# Research

My main research interest is the numerical treatment of structured matrices; in particular, I am interested in the case where low-rank structures appear. This phenomenon is encountered in several situations.

## Rank structures

Given some matrix $A \in \mathbb{C}^{m \times n}$, how can we find the best tall and skinny $U,V$ such that $\|A – UV^T\|_2$ is small as possible? Solving this problem is the task of low-rank approximation.

This problem becomes even more interesting when $A$ is a block of a larger matrix, which gives rise to hierarchical rank-structures.

My research focuses on studying the interplay of such structures with the solution of matrix equations and matrix functions.

## Matrix polynomials and polynomial rootfinding

Given a polynomial $p(x) = \sum_{j = 0}^n p_j x^j$, what is the “best” way to numerically compute its roots? The answers depends on several facts, for instance the basis in which the polynomial is expressed.

The same question can be posed when the coefficients $p_j$ are square matrices, and we aim at finding the values of $x$ for which $p(x)$ is singular; this is known as a polynomial eigenvalue problem.

In both frameworks, I study how to develop fast iterative methods, and structure-preserving linearizations that preserve any symmetry that might be present in the original polynomial. Such linearizations are often rank-structured, and this is a natural field for using structured eigenvalue solvers (such core-chasing methods).

## Toeplitz-like matrices

A Toeplitz matrix is a matrix with constant diagonal entries. Bi-infinite matrices with this structure form an algebra, but semi-infinite ones do not. Completing the set of semi-infinite Toeplitz matrices to be an algebra yields (under appropriate hypotheses on the topology of interest) the set of semi-infinite Toeplitz matrices plus compact corrections; the latter can be approximated by low-rank matrices, which provides a computationally friendly setting for dealing with infinite matrices.

Such matrices represent convolution-like operators, and find applications in the study of Markovian queues, in particular quasi-Birth-Death stochastic processes (QBD).