Snelle en robuuste methodes voor overgedetermineerde stelsels multivariate veeltermvergelijkingen met ruis

Linear models have long been the cornerstone of engineering and applied mathematics due to their simplicity and tractability. However, they fall short in capturing the nonlinear interactions that are inherent to many real-world problems. Nonlinear models, while expressive, are typically difficult to fully analyze and solve. Polynomial systems offer a middle ground: they are expressive enough to capture complex interactions, yet structured enough to admit exact solution methods, grounded in algebraic geometry, and numerical linear and multilinear algebra. In signal processing and related fields, the data is often only known up to measurement noise. Taking additional measurements can help improve the estimate, but this requires the solution of noisy overdetermined systems of multivariate polynomial equations. Extracting meaningful solutions from such systems requires methods that are both robust to noise and computationally efficient. Developing and analyzing these methods is the subject of this thesis. The first half of this thesis focuses on tailoring two resultant-based methods to noisy overdetermined systems. The Macaulay-based method constructs a structured matrix whose null space is used to form a tensor, from which the roots are recovered through a low-rank decomposition. Each step is designed to suppress noise, leveraging numerical techniques from linear and multilinear algebra. The second method builds on Bézoutians, which traditionally apply only to square systems. To use them in an noisy overdetermined setting, we propose a joint decomposition of multiple Bézoutians derived from square subsystems. The methods are evaluated on challenging signal processing problems, namely an underdetermined blind source separation problem with constant modulus and a blind multi-source localization problem. While these methods are effective, their computational cost can become prohibitive as the problem size grows. Therefore, the second half of this thesis focuses on accelerating the Macaulay-based method. The main contribution is an approach that constructs the Macaulay null space directly, without forming the Macaulay matrix itself. The resulting method, named MINI, builds the null space at increasing degrees by shifting and intersecting previously computed null spaces. This exploits the observation that the null space dimension grows much more slowly with the number of variables than the row space, enabling significant computational savings. Theoretical analysis and experimental validation demonstrate that MINI achieves substantial speedups over existing methods. Additionally, the classical concept of elimination is briefly revisited: when applied repeatedly and carefully, it can further reduce computation time while maintaining acceptable accuracy.