Asynchronous progressive iterative approximation method for least-squares fitting

TL;DR

This paper introduces an accelerated asynchronous LSPIA method using Chebyshev semi-iterative schemes and adaptive step sizes, significantly improving convergence speed for large-scale data fitting tasks.

Contribution

It proposes an innovative asynchronous LSPIA algorithm with Chebyshev-based adaptive step sizes, enhancing convergence speed over previous methods.

Findings

ALSPIA converges faster than original LSPIA.

The method is effective for both singular and nonsingular data fitting.

Numerical examples confirm the algorithm's feasibility and efficiency.

Abstract

For large-scale data fitting, the least-squares progressive-iterative approximation (LSPIA) methods were proposed by Lin et al. (SIAM Journal on Scientific Computing, 2013, 35(6):A3052-A3068) and Deng et al. (Computer-Aided Design, 2014, 47:32-44), where the constant step sizes were used. In this work, we further accelerate the LSPIA method in the sense of a Chebyshev semi-iterative scheme and present an asynchronous LSPIA (ALSPIA) method to fit data points. The control points in ALSPIA are updated by utilizing an extrapolated variant and an adaptive step size is chosen according to the roots of Chebyshev polynomials. Our convergence analysis reveals that ALSPIA is faster than the original LSPIA method in both cases of singular and nonsingular least-squares fittings. Numerical examples show that the proposed algorithm is feasible and effective.