Sequential Projected Newton method for regularization of nonlinear least squares problems

TL;DR

This paper introduces an efficient algorithm for regularizing nonlinear inverse problems using a projected Newton approach, incorporating the discrepancy principle and prior knowledge, demonstrated on X-ray imaging with reduced computational cost.

Contribution

The paper presents a novel projected Newton method for regularizing nonlinear least squares problems, enabling efficient solutions with inexact sub-problem solutions and improved computational performance.

Findings

Algorithm achieves similar reconstruction quality to state-of-the-art methods.

Significant reduction in computational time demonstrated.

Early iterations do not require high-accuracy solutions for progress.

Abstract

We develop a computationally efficient algorithm for the automatic regularization of nonlinear inverse problems based on the discrepancy principle. We formulate the problem as an equality constrained optimization problem, where the constraint is given by a least squares data fidelity term and expresses the discrepancy principle. The objective function is a convex regularization function that incorporates some prior knowledge, such as the total variation regularization function. Using the Jacobian matrix of the nonlinear forward model, we consider a sequence of quadratically constrained optimization problems that can all be solved using the Projected Newton method. We show that the solution of such a quadratically constrained sub-problem results in a descent direction for an exact merit function. This merit function can then be used to describe a formal line-search method. We also formulate a slightly more heuristic approach that simplifies the algorithm and allows for an inexact solution of the sequence of sub-problems. We illustrate the behavior of the algorithm using a number of numerical experiments, with Talbot-Lau X-ray phase contrast imaging as the main application. The numerical experiments confirm that the quadratically constrained sub-problems need not be solved with high accuracy in early iterations to make sufficient progress towards the solution. In addition, we show that the proposed method is able to produce reconstructions of similar quality compared to other state-of-the-art approaches with a significant reduction in computational time.