Seminar on Numerical Analysis

How to express the solution of an ODE as a linear system (in a suitable algebra) and exploit it for fast computation.

by Dr Stefano Pozza (Charles University)

Aula Mancini (Scuola Normale Superiore)

Aula Mancini

Scuola Normale Superiore dei Cavalieri, 7, 56126 Pisa PI

The solution of systems of non-autonomous linear ordinary differential equations is crucial in various applications, such as nuclear magnetic resonance spectroscopy. We introduced a new solution expression in terms of a generalization of the Volterra composition. Such an expression is linear in a particular algebraic structure of distributions, which can be mapped onto a subalgebra of infinite matrices.
It is possible to exploit the new expression to devise fast numerical methods for linear non-autonomous ODEs. As a first example, we present a new method for the operator solution of the generalized Rosen-Zener model, a system of linear non-autonomous ODEs from quantum mechanics. The new method’s computing time scales linearly with the model’s size in the numerical experiments.