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<p>This paper studies sparse nonlinear least squares problems, where the Jacobian matrices are unavailable or expensive to compute, yet have some underlying sparse structures. We construct the Jacobian models by the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( \ell _1 \)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> minimization subject to a small number of interpolation constraints with interpolation points generated from some certain distributions, and propose a derivative-free Levenberg–Marquardt algorithm based on such Jacobian models. It is proved that the Jacobian models are probabilistically first-order accurate and the algorithm converges globally almost surely. Numerical experiments indicate the efficiency of the proposed algorithm for sparse nonlinear least squares problems.</p>

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A derivative-free Levenberg–Marquardt method for sparse nonlinear least squares problems

  • 期刊论文

  • 发表日期：

    2026年4月2日

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  • Yuchen Feng,
  • Chuanlong Wang,
  • Jinyan Fan

摘要

This paper studies sparse nonlinear least squares problems, where the Jacobian matrices are unavailable or expensive to compute, yet have some underlying sparse structures. We construct the Jacobian models by the \( \ell _1 \) ℓ 1 minimization subject to a small number of interpolation constraints with interpolation points generated from some certain distributions, and propose a derivative-free Levenberg–Marquardt algorithm based on such Jacobian models. It is proved that the Jacobian models are probabilistically first-order accurate and the algorithm converges globally almost surely. Numerical experiments indicate the efficiency of the proposed algorithm for sparse nonlinear least squares problems.