the feasibility and scientificity of the proposed method through the illustrative example, sensitivity analysis, and comparative analysis.

Application of Local Integrated Radial Basis Function Method for Solving System of Fredholm Integro-Differential Equations

Application of Local Integrated Radial Basis Function Method for Solving System of Fredholm Integro-Differential Equations

Use of the Adaptive Cross Approximation for the Efficient Computation of the Reduced Matrix with the Characteristic Basis Function Method

A technique for the reduction in the CPU-time in the analysis of electromagnetic problems using the Characteristic Basis Function Method (CBFM) is presented here, allowing for analysis of electrically large cases where an iterative solution process cannot be avoided. This technique is based on the use of the Adaptive Cross Approximation (ACA) for the fast computation of the coupling matrix between CBFs belonging to adjacent blocks, as well as the Multilevel Fast Multipole Method (MLFMM) for the computation of matrix−vector products in the solution of the full system. This combination allows for a noticeable reduction in the computational resources during the analysis of electrically large and complex scenarios while maintaining a very good degree of accuracy. A number of test cases serve to validate the presented approach in terms of accuracy, memory and CPU-time compared with conventional techniques.

Bending analysis of quasicrystal plates using adaptive radial basis function method

A novel mesh-free kernel method is presented for solving function interpolation problems and partial differential equations (PDEs). Despite the simplicity and accuracy of the radial basis function (RBF) method, two main difficulties have been identified: the selection of the scale or shape parameter and the ill-conditioning issue. To address these challenges, an effective condition number (ECN) strategy is proposed for the scale parameter selection, utilizing the ill-conditioning issue to minimize errors. The study demonstrates that this method performs exceptionally well with severely ill-conditioned matrices, making it suitable challenging engineering problems. However, when the problem is severely ill-conditioned, an accurate and efficient method for determining the minimal singular values used in the ECN formulation is required. To tackle this, we utilize an efficient algorithm called minsv, which specifically computes the minimal singular value. The significance of accurately computing the smallest singular value is demonstrated through two interpolation examples, involving non-smooth and smooth test functions with various RBFs and data points. Furthermore, the RBF method is applied to solve a quasicrystal problem, which presents a challenging engineering problem with four unknown parameters, and excellent results are obtained. Finally, we illustrate that using MATLAB’s svd function can potentially produce completely incorrect minimal singular values. The codes implemented are available at https://doi.org/10.24433/CO.8716692.v1.

Temporal and spatial variations of terrestrial water storage in the northeastern Tibetan Plateau retrieved by GNSS observations

The variation of solid Earth's hydrologic loading could cause the elastic vertical deformation of the crust, and the Global Navigation Satellite System (GNSS) could effectively monitor the vertical displacement of surface loads. However, the widely used Green's function method does not work well in areas where GNSS sites are sparse. Here, the vertical displacement time series of GNSS stations and the Slepian basis function method have been applied to convert displacement signals into spatial spectrum signals. The elastic mass load theory is used to study the changes in terrestrial water storage on the Northeastern Tibetan Plateau (NETP). The temporal and spatial characteristics of seasonal water changes are well-represented by the GNSS, the Gravity Recovery and Climate Experiment (GRACE), and the Global Land Data Assimilation System (GLDAS). Several data points suggest that the change in water storage shows a gradual increase from the northeast to the southwest. The greatest annual amplitude of water storage retrieved by GNSS is ∼159 mm, which is greater than the ∼47 mm and ∼44 mm obtained by GRACE and GLDAS. These results demonstrate that GNSS is capable of capturing small-scale hydrological changes in this region, whereas GRACE and GLDAS data tend to underestimate seasonal variations in water storage. We also used GNSS to describe the hydrological drought conditions in NETP, showing that GNSS could be used as an independent method to characterize hydrological drought events. The findings suggest it could observe water storage with high spatial and temporal resolution and aid in comprehending regional hydrological trends with a sparse GNSS station network.

