Asynchronous progressive iterative approximation method for least squares fitting.

For large data fitting, the least squares progressive iterative approximation (LSPIA) methods have been proposed by Lin and Zhang (2013) and Deng and Lin (2014) , in which a constant step size is used. In this paper, we further accelerate the LSPIA method in terms of a Chebyshev semi-iterative scheme and present an asynchronous LSPIA (denoted by ALSPIA) method. The control points in ALSPIA are updated by using an extrapolated variant in which an adaptive step size is chosen according to the roots of Chebyshev polynomial. Our convergence analysis shows that ALSPIA is faster than the original LSPIA method in both singular and non-singular least squares fitting cases. Numerical examples show that the proposed algorithm is feasible and effective. • We propose a new variant of LSPIA (denoted by ALSPIA) by updating the control points with an adaptive step size. When the step size is the same constant, our method automatically reduces to the original LSPIA method. • We choose the step size according to the roots of Chebyshev polynomials. We prove that the ALSPIA method is convergent when the collocation matrix is rank deficient and of full-column rank. Convergence analysis reveals that ALSPIA has a faster convergence rate than LSPIA. • Numerical experiments show that the ALSPIA method is very robust and exhibits similar convergence behaviors with various selection orders for the step size, and outperforms classical LSPIA-type methods when fitting missing data, noisy data, and datasets of very large size. [ABSTRACT FROM AUTHOR]