In this paper we consider large scale nonlinear least-squares problems for which function and gradient are evaluated with dynamic accuracy and propose a Levenberg-Marquardt method for solving such problems. More precisely, we consider the case in which the exact function to optimize is not available or its evaluation is computationally demanding, but ap- proximations of it are available at any prescribed accuracy level. The proposed method relies on a control of the accuracy level, and imposes an improvement of function approximations when the accuracy is detected to be too low to proceed with the optimization process. We prove global and local convergence and complexity of our procedure and show encouraging numerical results on test problems arising in data assimilation and machine learning.
Dipartimento di Ingegneria Industriale, Università di Firenze, viale G.B. Morgagni 40, 50134 Firenze. ENSEEIHT, INPT, rue Charles Camichel, B.P. 7122 31071, Toulouse Cedex 7, France. 06/2017