112 research outputs found
Error bound for radial basis interpolation in terms of a growth function
We suggest an improvement of Wu-Schaback local error bound for radial basis interpolation by using a polynomial growth function. The new bound is valid without any assumptions about the density of the interpolation centers. It can be useful for the localized methods of scattered data fitting and for the meshless discretization of partial differential equation
Algorithms and error bounds for multivariate piecewise constant approximation
We review the surprisingly rich theory of approximation of functions of many vari- ables by piecewise constants. This covers for example the Sobolev-PoincarĀ“e inequalities, parts of the theory of nonlinear approximation, Haar wavelets and tree approximation, as well as recent results about approximation orders achievable on anisotropic partitions
Hierarchical Riesz bases for Hs(Omega), 1 < s < 5/2
On arbitrary polygonal domains , we construct hierarchical Riesz bases for Sobolev spaces . In contrast to an earlier construction by Dahmen, Oswald, and Shi (1994), our bases will be of Lagrange instead of Hermite type, by which we extend the range of stability from to . Since the latter range includes , with respect to the present basis, the stiffness matrices of fourth-order elliptic problems are uniformly well-conditioned
Scattered data fitting on surfaces using projected Powell-Sabin splines
We present C1 methods for either interpolating data or for fitting scattered data associated with a smooth function on a two-dimensional smooth manifold Ī© embedded into R3. The methods are based on a local bivariate Powell-Sabin interpolation scheme, and make use of local projections on the tangent planes. The data fitting method is a two-stage method. We illustrate the performance of the algorithms with some numerical examples, which, in particular, confirm the O(h3) order of convergence as the data becomes dens
On stable local bases for bivariate polynomial spline spaces
Stable locally supported bases are constructed for the spaces \cal S d r (\triangle) of polynomial splines of degree dā„ 3r+2 and smoothness r defined on triangulations \triangle , as well as for various superspline subspaces. In addition, we show that for rā„ 1 , in general, it is impossible to construct bases which are simultaneously stable and locally linearly independent
Overlap Splines and Meshless Finite Difference Methods
We consider overlap splines that are defined by connecting the patches of
piecewise functions via common values at given finite sets of nodes, without
using any partitions of the computational domain. It is shown that some
classical finite difference methods can be interpreted as collocation with
overlap splines. Moreover, several versions of the meshless finite difference
methods, such as the RBF-FD method, are equivalent to the collocation or
discrete least squares with overlap splines, for appropriately chosen patches
Error Bounds for a Least Squares Meshless Finite Difference Method on Closed Manifolds
We present an error bound for a least squares version of the kernel based
meshless finite difference method for elliptic differential equations on smooth
compact manifolds of arbitrary dimension without boundary. In particular, we
obtain sufficient conditions for the convergence of this method
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