Numerical Algorithms as Dynamcal Systems - Structure Preservation, Convergence Theory, and Rediscretization

The focus of the project is to improve iterative algorithms in numerical analysis by linking them to particular systems of differential equations. Thirty years ago, it was realized that the QR algorithm for calculating eigenvalues can be considered as a time-T map of the Toda lattice. This opens up the possibility of bringing the qualitative methods of dynamical systems to bear on the algorithm, and to potentially speed up the algorithm by more efficient discretizations. The crucial aspect of the Toda lattice is that it is a continuous conjugation that preserves upper Hessenberg form and therefore the eigenvalues. In recent research, the PI has achieved similar results with the calculation of the singular value decomposition by developing a Lotka-Volterra system that preserves bidiagonal structures, which has led to advances in computing the SVD. This project investigates other dynamical systems preserving symplectic and Hamiltonian structure, which are pivotal in many areas of applications. The ultimate goal is to investigate the connection between their geometric structures and existing numerical algorithms, to establish a rigorous mathematical theory on dynamical behaviors, and to develop possible structure-preserving re-discretizations to improve robustness, speed, and accuracy of iterations in numerical analysis.

Structure-preserving dynamical systems are natural and ubiquitous. Conservation laws in the physical world and constrained mechanics in engineered systems are just two examples. Structure preservation is also imperative in computation because it makes possible more efficient algorithms, improves physical feasibility and interpretability, and is more robust in long-term behavior. The proposed research recasts numerical algorithms as differential systems that mimic the structure-preserving properties of the corresponding iterative schemes. Understanding the overall dynamics of the continuum system can shed light on the convergence properties of the related discrete counterparts, and can also contribute to the re-discretization of the continuum system into a new algorithm with better numerical properties. A wide range of applications stands to benefit from the study of properties and geometric structure of these systems, which essentially include all disciplines that entail structure preservation, including classical and quantum mechanics, model reduction, reversible systems, and molecular dynamics.