2022/03/08 by M. H. A. Piro, J. S. Bell, Piro, M. H. A. +7 · 1 citation
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Advanced Measurement and Metrology Techniques #Applied mathematics #Calibration and Measurement Techniques #Computer science #Convergence (economics) #FOS: Mathematics #FOS: Physical sciences #Function (biology) #Inverse #Inverse problem #Jacobian matrix and determinant #Line search #Materials Science (cond-mat.mtrl-sci) #Mathematical optimization #Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Path (computing) #Rank (graph theory) #Trust region #cond-mat.mtrl-sci #cs.NA #math.NA
paper · pdf · doi:10.48550/arxiv.2203.04138
published in arXiv (Cornell University) (Cornell University)
arxiv created 2022/03/08 · openalex publication_date 2022/03/08 · arxiv updated 2022/03/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An effective numerical method is presented for optimizing model parameters that can be applied to any type of system of non-linear equations and any number of data-points, which does not require explicit formulation of the objective function or its partial derivatives. The numerics are reduced to solving a non-linear least squares problem, which uses the Levenberg-Marquardt algorithm and the Jacobian is approximated by applying rank-one updates using Broyden's method. An advantage of this methodology over conventional approaches is that the partial derivatives of the objective function do not have to be analytically calculated. For instance, there may be situations where one cannot formulate the partial derivatives, such as cases involving an objective function that itself contains a nested optimization problem. Moreover, a line search algorithm is also described that ensures that the Armijo conditions are satisfied and that convergence is assured, which makes the success of the approach insensitive to the initial estimates of the model parameters. The foregoing numerical methods are described with respect to the development of the Optima software to solve inverse problems, which are reduced to non-linear least squares problems. This computational approach has proven to be particularly useful at solving inverse problems of very complex physical models that cannot be optimized directly in a practical way.