Nesterov's Accelerated Gradient Method for Nonlinear Ill-Posed Problems with a Locally Convex Residual Functional
It provides a theoretical convergence guarantee for a fast optimization method in the context of ill-posed inverse problems, which is an incremental advance for researchers in inverse problems and optimization.
The paper analyzes Nesterov's Accelerated Gradient method for nonlinear ill-posed problems under a local convexity assumption, proving convergence and demonstrating effectiveness on numerical examples including a nonlinear diagonal operator and an auto-convolution inverse problem.
In this paper, we consider Nesterov's Accelerated Gradient method for solving Nonlinear Inverse and Ill-Posed Problems. Known to be a fast gradient-based iterative method for solving well-posed convex optimization problems, this method also leads to promising results for ill-posed problems. Here, we provide a convergence analysis for ill-posed problems of this method based on the assumption of a locally convex residual functional. Furthermore, we demonstrate the usefulness of the method on a number of numerical examples based on a nonlinear diagonal operator and on an inverse problem in auto-convolution.