Radial Basis Function Approximation Research Articles

Small dataset augmentation with radial basis function approximation for causal discovery using constraint‐based method

AbstractCausal analysis involves analysis and discovery. We consider causal discovery, which implies learning and discovering causal structures from available data, owing to the significance of interpreting causal relationships in various fields. Research on causal discovery has been primarily focused on constraint‐ and score‐based interpretable methods rather than on methods based on complex deep learning models. However, identifying causal relationships in real‐world datasets remains challenging. Numerous studies have been conducted using small datasets with established ground truths. Moreover, constraint‐based methods are based on conditional independence tests. However, such tests have a lower statistical power when applied to small datasets. To solve the small sample size problem, we propose a model that generates a continuous function from available samples using radial basis function approximation. We address the problem by extracting data from the generated continuous function and evaluate the proposed method on both real and synthetic datasets generated by structural equation modeling. The proposed method outperforms constraint‐based methods using only small datasets.

Applying a New Trigonometric Radial Basis Function Approximation in Solving Nonlinear Vibration Problems

Applying a New Trigonometric Radial Basis Function Approximation in Solving Nonlinear Vibration Problems

Parallel multi-objective optimization for expensive and inexpensive objectives and constraints

Expensive objectives and constraints are key characteristics of real-world multi-objective optimization problems. In practice, they often occur jointly with inexpensive objectives and constraints. This paper presents the Inexpensive Objectives and Constraints Self-Adapting Multi-Objective Constraint Optimization algorithm that uses Radial Basis function Approximations (IOC-SAMO-COBRA) for such problems. This is motivated by the recently proposed Inexpensive Constraint Surrogate-Assisted Non-dominated Sorting Genetic Algorithm II (IC-SA-NSGA-II). These algorithms and their counterparts that do not explicitly differentiate between expensive and inexpensive objectives and constraints are compared on 22 widely used test functions. The IOC-SAMO-COBRA algorithm finds significantly better (identical/worse) Pareto fronts in at least 78% (6%/16%) of all test problems compared to IC-SA-NSGA-II measured with both the hypervolume and Inverted Generational Distance+ performance metric. The empirical cumulative distribution functions confirm this advantage for both algorithm variants that exploit the inexpensive constraints. In addition, the proposed method is compared against state-of-the-art practices on a real-world cargo vessel design problem. On this 17-dimensional two-objective practical problem, the proposed IOC-SAMO-COBRA outperforms SA-NSGA-II as well. From an algorithmic perspective, the comparison identifies specific strengths of both approaches and indicates how they should be hybridized to combine their best components.

Radial basis function‐based Pareto optimization of an outer rotor brushless DC motor

AbstractThis paper presents the development of an optimization and modeling method for the objective functions of output power, efficiency and weight of an outer rotor permanent magnet brushless DC (BLDC) motor based on radial basis function (RBF) approximation technique. The proposed RBF‐based Pareto optimization method requires less knowledge about electric/magnetic formulas and can replace conventional optimizations based on these equations with higher accuracy. To apply the proposed optimization method, the initial design should be developed using such equations. Therefore, RBFs are used to model and predict engine behavior. To optimize the objective functions, we used a genetic algorithm optimization technique with nonlinear electric and magnetic constraints to find the Pareto front set. The design obtained by the proposed radial basis function Pareto optimization (RBFPO) method was finally verified by Ansoft Maxwell. The results of optimal design using the RBFPO method have higher output power and efficiency. Also, in addition to the advantage of a favorable accuracy, RBF‐based models are significantly faster than models available in simulation tools.

A novel global RBF direct collocation method for solving partial differential equations with variable coefficients

Conventional radial basis function (RBF) approximation techniques face two fundamental challenges: determining the optimal shape parameter and solving the ill-conditioned linear system, which result from the use of typical kernel functions such as the Multiquadric (MQ) RBF. The objective of this study is to address these challenges by employing a novel RBF kernel, termed the coupled MQ RBF (CMQ-RBF). The new kernel combines the MQ with an m-order conical spline, where m denotes the optimal value determined by the so-called back check method (BCM) introduced in this study. Leveraging the CMQ-RBF kernel, we present a novel global RBF collocation method that generates a well-conditioned linear system, enabling direct solver utilization and delivering highly accurate numerical results without the need for laborious shape parameter selection. Consequently, the previously challenging issues can be effectively circumvented by switching to the "better kernel" offered by the CMQ-RBF. The proposed methodology is characterized by its mathematical simplicity and ease of numerical implementation. Several benchmark examples are investigated to verify its performance.

