Standard Radial Basis Function Research Articles

RBF-based partition of unity methods for two-dimensional time-dependent PDEs: Numerical and theoretical aspects

This article discusses standard Radial Basis Function (RBF) partition of unity method and an enhanced version known as the direct RBF partition of unity method. We introduce the framework of these methods for the effective and accurate numerical analysis of certain two-dimensional time-dependent Partial Differential Equations (PDEs). Additionally, we conduct a convergence analysis of these methods and establish error bounds for local approximations. These error bounds are determined based on conditions associated with the eigenvalues of the Laplacian operator matrices generated by the proposed methods. To validate the accuracy and reliability of these approaches, we provide several numerical examples. We assess the consistency between theoretical and numerical results. The localized techniques introduced in this research demonstrate a significant reduction in computational cost and algorithmic complexity, achieved through the direct local procedure. Despite this computational advantage, both versions of the RBF partition of unity method exhibit a similar level of accuracy. Finally, we conduct a comparative analysis between the RBF partition of unity local methods and the RBF-generated finite differences method, offering valuable insights into the strengths and limitations of each approach.

An adaptive harmonic polynomial cell method for three-dimensional fully nonlinear wave-structure interaction with immersed boundaries

To accurately simulate wave-structure interaction based on fully nonlinear potential flow theory, a three-dimensional (3 D) high-order immersed-boundary adaptive harmonic polynomial cell (IB-AHPC) method is proposed. Both the free surface and body surface are immersed in background octree cells that are adaptively refined near the boundaries of interest, thereby dramatically reducing computational costs without loss of accuracy. We also propose an easy-to-implement IB strategy to deal with possible instabilities in the time-domain solution arising from the intersection of Dirichlet–Neumann boundaries. For a linearized problem of wave-wall interaction, a matrix-based stability analysis is performed, providing mathematical support for the robustness of the proposed IB strategy. In contrast to the two-dimensional HPC method, compressed cells are found to offer superior stability compared to stretched cells in the vertical direction, while equal mesh aspect ratio in the horizontal plane is superior. Cubic octree cells are, however, still preferred in practice. The free surface is primarily described by a set of massless background wave markers; however, to address the challenges of IB methods in tracking the free surface evolution near the structure, additional body-fitted wave markers are introduced close to the waterline. The information exchange between these two sets of wave markers is realized by radial basis function (RBF) interpolation. While standard RBF schemes have grid-size-dependent filtering performance, we propose a normalized RBF scheme, which is then optimized in terms of the number of neighboring nodes, a smoothing coefficient and the basis functions. Excellent accuracy properties of the proposed 3 D IB-AHPC method are demonstrated by studying fully nonlinear wave propagation. The method is further applied to study relevant fully nonlinear wave-structure interaction problems, including sloshing in 3 D rectangular tanks and wave diffraction of a bottom-mounted cylinder in regular waves. Satisfactory agreement is demonstrated with existing experimental and numerical results, suggesting that the proposed 3 D IB-AHPC method is a promising potential-flow method in marine hydrodynamics.

Mesh-free Solution of ADR Equations: A Comparison Between the Application of Hybrid and Standard Radial Basis Functions

Mesh-free Solution of ADR Equations: A Comparison Between the Application of Hybrid and Standard Radial Basis Functions

Lagrange radial basis function collocation method for boundary value problems in $ 1 $D

<abstract><p>This paper introduces the Lagrange collocation method with radial basis functions (LRBF) as a novel approach to solving 1D partial differential equations. Our method addresses the trade-off principle, which is a key challenge in standard RBF collocation methods, by maintaining the accuracy and convergence of the numerical solution, while improving the stability and efficiency. We prove the existence and uniqueness of the numerical solution for specific differential operators, such as the Laplacian operator, and for positive definite RBFs. Additionally, we introduce a perturbation into the main matrix, thereby developing the perturbed LRBF method (PLRBF); this allows for the application of Cholesky decomposition, which significantly reduces the condition number of the matrix to its square root, resulting in the CPLRBF method. In return, this enables us to choose a large value for the shape parameter without compromising stability and accuracy, provided that the perturbation is carefully selected. By doing so, highly accurate solutions can be achieved at an early level, significantly reducing central processing unit (CPU) time. Furthermore, to overcome stagnation issues in the RBF collocation method, we combine LRBF and CPLRBF with multilevel techniques and obtain the Multilevel PLRBF (MuCPLRBF) technique. We illustrate the stability, accuracy, convergence, and efficiency of the presented methods in numerical experiments with a 1D Poisson equation. Although our approach is presented for 1D, we expect to be able to extend it to higher dimensions in future work.</p></abstract>

