Substitution in Lie-Butcher Series (Ludwig Rahm, University of Bergen)

14.01.2025 14:00

Recent developments in numerical integration on homogeneous spaces has been understood through the formalism of the Munthe-Kaas--Wright (MKW) Hopf algebra, of planar rooted trees. In this talk we will focus on the substitution problem for Lie-Butcher series, which has applications to backwards error analysis and modified equations. One can see the substitution problem in Lie-Butcher series as the problem of describing the dual map of post-Lie algebra morphisms from the free post-Lie algebra into itself. Given an element $x$ in the free post-Lie algebra, which are the morphisms $f_y$ and elements $z$ such that $\langle f_y(z),x\rangle \neq 0$? Here we have identified $f_y(z)$ with its dual element using the dual basis. In order to solve this problem, we employ a construction by Foissy that makes a bialgebra from any operad. We describe the post-Lie operad in terms of rooted planar trees, and then apply Foissy's construction. We show how the coproduct of the resulting bialgebra can be used to describe the dual of post-Lie algebra morphisms, and we show how to compute the coproduct by using combinatorial operations on rooted planar trees.

Lieu

Bâtiment: Conseil Général 7-9

Room 1-05, Séminaire d'analyse numérique

entrée libre