Eigenstructures of LOw-RAnk matrix polynomials

Matrix polynomials appear as discretizations of differential and differential-algebraic equations coming from an extreme design in engineering (stability of bridges; vibrations in rails for high-speed trains; control of robotic arms), modelling of complex systems in computer science (network analysis, neural networks, autoencoders), and physics (viscous parallel flow,resonant frequencies in acoustics).

The dimensions of these polynomials grow rapidly as the complexity of the applications increases, resulting in a high demand for efficient methods that can perform dimensionality reduction and at the same time preserve the properties of the underlying physical system.At the heart of many dimensionality reduction techniques lies a low-rank approximation or factorization of the underlying matrices but for matrix polynomials such problems are intricate.

The spaces of low-rank matrix polynomials have a complicated geometry, that makes the requirements of structure-preservation, e.g., preservation of symmetries or eigenvalues with multiplicities, very hard to fulfill.The purpose of the project is to provide a framework for qualitative and quantitative analysis of low-rank matrix polynomials and their eigenstructures.

The project exploits the geometrical nature of the involved problems and is strongly motivated by computational needs.

It is mostly in the field of matrix analysis and numerical linear algebra with applications in control theory and deep learning.