A new control method for leader–follower consensus problem of uncertain constrained nonlinear multi-agent systems

The design of Adaptive Fuzzy Sliding-Mode Control (AFSMC) is discussed in this study for the distributed leader–follower consensus problem of nonlinear uncertain multi-agent systems with input and state constraints. Avoiding the critical circumstances related to the violation of input and state constraints is an important issue in the design of consensus control systems. The limitations of existing adaptive consensus controllers developed for full-state constrained systems motivated the present novel method without restrictive structural assumptions. To convert the original problem to an unconstrained consensus control problem, the proposed control system uses a barrier function-based state transformation method and the input-state linearization methodology. As a result, each agent’s adaptive fuzzy sliding-mode control input, which consists of a fuzzy system and a robust term, is constructed based on the derived representation of the system dynamics. The fuzzy system is used to approximate the dynamic equations of the agent, its neighbors, and maybe the leader in each local feedback control law structure, and the robust term is designed to compensate for the fuzzy approximation error. The magnitudes of the robust terms and the output vectors of the fuzzy systems are also determined using the proposed adaptation laws. The second Lyapunov theorem and Barbalat’s lemma are used to verify the closed-loop system’s asymptotic stability. The effectuality of the proposed methodology is confirmed by numerical simulations for several Autonomous Underwater Vehicles (AUVs). Simulation results show that by means of the proposed AFSMC, the leader–follower consensus is achieved without state constraint violation and performance degradation.

An Optimal Multiquadric Variable Shape Parameter for Boundary Value Problems Using Particle Swarm Optimization

The multiquadric radial basis function method has been widely used to solve partial differential equations-based problems regarding its flexibility and meshfree characteristics. The accuracy and stability of this method are derived and based on the use of a free-shape parameter that sensibly controls the comportment of the technique. Significant improvements have already been reported and show that variable shape parameters conduct the method to handle problems with striking results compared to global-based techniques. Nevertheless, choosing a suitable set of shape parameters is still an open topic because of the complexity of the method when the number of collocation points increases. The current work proposes a variant particle swarm optimization based on local displacement with attractors to determine the multi-quadratic function's ``best'' optimal variable shape parameter in solving boundary value problems. Based on an initially random set of variable shape parameters, the proposed algorithm first performs and evaluates the errors between the expected exact solution and the approximate solution thoroughly. In the first stage, the particle swarm algorithm search for an optimal set of shape parameters that minimize the error and the conditioning number of the radial basis system matrix. In the second stage, the obtained optimal set of shape parameters is applied to solve the considered problem. In this way, when the number of collocation points increases, the first stage based on particle swarm optimization stabilizes the strategy. It proposes an ``acceptable'' set of shape parameters for the given problem. The proposed method is applied to a set of well-known boundary value problems in one and two-dimensional spaces and compared to other techniques published in the literature. The results show that the proposed method achieves more accurate solutions than recently proposed in the literature.

An adaptive Bayesian approach for improved sensitivity in joint monitoring of mean and variance using Max-EWMA control chart

This article introduces an adaptive approach within the Bayesian Max-EWMA control chart framework. Various Bayesian loss functions were used to jointly monitor process deviations from the mean and variance of normally distributed processes. Our study proposes the mechanism of using a function-based adaptive method that picks self-adjusting weights incorporated in Bayesian Max-EWMA for the estimation of mean and variance. This adaptive mechanism significantly enhances the effectiveness and sensitivity of the Max-EWMA chart in detecting process shifts in both the mean and dispersion. The Monte Carlo simulation technique was used to calculate the run-length profiles of different combinations. A comparative performance analysis with an existing chart demonstrates its effectiveness. A practical example from the hard-bake process in semiconductor manufacturing is presented for practical context and illustration of the chart settings and performance. The empirical results showcase the superior performance of the Adaptive Bayesian Max-EWMA control chart in identifying out-of-control signals. The chart’s ability to jointly monitor the mean and variance of a process, its adaptive nature, and its Bayesian framework make it a useful and effective control chart.

Development of integrated radial basis function Kriging interpolation for linear and nonlinear parabolic integro-differential equations

In this study, we explore linear and nonlinear parabolic integro-differential equations in one and two dimensions. We employ a semi-implicit scheme to discretize the temporal variable and discretize the spatial variable using an integrated radial basis function based on the moving Kriging interpolation (MKI) method. Unlike the global integrated radial basis function (IRBF) method, our proposed approach is particularly well-suited for addressing large-scale problems. Moreover, issues such as the selection of shape parameters leading to matrices with high condition numbers can be mitigated by devising an algorithm to determine optimal shape parameters, resulting in lower condition numbers. To assess the accuracy of the proposed method, we utilize two criteria, namely, L2 and H1. We compare the computed results obtained from the IRBF-MKI method with those from two grid and finite element methods. To demonstrate the accuracy and efficiency of our approach, we consider irregular areas and scattered Halton points in the two-dimensional case.