Combination Of Radial Basis Functions Research Articles

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.

An Efficient Reliability Method with Multiple Shape Parameters Based on Radial Basis Function

Structural reliability analysis has an inherent contradiction between efficiency and accuracy. The metamodel can significantly reduce the computational cost of reliability analysis by a simpler approximation. Therefore, it is crucial to build a metamodel, which achieves the minimum simulations and accurate estimation for reliability analysis. Aiming at this, an effective adaptive metamodel based on the combination of radial basis function (RBF) model and Monte Carlo simulation (MCS) is proposed. Different shape parameters are first used to generate the weighted prediction variance, and the search for new training samples is guided by the active learning function that achieves a tradeoff of (1) being close enough to limit state function (LSF) to have a high reliability sensitivity; (2) keeping enough distance between the existing samples to avoid a clustering problem; and (3) being in the sensitive region to ensure the effectiveness of the information obtained. The performance of the proposed method for a nonlinear, non-convex, and high dimensional reliability analysis is validated by three numerical cases. The results indicate the high efficiency and accuracy of the proposed method.

Adaptive Weighted Strategy Based Integrated Surrogate Models for Multiobjective Evolutionary Algorithm

Although the integrated model has good convergence ability, it is difficult to solve the multimodal problem and noisy problem due to the lack of uncertainty evaluation. Radial basis function model performs best for different degrees of nonlinear problems with small-scale and noisy training datasets but is insensitive to the increase of decision-space dimension, while Gaussian process regression model can provide prediction fitness and uncertainty evaluation. Therefore, an adaptive weighted strategy based integrated surrogate models is proposed to solve noisy multiobjective evolutionary problems. Based on the indicator-based multiobjective evolutionary framework, our proposed algorithm introduces the weighted combination of radial basis function and Gaussian process regression, and U-learning sampling scheme is adopted to improve the performance of population in convergence and diversity and judge the improvement of convergence and diversity. Finally, the effectiveness of the proposed algorithm is verified by 12 benchmark test problems, which are applied to the hybrid optimization problem on the construction of samples and the determination of parameters. The experimental results show that our proposed method is feasible and effective.

Vibration optimization of laminated composites with curvilinear reinforcing fibers for shell structures

Tailored Fiber Placement (TFP) is an embroidery technique that enables it possible to design free fiber shapes on base material sheets. The TFP generally creates curved fibers on a flat surface. However, CFRP structures are often used as shell structures rather than as flat plates, and there is not enough research to design fiber shapes on curved surfaces. In this research, we applied our previous optimization method for curved fiber shapes to shell structures. Curved fiber shapes were created by the linear combination of radial basis functions (RBFs) to represent complex fiber shapes. The fiber shape on a curved surface was represented by projecting the fiber shape on a plane surface using a coordinate transformation technique. Using this method, we investigated optimum fiber shapes to improve the first natural frequency of shell structure composites. The results showed that the optimum fiber shape gave higher natural frequencies than the linear fiber orientation. Obtained optimum fiber shape resulted in higher natural frequencies than the linear fiber orientation.

Predicting coal price using time series methods and combination of radial basis function (RBF) neural network with time series

Predicting coal price using time series methods and combination of radial basis function (RBF) neural network with time series

A novel radial basis function method for 3D linear and nonlinear advection diffusion reaction equations with variable coefficients

The radial basis function method for 3D advection diffusion reaction equations with variable coefficients is presented. The proposed method implements the linear combination of radial basis functions which impose boundary conditions in advance, and thus such a combination with weighted parameters can be used to construct the final approximation. Furthermore, the weighted parameters are solved by substituting the approximation into governing equations. This method leads to crucial improvements in the feasibility and accuracy which can now be easily applied to general 3D nonlinear problems through linearized techniques. Finally, accuracy and efficiency of the proposed method are verified by several examples.

Meshless Approximation and Helmholtz-Hodge Decomposition of Vector Fields.

