12,043 research outputs found
Partition of unity interpolation using stable kernel-based techniques
In this paper we propose a new stable and accurate approximation technique
which is extremely effective for interpolating large scattered data sets. The
Partition of Unity (PU) method is performed considering Radial Basis Functions
(RBFs) as local approximants and using locally supported weights. In
particular, the approach consists in computing, for each PU subdomain, a stable
basis. Such technique, taking advantage of the local scheme, leads to a
significant benefit in terms of stability, especially for flat kernels.
Furthermore, an optimized searching procedure is applied to build the local
stable bases, thus rendering the method more efficient
A Discrete Adapted Hierarchical Basis Solver For Radial Basis Function Interpolation
In this paper we develop a discrete Hierarchical Basis (HB) to efficiently
solve the Radial Basis Function (RBF) interpolation problem with variable
polynomial order. The HB forms an orthogonal set and is adapted to the kernel
seed function and the placement of the interpolation nodes. Moreover, this
basis is orthogonal to a set of polynomials up to a given order defined on the
interpolating nodes. We are thus able to decouple the RBF interpolation problem
for any order of the polynomial interpolation and solve it in two steps: (1)
The polynomial orthogonal RBF interpolation problem is efficiently solved in
the transformed HB basis with a GMRES iteration and a diagonal, or block SSOR
preconditioner. (2) The residual is then projected onto an orthonormal
polynomial basis. We apply our approach on several test cases to study its
effectiveness, including an application to the Best Linear Unbiased Estimator
regression problem
RBF approximation of large datasets by partition of unity and local stabilization
We present an algorithm to approximate large dataset by Radial Basis Function
(RBF) techniques. The method couples a fast domain decomposition procedure with a
localized stabilization method. The resulting algorithm can efficiently deal with large
problems and it is robust with respect to the typical instability of kernel methods
Singular Higher-Order Complete Vector Bases for Finite Methods
This paper presents new singular curl- and divergence- conforming vector bases that incorporate the edge conditions. Singular bases complete to arbitrarily high order are described in a unified and consistent manner for curved triangular and quadrilateral elements. The higher order basis functions are obtained as the product of lowest order functions and Silvester-Lagrange interpolatory polynomials with specially arranged arrays of interpolation points. The completeness properties are discussed and these bases are proved to be fully compatible with the standard, high-order regular vector bases used in adjacent elements. The curl (divergence) conforming singular bases guarantee tangential (normal) continuity along the edges of the elements allowing for the discontinuity of normal (tangential) components, adequate modeling of the curl (divergence), and removal of spurious modes (solutions). These singular high-order bases should provide more accurate and efficient numerical solutions of both surface integral and differential problems. Sample numerical results confirm the faster convergence of these bases on wedge problems
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