A new study, carried out by researchers from ICMAT, the Universidad Nacional del Sur (Argentina), and the Friedrich-Alexander University Erlangen-Nürnberg (Germany), develops mathematical techniques to find optimal trajectories with given initial and final conditions. They propose an algorithm based on discrete variational methods, which preserve key properties of the physical system and allow for efficient implementation using parallel computing. The tool is being integrated into real navigation projects using meteorological and oceanographic data.

The new mathematical tools are being used to calculate real ship routes that reduce fuel consumption.

Ágata Timón (ICMAT)

In 1931, the German mathematician Ernst Zermelo asked how to navigate a ship from an initial point to a final destination in the shortest possible time, taking into account the influence of an external velocity field, such as wind or current. In most cases, it is impossible to find an exact solution to this classical question, so the goal is to find good numerical approximations. Moreover, if the initial and final conditions are sufficiently far apart, the problem becomes much more complex. Recently, an international group of researchers has obtained results that may facilitate finding solutions to these problems.

The theoretical methods developed in the new study, which are now published in the journal Journal of Computational Dynamics, can be applied to many other questions. For example, “finding navigation routes that use the least possible fuel, with potential applications in maritime navigation, astrodynamics, robotics…,” explains David Martín de Diego, CSIC researcher at ICMAT and author of the study, along with Sebastián Ferraro (Universidad Nacional del Sur and CONICET, Argentina) and Rodrigo Sato Martín de Almagro (Institute of Applied Dynamics at Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany). Their mathematical tools are being used in an interdisciplinary project to calculate real ship routes using oceanographic and meteorological data, significantly reducing the fuel used.

In addition, their algorithm can be implemented using parallel computing—where multiple computations are executed simultaneously—which improves its performance, meaning it is faster and uses fewer resources. “This is important because the main processors of a computer (multi-core CPUs), and especially graphics cards (GPUs), are designed for parallel computing,” says Martín de Diego.

In search of maxima and minima

From a mathematical point of view, these types of questions fall within the realm of variational problems, a branch of calculus that seeks to find the trajectory or configuration that minimizes (or maximizes) a certain quantity—in this case, travel time or fuel consumption. The formal approach involves defining a functional, that is, a function that assigns a real number to each trajectory, which can be, for example, the total time traveled along a given route. Solving the problem consists of finding the trajectory assigned the minimum (or maximum) value, considering constraints imposed by a velocity field, obstacles, or physical conditions of the environment.

Since solving these problems exactly is often unfeasible, numerical methods must be used to obtain approximate solutions. “A fundamental aspect in mathematics is to ensure that the solutions obtained by applying these techniques approximate the sought solution,” states Martín de Diego.

“There are numerous numerical methods that could be applied to a given problem, but some adapt better because they preserve certain properties of the original system,” says the mathematician. In particular, this work employs discrete variational methods, which represent trajectories through points separated by small time intervals. Instead of directly discretizing the differential equations that describe the system—which can lead to the loss of key properties, such as conservation of energy or momentum—these methods discretize the underlying variational principle, ensuring that the approximate solutions retain important geometric and physical characteristics of the original system.