The analysis of vector fields is crucial for the understanding of several physical phenomena, such as natural events (e.g., analysis of waves), diffusive processes, electric and electromagnetic fields. While previous work has been focused mainly on the analysis of 2D or 3D vector fields on volumes or surfaces, we address the meshless analysis of a vector field defined on an arbitrary domain, without assumptions on its dimension and discretisation. The meshless approximation of the Helmholtz-Hodge decomposition of a vector field is achieved by expressing the potential of its components as a linear combination of radial basis functions and by computing the corresponding conservative, irrotational, and harmonic components as solution to a least-squares or to a differential problem. To this end, we identify the conditions on the kernel of the radial basis functions that guarantee the existence of their derivatives. Finally, we demonstrate our approach on 2D and 3D vector fields measured by sensors or generated through simulation.

Automatic annotation of satellite images with multi class support vector machine

Automatic Image Annotation (AIA) is used in image retrieval systems to retrieve the images by predicting tags for images. To achieve image retrieval with high accuracy, an automatic image annotation approach by using Multiclass SVM with the hybrid kernel is proposed. The hybrid kernel is a combination of Radial Basis Function (RBF) and Polynomial Kernel which overcomes the drawbacks of single kernels such as less accuracy, high computational complexity, etc. This technique exploits the Linear Binary Pattern- Discrete Wavelet Transform (LBP-DWT) feature extraction technique to extract the features in horizontal, vertical, and diagonal directions. The experiments suggest that the multiclass SVM can attain a higher accuracy than other conventional SVM with any single kernels. The Multiclass SVM can achieve high accuracy as 95.61% and increases the accuracy by 3.26%, 1.79%, and Kappa coefficient by 3.22%, 2.27% when compared with SVM RBF kernel, polynomial kernel respectively.

MODEL RADIAL APPROXIMATIONS OF FILTERING BASED ON THE GREEN'S FUNCTIONS

The article discusses the problem of approximating solutions of differential equations describing the process of a two-dimensional fluid flow in porous media. The approximation is presented as a combination of radial basis functions based of the Green's function used to solve the Poisson equation with variable coefficients in the case of steady state filtration and parabolic equations in the transient regime. To illustrate the effectiveness of the proposed approximation obtained by the field pressure distribution in the reservoir with a network of injection and production wells. Comparing of approximated pressure and design points to a satisfactory accuracy of the results. It is revealed that the obtained model of approximation of the stationary filtration equation using Green as the basis functions provides the root-mean-square deviation of the approximated pressure from the calculated one at the level of 0.95 - 1.05 %. This type of basis functions is used in constructing approximating models for non-stationary modes of multiphase filtration. The maximum deviations are observed precisely in these cells, where the most abrupt (peak) pressure changes occur. The Neumann's boundary conditions, as follows from the presented results, are satisfied. It should be noted that the Neumann boundary conditions, especially for all boundaries, are the most complicated computational case. To approximate the nonstationary solution of the filtration equation, the Green function is used, which is shown as a product of two parts: one part only depends exponentially on time; the second part depends only on the spatial coordinates. The use of the background intensity of the sources made it possible to build approximators for the possibilities of non-stationary transformation of the intensity of the sources even with a non-zero equilibrium of the intensities.

A symplectic procedure for two-dimensional coupled elastic wave equations using radial basis functions interpolation

This study proposes a meshless symplectic algorithm for two-dimensional coupled elastic waves, which satisfy the equations of linear isotropic elasticity on a rectangular domain with appropriate initial and boundary conditions. This method is based on a combination of radial basis functions (RBFs) collocation and geometric integrators. The method involves two steps, first the spatial discretization by using RBFs interpolation transforms the seismic waves into a finite-dimensional Hamiltonian system, then evolves the semi-discretized system by using symplectic integrators. Furthermore, the conservations of energy and symplectic structure are investigated. The truncation error and global error of the method are estimated. Numerical experiments show that the scheme is efficient, easy to implement with the scattered knots and possesses a good behavior in the long time integration.

