B-spline functions, based on ordinary differential operators, are widely used in applied mathematics, with applications in computer-aided design (CAD), image processing, and the numerical solution of partial differential equations (PDEs). Their compact support, multiresolution properties, and ability to interact exactly with continuous-domain operations such as derivatives make them especially useful in practice.

Recently, vector-valued B-splines defined using matrices of ordinary differential operators (MDOs) have been introduced for applications such as vector-valued inverse problems, curve reconstruction, and trajectory estimation. Although the theoretical existence of these splines has been established, their numerical implementation has not yet been fully developed. The objective of this project is to design and implement a Python package for two-dimensional vector spline functions and related computational tools. Through this project, students will gain experience in translating mathematical theory into numerical algorithms. The implementation will be validated by solving either a challenging synthetic problem or a real-world application.

Moreover, this project will also focus on vector-valued inverse problems regularized with a MDO. Given a set of observations of correlated vector-valued signals, this project aims at developing methods to identify or learn a suitable MDO for reconstruction of such data.