Y. C. Hon and R. Schaback, On unsymmetric collocation by radial basis functions. Appl. Math. Comput. 119 (2--3) 177--186 (2001).  Home/Search   Document Details and Download   Summary   Related Articles   Check

....the interior points leads to the equations ,#) f(x i ) i= NB 1, N. This 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 ....

.... corresponding constraints on the expansion coe#cients will in many cases lead to a guaranteed non singular coe#cient matrix [16] However, for the elliptic problem, the non singularity cannot be guaranteed when unsymmetric collocation is employed, even if the correct polynomial term is added [13]. We have found that such polynomial terms make little or no di#erence in the resulting accuracy, so we have not considered them further. 3. For Method 3, it is not necessary to put the extra centers outside# , but it does not seem to be especially beneficial to put them inside. 4 A Cauchy ....

Y. C. Hon and R. Schaback, On unsymmetric collocation by radial basis functions. Appl. Math. Comput. 119 (2--3) 177--186 (2001).

....3. New domain type RBF schemes 3.1. Modified Kansa method based on the Green integral Kansa [7] introduced the first domain type RBF schemes which is now called as the Kansa s method. Despite great effort, the rigorous mathematical proof of the solvability of the Kansa s method is still missing [20]. One drawback in the Kansa s method is that the mixed boundary conditions may destroy the symmetric interpolation matrix. Fasshauer [8] presented the symmetric Hermite RBF collocation scheme which produces the symmetric matrix irrespective of the governing and boundary condition equations. In ....

Schaback, R. and Hon, Y.C., On unsymmetric collocation by radial basis functions, J. Appl. Math. Comp. 119, 2001, 177-186.

....are more attractive than the BEM in terms of accuracy, simplicity and efficiency. 3. Modified Kansa method based on the second Green identity, spline approximation FKM and direct MKM Despite great effort, the rigorous mathematical proof of the solvability of the Kansa s method is still missing [14]. The boundary conditions also destroy the symmetricity of its interpolation matrix. As an alternative, refs. 5 and 6 present the symmetric Hermite RBF collocation scheme with sound mathematical analysis of solvability. One common issue in the Kansa s method and symmetric Hermite method, however, ....

Schaback, P, Hon, Y.C. (2001) On unsymmetric collocation by radial basis functions. J. Appl. Math. Comp. 119, 177-186.

....i x j ,#) f(x i ) i=N B 1, N. This corresponds to a system of equations 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 ....

.... interpolation, adding a polynomial term of a certain degree to the RBF interpolant, and introducing corresponding constraints on the expansion coe#cients will in many cases lead to a guaranteed non singular coe#cient matrix [14] However, for the elliptic problem, this is not necessarily the case [11]. We have found that such polynomial terms make little or no di#erence in the resulting accuracy, so we have not considered them terms further. 6 Elisabeth Larsson and Bengt Fornberg 3. For Method 3, it is not necessary to put the extra centers outside# , but it does not seem to be especially ....

Y. C. Hon and R. Schaback, On unsymmetric collocation by radial basis functions. Appl. Math. Comput. 119 (2--3) 177--186 (2001).

.... OE is taken from an appropriate class of functions (e.g. strictly positive definite functions, but also multiquadrics or thin plate splines if additional care is taken) For the nonsymmetric method it was recently shown that there exist configurations of collocation points for which A is singular [13]. Since the nonsymmetric approach is easier to implement (especially for nonlinear problems) and since singular configurations seem to be quite rare (cf. 13] 2 we have decided to use this method in the present paper. The characterization of certain configurations of collocation points which ....

.... is taken) For the nonsymmetric method it was recently shown that there exist configurations of collocation points for which A is singular [13] Since the nonsymmetric approach is easier to implement (especially for nonlinear problems) and since singular configurations seem to be quite rare (cf. [13]) 2 we have decided to use this method in the present paper. The characterization of certain configurations of collocation points which guarantee nonsingularity of the collocation matrix for the nonsymmetric approach is an interesting open problem. In Sect. 2 we outline the operator Newton ....

Hon, Y. C. and R. Schaback, On unsymmetric collocation by radial basis functions, Appl. Math. Comput., to appear.

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