Radial Basis Functions Approximation Method for Time-Fractional FitzHugh–Nagumo Equation

In this paper, a numerical approach employing radial basis functions has been applied to solve time-fractional FitzHugh–Nagumo equation. Spatial approximation is achieved by combining radial basis functions with the collocation method, while temporal discretization is accomplished using a finite difference scheme. To evaluate the effectiveness of this method, we first conduct an eigenvalue stability analysis and then validate the results with numerical examples, varying the shape parameter c of the radial basis functions. Notably, this method offers the advantage of being mesh-free, which reduces computational overhead and eliminates the need for complex mesh generation processes. To assess the method’s performance, we subject it to examples. The simulated results demonstrate a high level of agreement with exact solutions and previous research. The accuracy and efficiency of this method are evaluated using discrete error norms, including L2, L∞, and Lrms.

Bayesian approach for radial kernel parameter tuning

In this paper we present a new fast and accurate method for Radial Basis Function (RBF) approximation, including interpolation as a special case, which enables us to effectively find the optimal value of the RBF shape parameter. In particular, we propose a statistical technique, called Bayesian optimization, that consists in modeling the error function with a Gaussian process, by which, through an iterative process, the optimal shape parameter is selected. The process is step by step self-updated resulting in a relevant decrease in search time with respect to the classical leave one out cross validation technique. Numerical results deriving from some test examples and an application to real data show the performance of the proposed method.

The Impact of Inclined Magnetic Field on Streamlines in a Constricted Lid-driven Cavity

The influence of oriented magnetic field on the incompressible and electrically conducting flow is investigated in a square cavity with a moving top wall and a no-slip constricted bottom wall. Radial basis function (RBF) approximation is employed to velocity-stream function-vorticity formulation of MHD equations. Numerical results are shown in terms of streamlines for different values of Hartmann number M, orientation angle of magnetic field ϴ and the height of the constricted bottom wall hc with a fixed Reynolds number. It is obtained that the number of vortices increases as either hc or M increases. However, the increase in ϴ leads to decrease the number of vortices. Formation of vortices depends on not only the strength and the orientation of the magnetic field but also the constriction of the bottom wall.

Meshfree methods for the nonlinear variable-order fractional advection–diffusion equation

The nonlinear variable-order fractional advection–diffusion equation is applied to simulate complex engineering problem that exhibits time memory and space global dependence, for example, anomalous diffusion. In this paper, Kansa’s method is used to solve the nonlinear time–space variable-order fractional advection–diffusion equation, in which the variable order of space fractional derivative relies on both time and space while the variable order of the time fractional derivative is determined by space. Moreover, the convection coefficients, diffusion coefficients and source term are nonlinear functions that depend on exact solution. The finite difference method is contracted for the time fractional derivative and Wendland’s C6 compactly supported radial basis functions are used to substitute the space fractional derivative in the problem. The nonlinear terms are approximated by using an explicit difference scheme. The Gauss-Jacobi quadrature rule is used to approximate the weakly singular integral of space fractional derivative after radial basis function approximation. To illustrate the effectiveness and accuracy of the method, three numerical examples are solved, in which 2D experiments include the cases of regular and irregular regions. The results show that Kansa’s method has obvious advantages for variable-order fractional advection–diffusion equation.

A new variable shape parameter strategy for RBF approximation using neural networks

The choice of the shape parameter highly effects the behaviour of radial basis function (RBF) approximations, as it needs to be selected to balance between the ill-conditioning of the interpolation matrix and high accuracy. In this paper, we demonstrate how to use neural networks to determine the shape parameters in RBFs. In particular, we construct a multilayer perceptron (MLP) trained using an unsupervised learning strategy, and use it to predict shape parameters for inverse multiquadric and Gaussian kernels. We test the neural network approach in RBF interpolation tasks and in a RBF-finite difference method in one and two-space dimensions, demonstrating promising results.

Solution of space–time fractional diffusion equation involving fractional Laplacian with a local radial basis function approximation

Solution of space–time fractional diffusion equation involving fractional Laplacian with a local radial basis function approximation

The analysis and optimization of thermal sensation of train drivers under occupational thermal exposure.