Combining Lipschitz and RBF surrogate models for high-dimensional computationally expensive problems

Standard evolutionary optimization algorithms assume that the evaluation of the objective and constraint functions is straightforward and computationally cheap. However, in many real-world optimization problems, these evaluations involve computationally expensive numerical simulations or physical experiments. Surrogate-assisted evolutionary algorithms (SAEAs) have recently gained increased attention for their performance in solving these types of problems. The main idea of SAEAs is the integration of an evolutionary algorithm with a selected surrogate model that approximates the computationally expensive function. In this paper, we propose a surrogate model based on a Lipschitz underestimation and use it to develop a differential evolution-based algorithm. The algorithm, called Lipschitz Surrogate-assisted Differential Evolution (LSADE), utilizes the Lipschitz-based surrogate model, along with a standard radial basis function surrogate model and a local search procedure. The experimental results on seven benchmark functions of dimensions 30, 50, 100, and 200 show that the proposed LSADE algorithm is competitive compared with the state-of-the-art algorithms under a limited computational budget, being especially effective for the very complicated benchmark functions in high dimensions.

A new approach for saturation height modelling in a clastic reservoir using response surface methodology and artificial neural network

• Brook-Corey's function was the best conventional water saturation prediction model. • The RSM model gave a better prediction fit than the conventional methods. • The independent variables had an influence on water saturation via 3D response surface plots. • Mathematical links improved the RSM model's performance in design space. • The RBFNN model was superior compared to the RSM model with mathematical links. Saturation Height Modelling (SHM) is an important reservoir characterization method for estimating the water saturation component which is a fraction of the total reservoir fluids in porous media. Although several methods have been used in calculating water saturation, little or no research has employed Response Surface Methodology (RSM) in predicting water saturation in field cores. In this paper, well datasets were tested with four conventional curve fitting models including Lambda, Thomeer, Leverette J, and Brooks-Corey of which Brook-Corey gave the best fit with an R 2 of 0.485. The RSM with the Central Composite Design (CCD) was used to obtain the mathematical and statistical relationship between the predictors and target as well as the variables’ interactions optimization. The model was analysed using Analysis of Variance (ANOVA) which validated the correlations with an F-value of 4.96, a p-value less than 0.05, and a Lack of Fit-value of 0.76 implying that the model developed was statistically significant. The RSM model gave a better fit with an R 2 value of 0.817 with mathematical links, almost twice that predicted by the conventional methods. Then, the testing performance of the RSM model was compared to the standard radial basis function neural network model ( R 2 of 0.9779). The results proved that both RSM and RBFNN models’ performance was accurate and reliable and could give a precise prediction of water saturation without any conventional curve fitting parameters.

Research on Temperature Compensation of Multi-Channel Pressure Scanner Based on an Improved Cuckoo Search Optimizing a BP Neural Network.

A multi-channel pressure scanner is an essential tool for measuring and acquiring various pressure parameters in aerospace applications. It is important to note, however, that the pressure sensor of each of these channels will drift significantly with the increase in the temperature range of the pressure measurement, and the output voltage of each of these channels will show nonlinear characteristics, which will constrain the improvements in the accuracy of the measurement. In the regression fitting process, it is difficult to fit nonlinear data with the traditional least-squares method, which leaves pressure measurement accuracy unsatisfactory. A temperature compensation method based on an improved cuckoo search optimizing a BP neural network for a multi-channel pressure scanner is proposed in this paper to improve pressure measurement accuracy in a wide temperature range. Using the chaotic simplex algorithm, we first improved the cuckoo search algorithm, then optimized the connection weights and thresholds of the BP neural network, and finally constructed an experimental calibration system to investigate the temperature compensation of the multi-channel pressure scanning valves in the −40 °C to 60 °C temperature range. The compensation test results show that the algorithm has a better compensation effect and is more suitable for the temperature compensation of multi-channel pressure scanners than the traditional least-squares method and the standard RBF and BP neural networks. The maximum full-scale error of all 32 channels is 0.02% FS (full-scale error) and below, which realizes its high-accuracy multi-point pressure measurement in a wide temperature range.

Adaptive RBF -SMC control for multi-point mooring system

A stable adaptive control scheme for multi-point mooring system (MPMS) with uncertain dynamics is proposed in this paper. The control scheme is designed by a hybrid controller based on RBF (Radial Basis Function) NN (Neural Network) and SMC (Sliding Mode Control), which learns the MPMS dynamic changes, and the compensation of external disturbances is realized through adaptive RBFNN control. Meanwhile the RBF-SMC control parameters are adapted by the Lyapunov method to minimize squares dynamic positioning (DP) error. The convergence of the hybrid controller is proved theoretically, and the proposed mooring control scheme is applied to the “Kantan3” mooring simulation system. Finally, the simulation results are compared with the traditional PID controller and standard RBF controller to demonstrate the effective mooring positioning performance of the control scheme for the MPMS.

