The aviation sector continues to grow rapidly, with projected revenues approaching 1 trillion USD by 2025. This growth raises significant concerns about CO2 emissions and the corresponding fuel consumption. In Europe alone, approximately 1.3 million people travel by air each day, contributing to nearly 240,000 tonnes of CO₂ emissions daily. Addressing these environmental challenges, as well as reducing the airlines' costs, necessitates innovative methods for reducing fuel usage. Here we focus on the implicit reduction by optimizing the routes that aircraft take, such that headwind can be avoided and tailwinds exploited.
This is routinely done in flight planning within discrete airway networks. For exploiting the wind conditions more fully, airspaces turn towards free flight, allowing arbitrary continuous trajectories.
Numerical studies suggest a potential of 2-3% fuel savings from moving to free flight.
This leads to the classic Zermelo navigation problem. The challenge here is the demand for global optima.
We present an algorithm capable of finding a continuous globally optimal trajectory for an aircraft in a stationary wind field. The algorithm solves a Hamilton-Jacobi-Bellman (HJB) equation associated with the flight trajectory optimization problem, employing linear finite elements and effective parallelization. Furthermore, we demonstrate a linear order of convergence of the discretization by giving an explicit bound for the error estimate of arrival time. In addition, this holds in the presence of singularities for HJB PDE, i.e., cut loci of the system. Finally, we discuss global optimality guarantees, and by combination with pre-existing optimal control approaches, we can find a globally optimal path within the desired accuracy.
The single-trajectory optimization also forms a building block of collaborative routing, a multi-player problem of airlines competing for capacity-bound airspace sectors in planning their flight trajectories. We report first results on a pricing strategy for solving this problem.
References
R. Borndörfer, F. Danecker, M. Weiser. A Discrete-Continuous Algorithm for Globally Optimal Free Flight Trajectory Optimization. In 22nd Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2022). 2022.
R. Borndörfer, A. Jocas, M. Weiser. An Eikonal Approach for Globally Optimal Free Flight Trajectories. arXiv:2603.11830, 2026.

Estimating the time of death is crucial in forensic investigations. One of the most reliable and widely accepted methods is to use the cooling of the corpse to infer the time of death. The prevailing approach is based on a purely phenomenological model of corpse cooling, however this leads to a lack of applicability to non-standard situations and difficulties in integrating multiple measurements.\\
Newer models are mechanistic and based on the computational simulation of cooling of the corpse by solving the heat equation on CT-generated personalised anatomies. This approach promises to provide higher accuracy as well as more reliable uncertainty quantification [2]. Moreover, this approach has the advantage that different situations can be taken into account by adjusting the physical parameters. These include, for example, the thermal properties of the different tissues, the environmental temperature and the exact point of measurement [1]. However, the exact values are not known. To address this, we formulate the time of death estimation as a Bayesian inverse problem [3]. Solving the inverse problem requires the repeated solution of the mechanistic model, using the finite element simulation to do so is computationally too expensive for real life applications.\\

We construct a projection-based reduced order model to replace the expensive finite element simulation. This is done by the method of snapshots and proper orthogonal decomposition. As the temperature at the point of measurement is the important quantity for estimating the time of death, we consider the point evaluation to be our output of interest. Then, an adjoint problem can be defined to derive an a posteriori error estimator for the output of interest. This error estimator can be used in a greedy algorithm to build the reduced model, such that the construction is problem specific and a certain accuracy can be guaranteed for all possible parameter values. Still, the reduced model makes an error compared to the finite element simulation. We investigate the covariance of this error and incorporate it in the formulation of the inverse problem.\\

References:

[1] J. S. Subramaniam, M. Hubig, H. Muggenthaler, S. Schenkl, J. Ullrich, G. Pourtier, M. Weiser, and G. Mall. Sensitivity of temperature-based time since death estimation on measurement location. International Journal of Legal Medicine, 137:1815 – 1837, 2023.

[2] M. Weiser, B. Erdmann, S. Schenkl, H. Muggenthaler, M. Hubig, G. Mall, and S. Zachow. Uncertainty in temperature-based determination of time of death. Heat and Mass Transfer, 54(9):2815– 2826, 2018.

[3] M. Weiser, Y. Freytag, B. Erdmann, M. Hubig, and G. Mall. Optimal design of experiments for
estimating the time of death in forensic medicine. Inverse Problems, 34(12), 2018.

In the recent years the problem of finite element approximation and analysis of elliptic problems where the forcing function is supported on a lower dimensional manifold, like a point or a curve in three dimensions Euclidian space have received a certain attention. There are many interesting and insightful results in this direction. However, the known results are not sufficient for obtaining optimal error estimates for optimal control problem with controls acting on curves in three dimensions. To obtain optimal (or nearly optimal) error estimates, we establish new weighted error estimates for the elliptic problems and applied them to the optimal control problems. In some sense, the established results generalized the technique of pointwise finite element error estimates.