On the geometric and algebraic properties of stochastic backward error analysis

Abstract: The exotic aromatic extension of Butcher series allowed the creation and study of integrators for the high-order sampling of the invariant measure of ergodic stochastic differential equations. In particular, the concept of backward error analysis, a key concept in geometric numerical integration, seemed to generalise in a certain sense for the study of stochastic dynamics using exotic aromatic B-series, though there was no general result beyond order 3. In this talk, we will detail the concept of backward error analysis, quickly present recent results on the Hopf algebra structures related to the composition and substitution laws of exotic aromatic series, and see that stochastic backward error analysis writes naturally and at any order with exotic aromatic B-series. Then, we shall show that the exotic aromatic formalism is precisely the right formalism for the formulation of backward error analysis, thanks to a universal geometric property of orthogonal equivariance. This is joint work with Eugen Bronasco (University of Geneva) and Hans Munthe-Kaas (University of Bergen and University of Tromsø).

machine learningcommutative algebraalgebraic geometryalgebraic topologycombinatoricscategory theoryoperator algebrasrings and algebrasrepresentation theory

Audience: researchers in the topic

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Algebraic and Combinatorial Perspectives in the Mathematical Sciences

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