Linearly convergent nonlinear conjugate gradient methods for a parameter identification problems

TL;DR

This paper develops and analyzes linearly convergent nonlinear conjugate gradient methods for parameter estimation in systems governed by nonlinear differential equations, establishing theoretical convergence rates and supporting them with numerical experiments.

Contribution

It introduces new linear convergence results for nonlinear conjugate gradient methods applied to non-convex inverse problems with constraints, under broad conditions.

Findings

Established linear convergence rates for the methods.

Proved convergence under Lipschitz gradient conditions.

Validated results with numerical experiments on nonlinear models.

Abstract

This paper presents a general description of a parameter estimation inverse problem for systems governed by nonlinear differential equations. The inverse problem is presented using optimal control tools with state constraints, where the minimization process is based on a first-order optimization technique such as adaptive monotony-backtracking steepest descent technique and nonlinear conjugate gradient methods satisfying strong Wolfe conditions. Global convergence theory of both methods is rigorously established where new linear convergence rates have been reported. Indeed, for the nonlinear non-convex optimization we show that under the Lipschitz-continuous condition of the gradient of the objective function we have a linear convergence rate toward a stationary point. Furthermore, nonlinear conjugate gradient method has also been shown to be linearly convergent toward stationary points where the second derivative of the objective function is bounded. The convergence analysis in this work has been established in a general nonlinear non-convex optimization under constraints framework where the considered time-dependent model could whether be a system of coupled ordinary differential equations or partial differential equations. Numerical evidence on a selection of popular nonlinear models is presented to support the theoretical results. Nonlinear Conjugate gradient methods, Nonlinear Optimal control and Convergence analysis and Dynamical systems and Parameter estimation and Inverse problem