Prolonged exposure of train drivers to thermal discomfort can lead to occupational safety and health (OSH) risks, causing physical and mental injuries. Traditional method of treating human skin as a wall surface fail to observe accurate skin temperature changes or obtain human thermal comfort that adapts to the thermal environment. This study employs the Stolwijk human thermal regulation model to investigate and optimize the thermal comfort of train drivers. To minimize the time-consuming design optimization, a pointer optimization algorithm based on radial basis function (RBF) approximation was utilized to optimize the train cab ventilation system design and enhance drivers' thermal comfort. The train driver thermal comfort model was developed using Star-CCM+ and 60 operating conditions were sampled using an Optimal Latin Hypercube Design (Opt LHD). We analyzed the effects of air supply temperature, air supply volume, air supply angle, solar radiation intensity and solar altitude angle on the local thermal sensation vote (LTSV) and overall thermal sensation vote (OTSV) of the train driver. Finally, the study obtained the optimal air supply parameters for the Heating Ventilation and Air Conditioning (HVAC) in the train cabin under extreme summer conditions, effectively improving the thermal comfort of the driver.

A localized meshfree technique for solving fractional Benjamin–Ono equation describing long internal waves in deep stratified fluids

This article proposes a local radial basis function partition of unity method to find the numerical solution of the time-fractional Benjamin–Ono equation. The proposed method is based on partitioning the original domain into several overlapping subdomains and employing the radial basis function approximation on each local subdomain. First, the time-discrete scheme of this equation is obtained by utilizing a finite difference approach. Then, spatial discretization is established by using the local radial basis function partition of unity method. Furthermore, the convergence and stability analysis of the proposed time-discrete scheme are investigated thoroughly. Also, numerical experiments are executed through some illustrated problems, and the results are compared with those acquired by the tanh approach and Kudryashov approach solutions to show the high accuracy and plausibility of the proposed method. Moreover, to demonstrate the physical significance, the conservation constants known as mass, momentum, and energy for this equation are also calculated by the proposed method.

Streamline analysis of MHD flow in a double lid-driven cavity

The magnetohydrodynamic flow of an incompressible, steady and electrically conducting fluid is considered in double lid-driven square cavity under an inclined magnetic field. The non-dimensional governing equations are solved numerically using a radial basis function (RBF) approximation meshless method in order to obtain the flow patterns. Hartmann number [Formula: see text] and the inclination angle of magnetic field [Formula: see text] are taken as a problem variables that produce various flow transformations by their variation for [Formula: see text]. Streamlines are examined topologically, and it has been observed that the inclination angle and strength of magnetic field have different influence on the bifurcation of flow structures.

Radial Basis Function Approximation with Distributively Stored Data on Spheres

Radial Basis Function Approximation with Distributively Stored Data on Spheres

An effective high-order five-point stencil, based on integrated-RBF approximations, for the first biharmonic equation and its applications in fluid dynamics

PurposeThe purpose of this paper is to present a new discretisation scheme, based on equation-coupled approach and high-order five-point integrated radial basis function (IRBF) approximations, for solving the first biharmonic equation, and its applications in fluid dynamics.Design/methodology/approachThe first biharmonic equation, which can be defined in a rectangular or non-rectangular domain, is replaced by two Poisson equations. The field variables are approximated on overlapping local regions of only five grid points, where the IRBF approximations are constructed to include nodal values of not only the field variables but also their second-order derivatives and higher-order ones along the grid lines. In computing the Dirichlet boundary condition for an intermediate variable, the integration constants are used to incorporate the boundary values of the first-order derivative into the boundary IRBF approximation.FindingsThese proposed IRBF approximations on the stencil and on the boundary enable the boundary values of the derivative to be exactly imposed, and the IRBF solution to be much more accurate and not influenced much by the RBF width. The error is reduced at a rate that is much greater than four. In fluid dynamics applications, the method is able to capture well the structure of steady highly non-linear fluid flows using relatively coarse grids.Originality/valueThe main contribution of this study lies in the development of an effective high-order five-point stencil based on IRBFs for solving the first biharmonic equation in a coupled set of two Poisson equations. A fast rate of convergence (up to 11) is achieved.

