A POTRA-TYPE METHOD WITH INVERSE OPERATOR APPROXIMATION FOR NONLINEAR LEAST SQUARES PROBLEMS

Анотація

This paper introduces difference method for the numerical solution of nonlinear least squares problems. Classical algorithms for nonlinear least squares, such as the Gauss-Newton method, are often hindered by the need to solve a dense linear system of normal equations at each iteration, a step that becomes a significant computational bottleneck for large-scale problems. To overcome this limitation, the proposed algorithm synergistically combines two powerful techniques. First, it employs a Potra-type iterative update, which achieves a convergence rate of 1.839... using only first-order divided differences, making the method applicable even to problems with non-differentiable operators. Second, to eliminate the matrix inversion bottleneck, the algorithm integrates a successive approximation of the inverse operator, replacing the costly linear system solve with more efficient matrix-matrix multiplications. We provide a local convergence analysis for this new method under standard Lipschitz conditions, formally establishing sufficient conditions for its convergence and confirming its high-order properties. The practical performance of the algorithm is then evaluated through numerical experiments on a suite of standard nonlinear least squares test problems of different types and complexity. A comparative analysis against a baseline Secant-type method that utilizes inverse operator approximation. The results empirically validate our central hypothesis - while the proposed algorithm may require slightly more iterations, it consistently achieves a lower total computation time by reducing the cost per iteration. This advantage is particularly pronounced for larger-scale problems, highlighting the method's potential as a robust and efficient tool for computationally demanding nonlinear least squares challenges.