Spectral Methods on Chebyshev Grids for Optimal Control and Differential Game Problems in Sports
Along with finite differences and finite elements, spectral methods are one of the three main technologies for solving partial differential equations and optimization problems on computers. If one wants to solve an optimization problem or a differential equation, and if the data defining the problem are smooth, then spectral methods are usually the best tool. They can often achieve ten digits of accuracy where a finite difference or finite element method would get two or three. At lower accuracies, they demand less computer memory than the alternatives.
The textbook "Spectral Methods in Matlab" gives an excellent practical introduction to the topic with downloadable Matlab Code. The relatec Matlab Chebfun Package provides implementations for the mathematical basis including Chebyshev expansions, barycentric interpolation, recursive zerofinding, and automatic differentiation.
Sound mathematical understanding is an important prerequisite for this project. Knowledge from lectures such as "Digital Signal Processing" is beneficial.
Primarily, practical problems from sport informatics, such as
- Parameter Estimation Problems
- Optimal design of cycling ergometer tests for the calibration of physiological odels
- Optimal Control Problems
- Minimum-time pacing strategies and steering for road cycling on 3-D tracks
- Minimum-time pacing strategies for running and swimming
- The Goddard rocket problem
- Differential Game Problems
- Target guarding problem
- Homicidal chauffeur problem
- The game of two cars
will be solved using Matlab's fmincon and bvp4c functions.