An interpolation method for coupling non-conforming patches in isogeometric analysis of vibro-acoustic systems

This paper presents an interpolation method for coupling non-conforming Non-Uniform Rational B-Splines (NURBS) patches in isogeometric analysis of vibro-acoustic systems. This coupling method is based on a master–slave formulation, in which a smooth interpolating function is first constructed to represent the sound pressure field of the master side and then is sampled to determine the discrete sound pressure of the slave side. The coupling constraints are studied by the interpolation matrix built in this process and imposed on the system by Lagrange multiplier method. For interfaces with weak non-conformity, the interpolating function is constructed by a direct linear combination of radial basis functions (RBF); for interfaces with strong non-conformity or curved shape, this basic interpolation method is enhanced by a moving least squares approximation to improve the coupling accuracy. The proposed method starts with non-conforming acoustic–acoustic interfaces and is extended to non-conforming structural–acoustic interfaces. In both cases, the RBF-based coupling constraints depend only on Euclidian distances between control points, allowing them to be generated in a problem-independent and dimension-insensitive way. The accuracy and convergence behavior of the proposed method are benchmarked by an eigenvalue analysis of a two-dimensional acoustic cavity. Its efficiency is further investigated through frequency response analyses of a three-dimensional car-like model, with both strong and weak structural–acoustic interactions.

Using radial basis functions to solve two dimensional linear stochastic integral equations on non-rectangular domains

The main goal of this paper is presenting an efficient numerical scheme to solve two dimensional linear stochastic integral equations on non-rectangular domains. The proposed method is based on combination of radial basis functions (RBFs) interpolation and Gauss–Legendre quadrature rule for double integrals. The most important advantage of proposed method is that it does not require any discretization and so it is independent of the geometry of the domains. Thus, many problems on the irregular domains can be solved. By using this method, the solution of consideration problem is converted to the solution of the linear system of algebraic equations which can be solved by a suitable numerical method. Also, the convergence analysis of this approach is discussed. Finally, applicability of the present method is investigated through illustrative examples.

Applying models of spatial interpolation for the approximation of responses in the refinement of parameters of finite element models

The numerical experiment on refining the parameters of the finite element model of the beam by the method of approximating the responses is presented in the article. As mathematical models of joint-stock companies are used: linear combinations of radial-basis functions, and Kriging-models. These models are generated in the work on the basis of Latin squares and depend on the parameters to be refined (the moduli of elasticity of finite element groups of the beam). To obtain optimal values of the parameters, a genetic optimization method was used. The results of solving the optimization problem showed a high level of coincidence of the parameter values with a combination of response models obtained from dynamic and static types of calculations. It was also shown that when solving the problems of finite element models, it is sufficient to use models constructed only on the basis of radial-basis functions.

The combination of meshless method based on radial basis functions with a geometric numerical integration method for solving partial differential equations: Application to the heat equation

In this paper a new scheme is investigated for solving partial differential equations via combination of radial basis functions (RBFs) and group preserving scheme (GPS), which takes advantage of two powerful methods. In this method, we use Kansas approach to approximate the spatial derivatives and then we apply GPS method to approximate first-order time derivative. An advantage of the developed method is that it can be applied to problems with non-regular geometrical domains. To show the efficiency of this method, some heat equations are solved in one, two and three dimension spaces. The two-dimensional version of heat equation on different geometries such as the rectangular, triangular and circular domains is solved. The three-dimensional case is solved on the cubical and spherical domains. To show the high accuracy of the method, a comparison study of the present method and method used in the paper of Dehghan [1] is given.

An Investigation of Radial Basis Function-Finite Difference (RBF-FD) Method for Numerical Solution of Elliptic Partial Differential Equations

The Radial Basis Function (RBF) method has been considered an important meshfree tool for numerical solutions of Partial Differential Equations (PDEs). For various situations, RBF with infinitely differentiable functions can provide accurate results and more flexibility in the geometry of computation domains than traditional methods such as finite difference and finite element methods. However, RBF does not suit large scale problems, and, therefore, a combination of RBF and the finite difference (RBF-FD) method was proposed because of its own strengths not only on feasibility and computational cost, but also on solution accuracy. In this study, we try the RBF-FD method on elliptic PDEs and study the effect of it on such equations with different shape parameters. Most importantly, we study the solution accuracy after additional ghost node strategy, preconditioning strategy, regularization strategy, and floating point arithmetic strategy. We have found more satisfactory accurate solutions in most situations than those from global RBF, except in the preconditioning and regularization strategies.

