Speaker
Description
We study the numerical solution of non-autonomous linear ODEs of the form
$$ \frac{d}{dt} \tilde{u}(t) = \tilde{A}(t)\tilde{u}(t), \quad \tilde{u}(a) = v,$$
where $\tilde{A}(t) \in \mathbb{C}^{N \times N}$ is analytic and often takes the form
$$ \tilde{A}(t) = \sum_{j=1}^k A_j f_j(t),$$
with large, sparse constant matrices $A_j$ and scalar analytic functions $f_j(t)$. Such equations commonly arise in quantum chemistry, particularly in spin dynamics.
In general, no closed-form solution exists, so we proposed a spectral method—called the **$\star$-approach**—which expands the solution in Legendre polynomials and approximates the coefficients by solving a structured matrix equation:
$$ X - F_1 X A_1^T - \dots - F_k X A_k^T = \phi v^T,$$
where the matrices $F_j$ encode the functions $f_j(t)$ in a structured algebra. This equation has favorable properties: (i) banded $F_j$, (ii) Kronecker-structured $A_j$, and (iii) a rank-1 right-hand side. These allow efficient iterative and low-rank methods.
We implemented this strategy in the so-called $\star$-method, shown to be highly efficient for the generalized Rosen-Zener model, with linear scaling in system size and strong performance on large time intervals (collaboration with Christian Bonhomme - Sorbonne University, and Niel Van Buggenhout - Universidad Carlos III). Preliminary results also show promise in extending the method to fractional ODEs, where similar matrix equations arise and exhibit low-rank structure (collaboration with Fabio Durastante — University of Pisa, and Pierre-Louis Giscard — ULCO).