A Radial Basis Function Network-Based Surrogate-Assisted Swarm Intelligence Approach for Fast Optimization of Power Delivery Networks

The design and optimization of Power Delivery Networks (PDNs) in Very Large Scale Integration (VLSI) systems is becoming very challenging with the increasing complexity of such systems. Decoupling capacitors are the key elements used in a PDN to minimize power supply noise and to maintain low impedance of the PDN to avoid system failure. In this paper, a novel approach using surrogate assisted swarm intelligence is presented for efficient and fast optimization of power delivery networks. For generating the surrogate models, a standard Radial Basis Function Network (RBFN) is used. Using the proposed approach, the decoupling capacitors are selected and placed optimally, eventually reducing the cumulative impedance of the PDN below the target impedance. The performance comparison between the conventional and the surrogate-assisted approach is presented. Three case studies are presented on a practical system to demonstrate the competence of the proposed approach. The results obtained by the proposed approach are also compared with the same obtained by the state-of-the-art approaches. For the proposed approach the run time is drastically reduced compared to the state-of-the-art approaches for the optimization problem without having any effect on the performance. The consistency of results in all of the case studies confirms the validity of the proposed approach.

A novel local meshless scheme based on the radial basis function for pricing multi-asset options

‎A novel local meshless scheme based on the radial basis function (RBF) is introduced in this article for price multi-asset options of even European and American types based on the Black-Scholes model‎. ‎The proposed approach is obtained by using operator splitting and repeating the schemes of Richardson extrapolation in the time direction and coupling the RBF technology with a finite-difference (FD) method that leads to extremely sparse matrices in the spatial direction‎. ‎Therefore‎, ‎it is free of the ill-conditioned difficulties that are typical of the standard RBF approximation‎. ‎We have used a strong iterative idea named the stabilized Bi-conjugate gradient process (BiCGSTAB) to solve highly sparse systems raised by the new approach‎. ‎Moreover‎, ‎based on a review performed in the current study‎, ‎the presented scheme is unconditionally stable in the case of independent assets when spatial discretization nodes are equispaced‎. ‎As seen in numerical experiments‎, ‎it has a low computational cost and generates higher accuracy‎. ‎Finally‎, ‎the proposed local RBF scheme is very versatile so that it can be used easily for Solving numerous models and obstacles not just in the finance sector‎, ‎as well as in other fields of engineering and science‎. ‎Finally‎, ‎we conclude that the proposed local RBF scheme is very versatile so that it can be used easily for Solving numerous models and obstacles not just in the finance sector‎, ‎as well as in other fields of engineering and science‎.

ADASYN and ABC-optimized RBF convergence network for classification of electroencephalograph signal

Electroencephalograph (EEG) is supposed to be a major challenge in the area of biomedical signal processing. Being one of the widely used invasive techniques, it is capable to find many cases of brain disorder problems like epileptic seizures and sleep disorder. This work follows the procedure of convergence computing where there are different computing techniques have been combined together to achieve our final goal with much perfection. radial basis function (RBF) being one of the simplest forms of neural network (NN) can be used for the purpose of EEG signal classification, where the network uses the radial basis function as an activation function. In this study, artificial bee colony (ABC) technique was applied for optimizing the parameters that were required in the RBF network for the EEG signal classification. Adaptive synthetic oversampling (ADASYN) process was considered for improving the learning method of managing the class imbalance problem in the EEG dataset. An EEG dataset for normal and epileptic person is typically seen and is processed where five datasets are combined together in three different ways. The new datasets were grouped and were created as set 1 (A+D & E), set 2 (B+D & E), and set 3 (C+D & E). The ABC optimized RBF classifier is applied along with ADASYN method where the resolution for the above stated datasets states that the classification accuracy for set 1 and set 3 were increasing significantly as compared to the standard RBF classifier network.

