E. J. Kansa and Y. C. Hon, Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial di#erential equations. Computers Math. Applic. 39 (7-8) 123--137 (2000).  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the same: The ability to elegantly and accurately approximate scattered data without using any mesh. There have been some concerns about the computational cost and stability of the RBF methods, but many di#erent viable approaches to overcome these di#culties have been proposed, see for example [12, 13, 14, 15, 16] and the references therein. There are two main groups of radial basis functions, piecewise smooth and infinitely smooth. Some examples of both are given in Table 1. Typically, the piecewise smooth RBFs lead to an algebraic rate of convergence to the desired function as the number of points ....

E. J. Kansa and Y. C. Hon, Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial di#erential equations. Computers Math. Applic. 39 (7-8) 123--137 (2000).

....corresponds to a system of equations with an unsymmetric coe#cient matrix, schematically structured as 5 2 f 3 . It has been shown (by example) that, in rare cases, the coe#cient matrix may become singular [13] However, practical experience shows that in general the method works well [14]. Method 2. Symmetric collocation A variation that leads to a symmetric coe#cient matrix was derived by Wu, 1992 [6] see also Fasshauer, 1996 [7] It has been shown for this method that the symmetry assures a non singular system of equations [6] The idea is to modify the basis functions in the ....

E. J. Kansa and Y. C. Hon, Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial di#erential equations. Comput. Math. Appl. 39 (7-8) 123--137 (2000).

....FKM discretizing PDEs. It is worth pointing out that unlike the FEM, the FKM holds the symmetric matrix with radically symmetric radial basis function even if the differential operator lacks the self adjoint property. In addition, the FKM eases severe ill conditioning of the global RBF schemes [26]. Therefore, the FKM is a very competing technique to handle a broader range of large size problems. By applying local symmetric geometry partitioning, the factorization merit of symmetric centrosymmetric matrix can lead to further significantly reducing the computational effort, preserving the ....

Kansa, E.J. and Hon, Y.C., Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations. Comput. Math. Appls. 39, 2000, 123-137.

....with an unsymmetric coe#cient matrix, schematically structured as 2 4 # ## 3 5 2 4 # 3 5 = 2 4 g f 3 5 . It has been shown (by example) that, in rare cases, the coe#cient matrix may become singular [11] However, practical experience shows that in general the method works well [12]. Method 2. Symmetric collocation A variation that leads to a symmetric coe#cient matrix was derived by Wu, 1992 [4] see also Fasshauer, 1996 [5] It has been shown for this method that the symmetry assures a non singular system of equations [4] The idea is to modify the basis functions in the ....

E. J. Kansa and Y. C. Hon, Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial di#erential equations. Comput. Math. Appl. 39 (7-8) 123--137 (2000).

....of a constant multiquadric scaling for the interpolation of various test functions. They concluded that test functions with large curvature such as the surface of a sphere require large scales, while those with considerable variation require smaller scales. Based upon their results, Hon and Kansa [10] conjectured that the scaling of multiquadrics should be proportional to the local radius of curvature. Galperin and Zheng [8] suggested to optimize local scaling factors along with the data points. There is some need for data dependent strategies for optimal choice of data locations and scales. ....

Y.C. Hon and E.J. Kansa. Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial dierential equations. Comput. Math. Applic., 39:123-127, 2000.

....to the applicability of these methods (see [40, 49] The only way to make things work acceptably with globally supported RBFs for large problems is to employ some kind of localization scheme. Various techniques are currently being investigated. Among them are domain decomposition (see e.g. [13, 37]) localization of the basis functions (see e.g. 2, 37] the ideas for which are akin to fast multipole methods popular in quantum chemistry and physics (cf. 28] and most recently (see e.g. 20, 46] the use of these techniques within an iterative scheme which is likened to Newton s method ....

....The only way to make things work acceptably with globally supported RBFs for large problems is to employ some kind of localization scheme. Various techniques are currently being investigated. Among them are domain decomposition (see e.g. 13, 37] localization of the basis functions (see e.g. [2, 37]) the ideas for which are akin to fast multipole methods popular in quantum chemistry and physics (cf. 28] and most recently (see e.g. 20, 46] the use of these techniques within an iterative scheme which is likened to Newton s method for polynomial interpolation in [46] All of these ....

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Kansa, E. J. and Y. C. Hon, Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations, preprint, 1998.

....Chen and He [25] conjectured that the RBF based numerical scheme may circumvent the curse of dimensionality like the Monte Carlo method. To be more precise, the computational effort in using the RBF on solving higher dimensional problems only grows linearly instead of exponentially. Kansa and Hon [26] in their numerical tests also observed that the RBF collocation 15 method seems to enjoy this computational advantage. For numerical verifications, it can be seen from Tables 4 1 and 4 2 that the BKM with only tens points produced rather accurate solutions for the 3D problems. This indicates ....

....to attain the reliable solutions by the boundary type discretization schemes. 4.2. Convergence and conditioning number Like all global numerical schemes, the large dense interpolation matrix resulted from using the RBF based scheme usually suffers from severe ill conditioning inefficiency [3,26]. To further investigate this issue, Table 6 displays the average relative errors and conditioning numbers of the BKM solutions of the previous 2D homogeneous Helmholtz and convection diffusion problems, with the columns under Err and Cond with lower case and c respectively; and L is the total ....

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E.J. Kansa and Y.C. Hon, 'Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations', Cornput. Math. Appls., 39, 123-137 (2000).

.... problem of using RBFs for solving partial di erential equations (PDEs) For instances, domain decomposition is a common technique used in traditional numerical schemes and had been shown to reduce the condition number and increase eciency in using the RBFs for solving PDEs by Kansa and Hon [19] and Wong et al. 25] However, no theoretical discussion of convergence and condition number of the preconditioner is given by these methods. Wu [28] recently developed classes of compactly supported radial basis functions (CSRBFs) in which the unique existence of the solution was assured. It had ....

Kansa E. J. and Y. C. Hon, "Circumventing the Ill-Conditioning Problem with Multiquadric Radial Basis Functions: Applications to Elliptic Partial Dierential Equations", Comput. Math. Applic., in press.

....multi grid schemes. The MQ RBFs are attractive for prewavelet construction due to exceptional rates of convergence and their infinite 2 differentiability. Franke and Schaback paper [18] provides the first theoretical analysis for solving PDEs by collocation using the RBF methods. Kansa and Hon [29] studied several methods for solving linear systems that arise from the MQ collocation problems. They studied the 2D Poisson equation, and showed that ill conditioning of the system could be circumvented by using block partitioning methods. Kansa [28] introduced the concept of variable shape ....

....problems. They studied the 2D Poisson equation, and showed that ill conditioning of the system could be circumvented by using block partitioning methods. Kansa [28] introduced the concept of variable shape parameters c j in the MQ scheme that appeared to work well in some cases. In Kansa and Hon [29], a recipe for selecting c j based upon the local radius of curvature of the solution surface was found to give better results than a constant c j MQ scheme. Kansa and Hon [29] tested the MQ method for the 2D Poisson equation with a set of exact solutions F = exp(ax by) cos(ax by) sin(ax ....

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E.J. Kansa, Y.C. Hon, Circumventing the ill-conditioning problem with Multiquadrics radial basis functions: Applications to elliptic partial differential equations, Adv. Comp. Math. (1998).

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