A Fast Treecode for Multiquadric Interpolation with Varying Shape Parameters

Quan Deng, Tobin A. Driscoll
2012 SIAM Journal on Scientific Computing  
A treecode algorithm is presented for the fast evaluation of multiquadric radial basis function (RBF) approximations. The method is a dual approach to one presented by Krasny and Wang, which applies far-field expansions to clusters of RBF centers (source points). The new approach clusters evaluation points instead and is therefore easily able to cope with basis functions that have different multiquadric shape parameters. The new treecode is able to evaluate an approximation on N centers at M
more » ... n N centers at M points in O((N + M ) log M ) time in the ideal case when evaluation points are uniformly distributed. When coupled with a two-level restricted additive Schwarz preconditioner for GMRES iterations, the treecode is well suited for use within an adaptive RBF iteration, previously described by Driscoll and Heryudono, as is demonstrated by experiments on test functions. A1127 where the interpolant has good approximation properties, whether or not supplemental terms are included; hence, we omit them for brevity. The basic function, shape parameters, and center locations have major effects on the nature of the linear system and the approximation properties of the resulting interpolant [10, 16, 31] . Throughout this work we focus on the multiquadric basic function φ(r) = √ 1 + r 2 . Multiquadrics are popular because they give high-order or spectral convergence [16, 26] and are observed to work very efficiently, in terms of the number of centers required, for both interpolation and differential equations [12, 17] . The main disadvantage of multiquadrics, compared to other choices of φ that have compact support or rapid decay, is that they lead to full matrices to solve for the coefficients w j in (1.2). For equivalent resolving power, the multiquadric matrices can often be far smaller than in the sparse cases, and the most efficient route to achieve a desired interpolation accuracy is far from clear, even when direct linear algebra is applied. When N is required to be large, however, the O(N 3 ) requirements of a direct solution of (1.2) are unacceptable, and one must turn to iterative methods. For these to be successful, one requires two key elements: a method to evaluate the RBF sum (1.1) rapidly at the interpolation nodes, and a good preconditioner. Fast evaluation strategies for RBFs have a substantial history [2, 4, 5, 15, 22, 23, 14] , but to the best of our knowledge, all methods presented to date have been demonstrated only for the case of constant shape parameter, j ≡ . However, allowing shape parameters to vary, often over orders of magnitude, is important to achieve optimal interpolation accuracy at a particular set of centers. In this work we modify the approach of Krasny and Wang [22] , who applied a treecode summation based on particle-cluster interactions to multiquadric RBF sums. We show that a dual approach of cluster-particle interactions, grouping together evaluation points rather than basis functions (centers), makes the use of varying shape parameters simple. We also show how to apply the treecode summation method within an adaptive strategy for selecting RBF centers and shape parameters described by Driscoll and Heryudono [10, 18] . The adaptive iteration pairs well with the fast summation since it constructs centers and nodes within a geometric quadtree/octree structure that can be used for the treecode as well. We also describe a two-level restricted additive Schwarz preconditioner based on domain decomposition, following Cai and Sarkis [8] and Beatson, Light, and Billings [6]. The effectiveness of the fast summation and preconditioned GMRES for adaptive interpolation are all demonstrated in numerical experiments. A new treecode for RBF summation. In this section we describe an asymptotically fast method for computing the RBF sum (1.1) at a set of nodes x i , i = 1, . . . , M. In the application to Krylov iterations such as GMRES for the linear system (1.2), we have M = N , and the node set is equal to the center set, but our description does not make this assumption. Use of far-field expansions. Krasny and Wang [22] proposed a treecode method for this task. The treecode builds a quadtree/octree structure T on the center set. Each node of T is associated with a box or cell in R d . The root of T contains all the center points y j . The 2 d children of a node are defined by bisection of the node's cell in each dimension, and each leaf of T contains no more than N 0 centers, where N 0 = 200 is a common choice. Assuming a uniform distribution of the N centers, the height of the tree is O(log N ) on average. Krasny and Wang's method evaluates (1.1) sequentially at each evaluation node x i by applying far-field Taylor expansions to cells containing centers that are "well-Downloaded 06/28/13 to Redistribution subject to SIAM license or copyright; see
doi:10.1137/110836225 fatcat:w5wntziyzfd5fa2loyujgeqpuu