Adaptive Sparse Approximations of Scattered Data

In this paper, two adaptive approximations, i.e., sparse residual tree (SRT) and sparse residual forest (SRF), are proposed for multivariate scattered data. SRT not only leads to sparse and stable approximations in areas where the data is sufficient or redundant, but also points out the possible local regions where the data is inadequate. And SRF is a combination of SRT predictors to improve the approximation accuracy and stability according to the error characteristics of SRTs. The hierarchical parallel SRT algorithm is based on both tree decomposition and adaptive radial basis function (RBF) explorations, whereby for each child a sparse and proper RBF refinement is added to the approximation by minimizing the norm of the residual inherited from its parent. The convergence results are established for both SRTs and SRFs. The worst case time complexity of SRTs is 𝒪(Nlog _2N) for the initial work and 𝒪(log _2N) for each prediction, meanwhile, the worst case storage requirement is 𝒪(Nlog _2N) , where the N data points can be arbitrary distributed. Numerical experiments are performed for several illustrative examples.