A radial basis function approximation method for conservative Allen–Cahn equations on surfaces

In this paper, we present a meshless radial basis function method to solve conservative Allen–Cahn equation on smooth compact surfaces embedded in R3, which can inherits the mass conservation property. The proposed method is established on the operator splitting scheme. We approximate the surface Laplace–Beltrami operator by an iterative radial basis function approximation method and discretize the diffusion equation in time by the Euler method. The reaction equation containing the nonlinear function is solved analytically. Moreover, to make the mass conservation, we employ a kernel-based quadrature formula to approximate the Lagrange multiplier. The salient feature of the meshless conservative scheme is that it is explicit and more efficient than narrow band methods since few scattered nodes on the surface are adopted in spatial approximation. Several numerical experiments are performed to illustrate the accuracy and the conservation property of the scheme on spheres and other surfaces.

Design optimization of a transonic compressor blade with sweep and lean integrated with axial slot casing treatment

Two multi-objective optimization designs of the rotor blade for an axial transonic compressor are carried out in this study. Among them, the optimization of the blade alone is called blade-only optimization. The optimization of the blade integrated with axial slot casing treatment (ASCT) is called blade-integrated optimization. During the optimization stage, the values of sweep and lean in the tip region are used as the optimal variables. The stall margin improvement and the efficiency at the design-operating-point are adopted as the optimal objectives. The Radial Basis Function (RBF) approximation is used to construct the surrogate model, and the non-dominated sorting genetic algorithm II (NSGA-II) is selected to search for Pareto fronts. The optimal individuals selected from the Pareto fronts of the two optimizations are compared and analyzed. The coupling effect between the blade and ASCT is studied in this paper to illustrate the advantage of the integrated optimization. The results show that the blade-integrated optimization can optimize the coupling effect, which strengthens the ability of the ASCT to eliminate the blockage while ensuring little efficiency loss at the design point. However, the application of ASCT for the blade-only optimized blade may reduce efficiency and lead to a premature stall, which is caused by the failure to eliminate the negative effect of the coupling. Compared with the blade-only optimized blade, the application of ASCT for the blade-integrated optimized blade increases the stall margin improvement from 10.77% to 14.88%, and increases the efficiency improvement from −0.3% to −0.12%.

Spherical radial basis function approximation of some physical quantities measured

The article treats the spherical interpolation and approximation, i.e., a way of measured data processing in case when the scalar data sampling is performed at nodes on the unit 2D sphere surface in the 3D Euclidean space (in general, on the (d−1)-dimensional sphere surface in the d-dimensional space). We use the spherical radial basis function interpolation with an inverse multiquadric basis function and a second degree polynomial in Cartesian coordinates as a trend.The formulae of this type can be useful in the interpretation of various physical measurements and have wide applications in geosciences, e.g., in the treatment of anisotropy of magnetic susceptibility data measured. We present the advantages of the formula chosen on numerical examples. However, in practical computation with high number of sampling nodes, the matrix of the system for determining interpolation coefficients may be ill-conditioned. Then the standard double precision LU factorization does not give a reliable solution due to the effect of round-off error accumulation and some more sophisticated means of solving the system are to be used.

Order enhanced finite volume methods through non-polynomial approximation

In this paper, we introduce an approximation method that establishes certain order enhancements by leveraging radial basis functions (RBFs) in the numerical solution of conservation laws. The use of RBFs for interpolation and approximation is a well developed area of research. Of particular interest in this work is the development of high order finite volume (FV) weighted essentially non-oscillatory (WENO) methods, which utilize RBF approximations to obtain required data at cell interfaces. The aforementioned improvement in the order of accuracy is addressed through an analysis of the truncation error, resulting in expressions for the shape parameters appearing in the basis. This paper seeks to address the practical elements of the approach, including the evaluations of shape parameters as well as a hybrid implementation. To highlight the effectiveness of the non-polynomial basis in shock-capturing, the proposed methods are applied to systems of one-dimensional hyperbolic and weakly hyperbolic conservation laws and compared with several well-known WENO schemes in the literature. We also include a two-dimensional example for a scalar problem that demonstrates an extension to multiple dimensions. In the case of the non-smooth, weakly hyperbolic test problem, notable improvements are observed in predicting the location and height of the finite time blowup. The numerical results demonstrate that the proposed schemes attain notable improvements in accuracy, as indicated by the analysis of the reconstructions. A key contribution of this work is the development of robust third-order WENO method, which further demonstrates the effectiveness of the non-polynomial basis.