Efficient radial basis function mesh deformation methods for aircraft icing

This paper presents an approach to update the moving ice boundary resulting from aircraft icing simulations using radial basis function mesh deformation techniques. State-of-the-art surface and volume point reduction schemes are used to reduce the computational cost of the mesh deformation. The data reduction schemes which are utilised include multi-level greedy surface point selection and volume point reduction. The multi-level greedy surface point selection reduces the number of control points to increase the efficiency of the interpolation operation. While the volume point reduction improves the computational cost of the volume mesh update operation which is important for large data sets. The paper assesses the capabilities of radial basis function mesh deformation for both two and three-dimensional problems. Furthermore, the effectiveness of the deformation technique is assessed for both local, non-smooth deformations and global, smooth deformations. The convergence history of the multi-level greedy point selection is assessed in terms of number of control points and computational cost. The location of the selected control points near the ice accretion illustrates the efficacy of the method for localised deformations. The results show that the radial basis function mesh deformation performs well for both the two and three-dimensional problems. The data-reduction schemes utilised in this work represent a significant improvement to standard radial basis function mesh deformation for aircraft icing problems comprising of large data-sets typical of three-dimensional problems.

A stable kernel-based technique for solving linear Fredholm integral equations of the second kind and its applications

Meshless methods based on infinitely smooth radial kernels have the potential to provide spectrally accurate function approximations with enormous geometric flexibility in any number of dimensions. The highest accuracy can often be obtained when the shape parameter in the basis function is small. But as the shape parameter goes to zero, the standard RBF interpolant matrix becomes severely ill-conditioned. The ill-conditioning can be reduced using alternate bases. One of these alternative bases is the Hilbert–Schmidt SVD basis. The Hilbert–Schmidt SVD method offers a stable mechanism for converting a set of near-flat kernels with scattered centres to a well-conditioned base for exactly the same space. In this paper, we apply the Gaussian Hilbert–Schmidt SVD basis functions method for solving the linear Fredholm integral equations of the second kind. The method estimates the solution by the discrete collocation method based on Gaussian Hilbert–Schmidt SVD basis functions constructed on a set of disordered data. This approach reduces the solution of the problem under study to the solution of a linear system of algebraic equations. Also, the convergence of the proposed approach is analyzed. Lastly, several numerical experiments are presented to test the stability and accuracy of the proposed method.

Application of Hilbert–Schmidt SVD approach to solve linear two-dimensional Fredholm integral equations of the second kind

Meshfree techniques based on infinitely smooth radial kernels have the great potential to provide spectrally accurate function approximations with irregular domain in high dimensions. The maximum accuracy can mostly be found when the RBF shape parameter is small, i.e., when the radial kernel is relatively smooth. However, as the shape parameter goes to zero, the standard RBF interpolant matrix will be very ill-conditioned. The ill-conditioning can be alleviated using alternate bases. One of these alternative bases is the Hilbert–Schmidt SVD basis. The Hilbert–Schmidt SVD approach suggests a stable mechanism for replacing a set of near-flat kernels with scattered centres to a well-conditioned base for exactly the same space. In this work, the Gaussian Hilbert–Schmidt SVD basis functions method is presented to numerically solve the linear two-dimensional Fredholm integral equations of the second kind. The method estimates the solution by the discrete collocation method based on Gaussian Hilbert–Schmidt SVD basis functions constructed on a set of scattered points. The emerged integrals in the scheme are approximately computed by the Gauss–Legendre quadrature rule. This approach reduces the problem under study to a linear system of algebraic equations which can be solved easily via applying an appropriate numerical technique. Also, the convergence of the proposed approach is established. Finally, numerical results are compared with standard RBF method to indicate the accuracy and efficiency of the suggested approach.

RETRACTED ARTICLE: Monitoring the quality of water in shrimp ponds and forecasting of dissolved oxygen using Fuzzy C means clustering based radial basis function neural networks

The success of shrimp farming lies in the proper monitoring of water quality in the pond. Of all the water quality parameters the dissolved oxygen plays a vital role in the proper growth of shrimps and influences mortality to a great extent. Traditional methods of sensing the water quality especially dissolved oxygen is done in laboratories far away from the actual pond site and it is often time consuming. Hence an accurate forecasting model which predicts the changes in the DO content is quintessential to decrease mortality, yield export quality shrimps and reduce operational costs. Water quality datasets are collected by using devices fabricated with wireless water quality sensors which floats in the ponds. Fuzzy C means was chosen as the clustering method based on the dataset collected and radial basis functions neural networks were constructed with accurate number of hidden neurons to predict the changes in the dissolved oxygen. Root Mean Square Error (RMSE), Mean Absolute Error (MAE), Correlation Coefficient (R) and Willmott index of agreement (WIA) were used as the evaluation parameters in comparison with the Multilayer Perceptron based Backpropagation Neural Network and Standard Radial Basis Function Neural Network for prediction of the DO content. The results show that the proposed method is effective with reduced errors and correlation coefficient close to unity. This accurate prediction of the dissolved oxygen helps the farmers to take corrective action when required and decrease the operational costs and produce export quality shrimps.

