Piece-wise moving least squares approximation

The standard moving least squares (MLS) method might have an expensive computational cost when the number of test points and the dimension of the approximation space are large. To reduce the computational cost, this paper proposes a piece-wise moving least squares approximation method (PMLS) for scattered data approximation. We further apply the PMLS method to solve time-dependent partial differential equations (PDE) numerically. It is proven that the PMLS method is an optimal design with certain localized information. Numerical experiments are presented to demonstrate the efficiency and accuracy of the PMLS method in comparison with the standard MLS method in terms of accuracy and efficiency.