Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are ...Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C<sup>∞</sup>-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C<sup>∞</sup>-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.展开更多
We solve numerically an eigenvalue elliptic partial differential equation(PDE)ranging from two to six dimensions using the generalized multiquadric(GMQ)radial basis functions(RBFs).Two discretization methods are em-pl...We solve numerically an eigenvalue elliptic partial differential equation(PDE)ranging from two to six dimensions using the generalized multiquadric(GMQ)radial basis functions(RBFs).Two discretization methods are em-ployed.The first method is similar to the classic mesh-based discretization method requiring n centers per dimension or a total ndpoints.The second method is based upon n randomly generated points in dℜrequiring far fewer data centers than the classic mesh method.Instead of having a crisp boundary,we form a“fuzzy”boundary.Using the analytic or numerical in-verse interior and boundary operators,we find the local and global minima and maxima to cull discretization points.We also find that the GMQ-RBF“flatness”can be controlled by increasing the GMQ exponential,β.We per-form a search to find the smallest root mean squared error(RMSE)by varying the exponent,the maximum,the minimum range of the GMQ shape parame-ter,and boundary influence,with the exponential having the most influence.Because the GMQ-RBFs are essentially nonlinear,it follows that the starting point of the simple search influences the end result.The optimal algorithm for high dimensional PDEs is still the subject of much research and may wait for the common place availability of massively parallel quantum computers for even higher dimensional PDEs and integral equations(IEs).展开更多
文摘Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C<sup>∞</sup>-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C<sup>∞</sup>-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.
文摘We solve numerically an eigenvalue elliptic partial differential equation(PDE)ranging from two to six dimensions using the generalized multiquadric(GMQ)radial basis functions(RBFs).Two discretization methods are em-ployed.The first method is similar to the classic mesh-based discretization method requiring n centers per dimension or a total ndpoints.The second method is based upon n randomly generated points in dℜrequiring far fewer data centers than the classic mesh method.Instead of having a crisp boundary,we form a“fuzzy”boundary.Using the analytic or numerical in-verse interior and boundary operators,we find the local and global minima and maxima to cull discretization points.We also find that the GMQ-RBF“flatness”can be controlled by increasing the GMQ exponential,β.We per-form a search to find the smallest root mean squared error(RMSE)by varying the exponent,the maximum,the minimum range of the GMQ shape parame-ter,and boundary influence,with the exponential having the most influence.Because the GMQ-RBFs are essentially nonlinear,it follows that the starting point of the simple search influences the end result.The optimal algorithm for high dimensional PDEs is still the subject of much research and may wait for the common place availability of massively parallel quantum computers for even higher dimensional PDEs and integral equations(IEs).
