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Just as the damped Newton method for the numerical solution of nonlinear algebraic problems can be interpreted as a forward Euler timestepping on the Newton flow equations, the damped Gauß–Newton method for nonlinear least squares problems is equivalent to forward Euler timestepping on the corresponding Gauß–Newton flow equations. We highlight the advantages of the Gauß–Newton flow and the Gauß–Newton method from a statistical and a numerical perspective in comparison with the Newton method, steepest descent, and the Levenberg–Marquardt method, which are respectively equivalent to Newton flow forward Euler, gradient flow forward Euler, and gradient flow backward Euler. We finally show an unconditional descent property for a generalized Gauß–Newton flow, which is linked to Krylov–Gauß–Newton methods for large-scale nonlinear least squares problems. We provide numerical results for large-scale problems: An academic generalized Rosenbrock function and a real-world bundle adjustment problem from 3D reconstruction based on 2D images.

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Accurate state-estimation is a vital prerequisite for fast feedback control methods such as Nonlinear Model Predictive Control (NMPC). For efficient process control, it is of great importance that the estimation process is carried out as fast as possible to provide the feedback mechanism with fresh information and enable fast reactions in case of any disturbances. We discuss how Multi-Level Iterations (MLI), known from NMPC, can be applied to the Moving Horizon Estimation (MHE) method for estimating the states and parameters of a system described by a Differential Algebraic Equation model. A challenging field of application for the proposed MLI-MHE method are electric microgrids. These push current control approaches to their limits due to the rising penetration of volatile renewable energy sources and the fast electrical system dynamics. We investigate the closed-loop control performance of the proposed MLI-MHE algorithm in combination with an NMPC controller for a realistic sized microgrid as a numerical example.