Structural data constraints for implicit modeling of folds

A recent method for modeling folds uses a fold frame with coordinates based on the structural geology of folds: fold axis direction, fold axial surface and extension direction. The fold geometry can be characterised by rotating the fold frame by the pitch of the fold axis in the axial surface and the angle between the folded foliation and the axial surface. These rotation angles can be expressed as 1D functions of the fold frame coordinates. In this contribution we present methods for extracting and automatically modeling the fold geometries from structural data. The fold rotation angles used for characterising the fold geometry can be calculated locally from structural observations. The fold rotation angles incorporate the structural geology of the fold and allow for individual structural measurements to be viewed in the context of the folded structure. To filter out the effects of later folding the fold rotation angles are plotted against the coordinates of the fold frame. Using these plots the geometry of the folds can be interpolated directly from structural data where we use a combination of radial basis function and harmonic analysis to interpolate and extrapolate the fold geometry. This contribution addresses a major limitation in existing methods where the fold geometry was not constrained from structural data. We present two case studies: a proof of concept synthetic model of a non-cylindrical fold and an outcrop of an asymmetrical fold within the Lachlan Fold belt at Cape Conran, Victoria, Australia.

Representing 3D shapes based on implicit surface functions learned from RBF neural networks

We propose to represent the shape of 3D objects using a neural network classifier. The 3D shape is learned from a neural network, where Radial Basis Function (RBF) is applied as the activation function for each perceptron. The implicit functions derived from the neural network is a combination of radial basis functions, which can represent complex shapes. The use of RBF provides a rotation, translation and scaling invariant feature to represent the shape. We conduct experiments on a new prostate dataset and public datasets. Our testing results show that our neural network-based method can accurately represent various shapes.

Computer vision for Ethiopian agricultural crop pest identification

<p>Crop pest is an organism that creates damage on to the agriculture by feeding crops. The research focuses on four major types of crop pest which occurs on teff, wheat, sorghum, barley and maize these are Black tef beetles, Ageda korkur, Degeza and Yesinde Kish Kish. The aim of this paper is identification of the four types of agricultural crop pest using a computer vision technique. The image of crop pest were taken from Amhara regions of Ethiopia i.e. Adiet, Dejen, Gonder, Debremarkos (places where images were taken). In this paper, artificial neural network (ANN), a hybrid of self organizing map (SOM) with Radial basis function (RBF) and a hybrid of support vector machine (SVM) with Radial basis function (RBF) are used, and also we used Otsu and Kmeans segmentation techniques. Finally we selected Kmeans techniques for segmenting crop pest. In general, the overall result showed that using kmeans segmentation in RBF and SVM perform better than using otsu method in the other algorithm and the recognition performance of the combination of RBF (Radial basis function) and SVM (support vector machine) is 93.33%.</p>

A meshless method based on the dual reciprocity method for one‐dimensional stochastic partial differential equations

This article describes a new meshless method based on the dual reciprocity method (DRM) for the numerical solution of one‐dimensional stochastic heat and advection–diffusion equations. First, the time derivative is approximated by the time–stepping method to transforming the original stochastic partial differential equations (SPDEs) into elliptic SPDEs. The resulting elliptic SPDEs have been approximated with the new method, which is a combination of radial basis functions (RBFs) method and the DRM method. We have used inverse multiquadrics (IMQ) and generalized IMQ (GIMQ) RBFs, to approximate functions in the presented method. The noise term has been approximated at the source points, at each time step. The developed formulation is verified in two test problems with investigating the convergence and accuracy of numerical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 292–306, 2016

Automatic Curve Fitting Based on Radial Basis Functions and a Hierarchical Genetic Algorithm

Curve fitting is a very challenging problem that arises in a wide variety of scientific and engineering applications. Given a set of data points, possibly noisy, the goal is to build a compact representation of the curve that corresponds to the best estimate of the unknown underlying relationship between two variables. Despite the large number of methods available to tackle this problem, it remains challenging and elusive. In this paper, a new method to tackle such problem using strictly a linear combination of radial basis functions (RBFs) is proposed. To be more specific, we divide the parameter search space into linear and nonlinear parameter subspaces. We use a hierarchical genetic algorithm (HGA) to minimize a model selection criterion, which allows us to automatically and simultaneously determine the nonlinear parameters and then, by the least-squares method through Singular Value Decomposition method, to compute the linear parameters. The method is fully automatic and does not require subjective parameters, for example, smooth factor or centre locations, to perform the solution. In order to validate the efficacy of our approach, we perform an experimental study with several tests on benchmarks smooth functions. A comparative analysis with two successful methods based on RBF networks has been included.