Radial Basis Function methods for scattered data interpolation and for the numerical solution of PDEs were originally implemented in a global manner. Subsequently, it was realized that the methods could be implemented...Radial Basis Function methods for scattered data interpolation and for the numerical solution of PDEs were originally implemented in a global manner. Subsequently, it was realized that the methods could be implemented more efficiently in a local manner and that the local approaches could match or even surpass the accuracy of the global implementations. In this work, three localization approaches are compared: a local RBF method, a partition of unity method, and a recently introduced modified partition of unity method. A simple shape parameter selection method is introduced and the application of artificial viscosity to stabilize each of the local methods when approximating time-dependent PDEs is reviewed. Additionally, a new type of quasi-random center is introduced which may be better choices than other quasi-random points that are commonly used with RBF methods. All the results within the manuscript are reproducible as they are included as examples in the freely available Python Radial Basis Function Toolbox.展开更多
Solving large radial basis function （RBF） interpolation problem with non-customized methods is computationally expensive and the matrices that occur are typically badly conditioned. In order to avoid these difficult...Solving large radial basis function （RBF） interpolation problem with non-customized methods is computationally expensive and the matrices that occur are typically badly conditioned. In order to avoid these difficulties, we present a fitting based on radial basis functions satisfying side conditions by least squares, although compared with interpolation the method loses some accuracy, it reduces the computational cost largely. Since the fitting accuracy and the non-singularity of coefficient matrix in normal equation are relevant to the uniformity of chosen centers of the fitted RBE we present a choice method of uniform centers. Numerical results confirm the fitting efficiency.展开更多
It takes more time and is easier to fall into the local minimum value when using the traditional full-supervised learning algorithm to train RBFNN. Therefore, the paper proposes one algorithm to determine the RBFNN’s...It takes more time and is easier to fall into the local minimum value when using the traditional full-supervised learning algorithm to train RBFNN. Therefore, the paper proposes one algorithm to determine the RBFNN’s data center based on the improvement density method. First it uses the improved density method to select RBFNN’s data center, and calculates the expansion constant of each center, then only trains the network weight with the gradient descent method. To compare this method with full-supervised gradient descent method, the time not only has obvious reduction (including to choose data center’s time by density method), but also obtains better classification results when using the data set in UCI to carry on the test to the network.展开更多
文摘Radial Basis Function methods for scattered data interpolation and for the numerical solution of PDEs were originally implemented in a global manner. Subsequently, it was realized that the methods could be implemented more efficiently in a local manner and that the local approaches could match or even surpass the accuracy of the global implementations. In this work, three localization approaches are compared: a local RBF method, a partition of unity method, and a recently introduced modified partition of unity method. A simple shape parameter selection method is introduced and the application of artificial viscosity to stabilize each of the local methods when approximating time-dependent PDEs is reviewed. Additionally, a new type of quasi-random center is introduced which may be better choices than other quasi-random points that are commonly used with RBF methods. All the results within the manuscript are reproducible as they are included as examples in the freely available Python Radial Basis Function Toolbox.
基金Supported by National Natural Science Youth Foundation (10401021).
文摘Solving large radial basis function （RBF） interpolation problem with non-customized methods is computationally expensive and the matrices that occur are typically badly conditioned. In order to avoid these difficulties, we present a fitting based on radial basis functions satisfying side conditions by least squares, although compared with interpolation the method loses some accuracy, it reduces the computational cost largely. Since the fitting accuracy and the non-singularity of coefficient matrix in normal equation are relevant to the uniformity of chosen centers of the fitted RBE we present a choice method of uniform centers. Numerical results confirm the fitting efficiency.
文摘It takes more time and is easier to fall into the local minimum value when using the traditional full-supervised learning algorithm to train RBFNN. Therefore, the paper proposes one algorithm to determine the RBFNN’s data center based on the improvement density method. First it uses the improved density method to select RBFNN’s data center, and calculates the expansion constant of each center, then only trains the network weight with the gradient descent method. To compare this method with full-supervised gradient descent method, the time not only has obvious reduction (including to choose data center’s time by density method), but also obtains better classification results when using the data set in UCI to carry on the test to the network.
