Solvability of partial differential equations by meshless kernel methods

Hon, Y.C.; Schaback, Robert

doi:10.1007/s10444-006-9023-2

Cited by 18 publications

(8 citation statements)

References 42 publications

(34 reference statements)

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“…In order to carry out some mathematical analysis, it is necessary to make further assumptions and modify the formulation. In some of our earlier works [16][17][18], Kansa's method was modified in such a way that solvability is guaranteed. Later, we proposed in [19,20] another variant of the method so that error bounds become possible.…”

Section: Introductionmentioning

confidence: 99%

On convergent numerical algorithms for unsymmetric collocation

2008

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In this paper, we are interested in some convergent formulations for the unsymmetric collocation method or the so-called Kansa's method. We review some newly developed theories on solvability and convergence. The rates of convergence of these variations of Kansa's method are examined and verified in arbitraryprecision computations. Numerical examples confirm with the theories that the modified Kansa's method converges faster than the interpolant to the solution; that is, exponential convergence for the multiquadric and Gaussian radial basis functions (RBFs). Some numerical algorithms are proposed for efficiency and accuracy in practical applications of Kansa's method. In double-precision, even for very large RBF shape parameters, we show that the modified Kansa's method, through a subspace selection using a greedy algorithm, can produce acceptable approximate solutions. A benchmark algorithm is used to verify the optimality of the selection process.

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Section: Introductionmentioning

confidence: 99%

On convergent numerical algorithms for unsymmetric collocation

2008

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“…In some of our earlier works [2,8,9], Kansa's method was modified in such a way that solvability is guaranteed. Later, we proposed in [10,11] another variant of the method so that error bounds become possible.…”

Section: Introductionmentioning

confidence: 99%

Arbitrary Precision Computations of Variations of Kansa's Method

Ling

2009

Progress on Meshless Methods

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“…The development of radial basis function (RBF) as a special case of RKHS has proven to be efficient and effective in multivariate scattered data interpolation [15,50] and boundary value problems solver [18,21]. It is noted that the RKHS and RBF have recently been widely utilized in smoothing spline [17] and solving partial differential equations [11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Multiscale Support Vector Approach for Solving Ill-Posed Problems

2014

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Based on the use of compactly supported radial basis functions, we extend in this paper the support vector approach to a multiscale support vector approach (MSVA) scheme for approximating the solution of a moderately ill-posed problem on bounded domain. The Vapnik's -intensive function is adopted to replace the standard l 2 loss function in using the regularization technique to reduce the error induced by noisy data. Convergence proof for the case of noise-free data is then derived under an appropriate choice of the Vapnik's cut-off parameter and the regularization parameter. For noisy data case, we demonstrate that a corresponding choice for the Vapnik's cut-off parameter gives the same order of error estimate as both the a posteriori strategy based on discrepancy principle and the noise-free a priori strategy. Numerical examples are constructed to verify the efficiency of the proposed MSVA approach and the effectiveness of the parameter choices.

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Solvability of partial differential equations by meshless kernel methods

Cited by 18 publications

References 42 publications

On convergent numerical algorithms for unsymmetric collocation

On convergent numerical algorithms for unsymmetric collocation

Arbitrary Precision Computations of Variations of Kansa's Method

Multiscale Support Vector Approach for Solving Ill-Posed Problems

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