Semisupervised classification of hyperspectral images with low-rank representation kernel.

A semisupervised deformed kernel function, using low-rank representation with consideration of local geometrical structure of data, is presented for the classification of hyperspectral images. The proposed method incorporates the wealth of unlabeled information to deal with the limited labeled samples situation as a common case in HSIs applications. The proposed kernel needs to be computed before training the classifier, e.g., a support vector machine, and it relies on combining the standard radial basis function kernel based on labeled information and the low-rank representation kernel obtained using all available (labeled and unlabeled) information. The low-rank representation kernel can overcome the difficulties of clustering methods that are used to construct the kernels such as bagged kernel and multi-scale bagged kernel. The experimental results of two well-known HSIs data sets demonstrate the effectiveness of the proposed method in comparison with cluster kernels obtained using traditional clustering methods and graph learning methods.

An Evolutionary Self-organizing Cost-Sensitive Radial Basis Function Neural Network to Deal with Imbalanced Data in Medical Diagnosis

Class imbalance is a common issue in medical diagnosis. Although standard radial basis function neural network (RBF-NN) has achieved remarkably high performance on balanced data, its ability to classify imbalanced data is still limited. So far as we know, cost-sensitive learning is an advanced imbalanced data processing method. However, few studies have focused on the combination of RBF-NN and cost sensitivity. From our knowledge, only one paper has proposed a cost-sensitive RBF-NN for software defect prediction. However, the authors implemented a fixed RBF-NN structure. In this paper, a novel cost-sensitive RBF-NN that optimizes structure and parameters simultaneously is proposed to handle medical imbalanced data. Genetic algorithm (GA) and improved particle swarm optimization (IPSO) are used to optimize the structure and parameters of cost-sensitive RBF-NN respectively, and the optimization of cost-sensitive RBF-NN based on dynamic structure is realized. A cost-sensitive function determined adaptively by the sample distribution as the objective function of RBF-NN, so that it can adapt to datasets with different sample distributions. Experimental results show that the proposed cost-sensitive RBF-NN outperforms other state-of-the-art representative algorithms for five imbalanced medical diagnostic datasets in term of accuracy and area under curve (AUC). It can improve the accuracy of medical diagnosis and reduce the error rate of medical decisions.

An accurate and stable RBF method for solving partial differential equations

Most standard radial basis function (RBF) methods strongly depend on a shape parameter in accuracy and stability. However, the selection of optimal shape parameter in the RBF methods has long been a challenging task. In this paper, a new global RBF, known as coupled radial basis function (CRBF), is presented based on coupling the infinitely smooth RBFs with the conical spline. It is found that the Kansa’s method with the CRBF inherits the strengths of the aforementioned two types of RBFs, and achieves an accurate and stable numerical solution almost independent of the shape parameter values. Therefore, the proposed methodology completely overcome the troublesome issue of the optimal shape parameter that is a formidable obstacle to global RBF schemes. Two test examples verify the accuracy and robustness of the proposed CRBF method.

Unsteady aerodynamic modeling based on fuzzy scalar radial basis function neural networks

In this paper, a fuzzy scalar radial basis function neural network is proposed, in order to overcome the limitation of traditional aerodynamic reduced-order models having difficulty in adapting to input variables with different orders of magnitude. This network is a combination of fuzzy rules and standard radial basis function neural network, and all the basis functions are defined as scalar basis functions. The use of scalar basis function will increase the flexibility of the model, thus enhancing the generalization capability on complex dynamic behaviors. Particle swarm optimization algorithm is used to find the optimal width of the scalar basis function. The constructed reduced-order models are used to model the unsteady aerodynamics of an airfoil in transonic flow. Results indicate that the proposed reduced-order models can capture the dynamic characteristics of lift coefficients at different reduced frequencies and amplitudes very accurately. Compared with the conventional reduced-order model based on recursive radial basis function neural network, the fuzzy scalar radial basis function neural network shows better generalization capability for different test cases with multiple normalization methods.

Generalized RBF kernel for incomplete data

We construct genRBF kernel, which generalizes standard Gaussian RBF kernel to the case of incomplete data. Instead of using typical imputation techniques, which fill missing attributes by single values, we model possible outcomes at missing coordinates using data distribution. This allows to derive analytical formula for the expected value of RBF kernel taken over all possible imputations, which is a basic idea behind our method. In particular, for complete observations genRBF reduces to standard RBF kernel. Experiments show that introduced kernel applied to SVM classifier and regressor gives better results than state-of-the-art methods, especially in the case when large number of features is missing. Moreover, genRBF is easy to implement and can be used together with any kernel approach without any additional modifications.