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    Area of Science:

    • Numerical Analysis
    • Optimization
    • Scientific Computing

    Background:

    • Nonlinear least squares problems are common in various scientific and engineering fields.
    • Separating variables into linear and nonlinear components can simplify complex problems.
    • Existing methods often struggle with efficiency and stability for these problems.

    Purpose of the Study:

    • To compare the performance of four different algorithms for solving nonlinear least squares problems with separated variables.
    • To evaluate the impact of using full versus simplified Jacobian matrices in variable projection algorithms.
    • To assess the effectiveness of combining variable projection with different optimization methods.

    Main Methods:

    • Variable Projection (VP) algorithm with a full Jacobian matrix (Golub and Pereyra).
    • VP algorithms with simplified Jacobian matrices (Kaufman and Ruano et al.).
    • An algorithm utilizing only the gradient of the reduced problem.
    • Monte Carlo simulations for performance comparison.

    Main Results:

    • The simplified Jacobian proposed by Ruano et al. can hinder convergence for the VP algorithm.
    • The gradient-only algorithm shows moderate performance.
    • The VP algorithm with the full Jacobian matrix demonstrates greater stability than with Kaufman's simplified Jacobian.
    • Combining the VP algorithm with the Levenberg-Marquardt method is more effective than with the Gauss-Newton method.

    Conclusions:

    • The choice of Jacobian matrix significantly impacts the performance and stability of VP algorithms.
    • The full Jacobian approach is recommended for improved stability in VP algorithms.
    • Levenberg-Marquardt optimization is a more effective choice than Gauss-Newton when combined with VP for these problems.