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Asynchronous progressive iterative approximation method for least-squares fitting

Nian-Ci Wu, Cheng-Zhi Liu

TL;DR

This work introduces ALSPIA, an asynchronous, Chebyshev semi-iterative acceleration of LSPIA for least-squares fitting of curves and surfaces. The method updates control points with adaptive steps ω_k in the form p^{(k+1)} = p^{(k)} + ω_k A^T ( q - A p^{(k)} ), and derives ω_k from Chebyshev polynomials to optimize convergence. The authors provide rigorous convergence results: sub-linear rates in the rank-deficient case using singular A^T A and linear rates in the full-rank case, with explicit bounds involving the extreme eigenvalues of A^T A (and tensor-product variants for surfaces). Numerical experiments show substantial iteration and CPU-time speed-ups over LSPIA, including cases where LSPIA fails to converge within a 10^4-iteration cap, highlighting ALSPIA's robustness and scalability for large-scale geometric fitting with blending bases.

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.

Asynchronous progressive iterative approximation method for least-squares fitting

TL;DR

This work introduces ALSPIA, an asynchronous, Chebyshev semi-iterative acceleration of LSPIA for least-squares fitting of curves and surfaces. The method updates control points with adaptive steps ω_k in the form p^{(k+1)} = p^{(k)} + ω_k A^T ( q - A p^{(k)} ), and derives ω_k from Chebyshev polynomials to optimize convergence. The authors provide rigorous convergence results: sub-linear rates in the rank-deficient case using singular A^T A and linear rates in the full-rank case, with explicit bounds involving the extreme eigenvalues of A^T A (and tensor-product variants for surfaces). Numerical experiments show substantial iteration and CPU-time speed-ups over LSPIA, including cases where LSPIA fails to converge within a 10^4-iteration cap, highlighting ALSPIA's robustness and scalability for large-scale geometric fitting with blending bases.

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.
Paper Structure (13 sections, 6 theorems, 69 equations, 7 figures, 4 tables)

This paper contains 13 sections, 6 theorems, 69 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The ALSPIA method
  3. The case of curves
  4. The case of surfaces
  5. Convergence analyses of ALSPIA
  6. The ALSPIA method for curve fitting
  7. Case 1: ${\bm A}^T{\bm A}$ is singular
  8. Case 2: ${\bm A}^T{\bm A}$ is nonsingular
  9. The ALSPIA method for surface fitting
  10. Numerical experiments
  11. Curve fitting
  12. Surface fitting
  13. Conclusions

Key Result

Theorem 3.1

Let $\left\{\mu_i(x)\right\}_{i=0}^n$ be a blending basis and ${\bm A}$ be the corresponding collocation matrix on the real increasing sequence $\left\{x_j \right\}_{j=0}^m$. Suppose that the step size sequence $\left\{\omega_\ell\right\}_{\ell=0}^{k-1}$ depends on the roots of the Chebyshev polynom for a fixed number of iterations $k$, where $v$ is the largest eigenvalue of $\bar{{\bm A}}={\bm A}

Figures (7)

  • Figure 1: Geometric interpretation of ALSPIA for curve fitting.
  • Figure 2: The initial data points, the cubic B-spline fitting curves, and the convergence behaviors of relative fitting error versus CPU time given by LSPIA and ALSPIA with $m=8000$ and $n=1000$ for Example \ref{['ex:ALSPIA+curve1']}.
  • Figure 3: The initial data points, the cubic B-spline fitting curves, and the convergence behaviors of relative fitting error versus CPU time given by LSPIA and ALSPIA with $m=8000$ and $n=1000$ for Example \ref{['ex:ALSPIA+curve2']}.
  • Figure 4: The initial data points, the cubic B-spline fitting curves, and the convergence behaviors of relative fitting error versus CPU time given by LSPIA and ALSPIA with $m=14600$ and $n=2000$ for Example \ref{['ex:ALSPIA+curve3']}.
  • Figure 5: The initial data points, the cubic B-spline fitting curves, and the convergence behaviors of relative fitting error versus CPU time given by LSPIA and ALSPIA with $m=18897$ and $n=3000$ for Example \ref{['ex:ALSPIA+curve4']}.
  • ...and 2 more figures

Theorems & Definitions (16)

  • Remark 3.1
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Corollary 3.1
  • Corollary 3.2
  • Example 4.1
  • Example 4.2
  • ...and 6 more