Non-linear algorithms for least squares measurement problems

Research project
This project aims at combining linear estimation theory with non-linear optimization theory to solve non-linear estimation problems found in Photogrammetry and Space Physics.

Many measurement problems of real-world phenomena are solved using linear least squares methods. However, real-world problems are often non-linear, and linear estimation methods generally lack strong convergence properties, something necessary to obtain a solution. On the other hand, non-linear optimization theory, whose methods do guarantee a solution, usually ignore the statistical properties of the problem, something necessary to estimate the quality of the obtained solution. By combining theories from both fields, methods with superior convergence and statistical properties can be constructed. Two application examples are position measurement in images used in photogrammetry and measurements of current sheets in magnetospherical data.

Many measurement problems of real-world phenomena are solved using linear least squares methods. If the problem is linear and the measurement errors are normally distributed with known covariance, linear estimation theory suggests how to estimate values from observations. Furthermore, error bounds on the estimates are easily derived. However, many measurement problems are non-linear and must be solved by iterative algorithms. Although first-order error bounds are easily derived, these may not be sufficient near singularities or strong non-linearities. Furthermore, there is the question of whether the algorithm converges or not.

This problem domain is at the intersection of \emph{non-linear} optimization and \emph{linear} estimation theory. However, linear estimation theory applied to non-linear problems usually has a simplistic ``linearize and iterate''-approach that does not guarantee convergence. On the other hand, non-linear optimization methods usually focus on guaranteeing convergence, but do not exploit the statistical properties of the problem. In both cases, the two techniques are not used to their full combined potential.

Two research areas that would benefit from algorithms with a unified treatment of non-linear optimization and linear estimation theory are Photogrammetry and Space Physics.