Results on meshless collocation techniques

Stable and Convergent Unsymmetric Meshless Collocation Methods

“…The first question requires that if n >> m test functionals are fixed and are linearly independent, the system has rank m, if the trial functions are chosen properly. This was proven in [11], while the earlier paper [13] contained an asymptotic analysis for sufficiently many test functionals and trial functions. By a rather complicated and abstract machinery, a fairly general theory leading to error bounds and convergence results for the unsymmetric meshless collocation method was finally provided in [14].…”

“…Recall that the new adaptive linear optimization scheme uses the selected m trial centers and finally performs an ℓ ∞ fit to all N >> m available test functionals. In contrast, the original adaptive greedy scheme of [11] calculates an exact interpolation using the selected m trial functions and m = n << N selected test functionals.…”

“…Going for the largest possible value can be called a "greedy" strategy. This is the common choice in linear optimization, and it has been used favourably [11] also in meshless adaptive PDE solvers based on unsymmetric collocation. But note that (12) is satisfied if we only make sure that our new functional λ has the property…”

“…First, the test problem is run on Ω = [−1, 1] 2 with different shape parameters c in order to demonstrate the performance of the adaptive greedy scheme in [11] and the proposed adaptive greedy linear optimization scheme of this paper. The number of trial functions and test functionals offered to the algorithms for selection are M = 8822 and N = 2129, respectively, but note that the algorithms select much smaller subsets adaptively.…”

A Generalised RBF Finite Difference Approach to Solve Nonlinear Heat Conduction Problems on Unstructured Datasets

“…This is a consequence of ill-conditioning in the determination of RBF weighting coefficients (as demonstrated in Driscoll & Fornberg (2002)), and is described by Robert Schaback Schaback (1995) as the uncertainty relation; better conditioning is associated with worse accuracy, and worse conditioning is associated with improved accuracy. Many techniques have been developed to reduce the effect of the uncertainty relation in the traditional RBF formulation, such as RBF-specific preconditioners Baxter (2002); Beatson et al (1999); Brown (2005); Ling & Kansa (2005), or adaptive selection of data centres Ling et al (2006); Ling & Schaback (2004). However, at present the only reliable methods of controlling numerical ill-conditioning and computational cost as problem size increases are domain decomposition Hernandez Rosales & Power (2007); Wong et al (1999); Zhang (2007); Zhou et al (2003), or the use of locally supported basis functions Fasshauer (1999); Schaback (1997); Wendland (1995); Wu (1995).…”