Recovery of functions from weak data using unsymmetric meshless kernel-based methods

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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“…• Least squares optimization is much easier to implement, but it has a much more complicated mathematical background [26,27].…”

Section: Introductionmentioning

confidence: 99%

On convergent numerical algorithms for unsymmetric collocation

2008

Self Cite

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“…Some mathematical issues about those items may be found in available literature. A general mathematical framework for establishing the error bounds and convergence of the unsymmetric meshless methods similar to the MLPG approach has been provided in [27,28]. The useful information on the mathematical background of unsymmetric collocation methods may be found in [29].…”

Section: Discretizationmentioning

confidence: 99%

Mixed meshless formulation for analysis of shell-like structures

Sorić

Jarak

2010

Computer Methods in Applied Mechanics and Engineering

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“…Arcangéli, et al [1] used Theorem 1.1 to derive error bounds for interpolating and smoothing (m, s)-splines, an application which we do not consider. Instead we are interested in another major application of these Sobolev estimates: Schaback's framework for unsymmetric meshless methods for operator equations [12], see also the earlier version [13]. A sampling inequality is necessary for unsymmetric meshless methods, such as Schaback's modification of Kansa's method [12,13], which involve an overdetermined system of equations in general.…”

Section: Introductionmentioning

confidence: 99%

A sampling inequality for fractional order Sobolev semi-norms using arbitrary order data

Corrigan,

Wallin,

Wanner

2008

Preprint

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To improve convergence results obtained using a framework for unsymmetric meshless methods due to Schaback (Preprint Göttingen 2006), we extend, in two directions, the Sobolev bound due to Arcangéli et al. (Numer Math 107, 181-211, 2007), which itself extends two others due to Wendland and Rieger (Numer Math 101, 643-662, 2005) and Madych (J. Approx Theory 142, 116-128, 2006). The first is to incorporate discrete samples of arbitrary order derivatives into the bound, which are used to obtain higher order convergence in higher order Sobolev norms. The second is to optimally bound fractional order Sobolev semi-norms, which are used to obtain more optimal convergence rates when solving problems requiring fractional order Sobolev spaces, notably inhomogeneous boundary value problems.

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Recovery of functions from weak data using unsymmetric meshless kernel-based methods

Cited by 8 publications

References 23 publications

On convergent numerical algorithms for unsymmetric collocation

On convergent numerical algorithms for unsymmetric collocation

Mixed meshless formulation for analysis of shell-like structures

A sampling inequality for fractional order Sobolev semi-norms using arbitrary order data

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