A discrete adapted hierarchical basis solver for radial basis function interpolation
A discrete adapted hierarchical basis solver for radial basis function interpolation
复制标题
用于径向基函数插值的离散自适应分层基求解器
DOI：
10.1007/s10543-012-0397-x
复制
发表时间：
2011
影响因子：
1.5
通讯作者：
V. Eijkhout
中科院分区：
文献类型：
--
作者：
J. Castrillón;Jun Yu Li;V. Eijkhout
本文提出了一种离散分层基（HB）算法来有效地解决变多项式次数的径向基函数（RBF）插值问题。HB形成一个正交集，并适应于内核种子函数和插值节点的位置。此外，这个基础是正交的多项式的一组上定义的插值节点上的给定程度。因此，我们能够解耦的RBF插值问题的任何次数的多项式插值，并解决它在两个步骤：（1）多项式正交RBF插值问题是有效地解决了转换HB基与GMRES迭代和对角（或块SSOR）预条件。(2)然后将残差投影到正交多项式基上。我们应用我们的方法在几个测试案例中，研究其有效性。
In this paper we develop a discrete Hierarchical Basis (HB) to efficiently solve the Radial Basis Function (RBF) interpolation problem with variable polynomial degree. 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 degree defined on the interpolating nodes. We are thus able to decouple the RBF interpolation problem for any degree 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.