Ecalle's arborification-coarborification transforms and Connes-Kreimer Hopf algebra

Fauvet, Frédéric; Menous, Frédéric

doi:10.24033/asens.2315

Cited by 14 publications

(1 citation statement)

References 24 publications

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“…This work proposes a new application of the Butcher–Connes–Kreimer Hopf algebra [16, 31] to dispersive PDEs. It gives a new light on structures that have been used in various fields such as numerical analysis, renormalisation in quantum field theory, singular SPDEs and dynamical systems for classifying singularities via resurgent functions introduced by Jean Ecalle (see [34, 38]). This is another testimony of the universality of this structure and adds a new object to this landscape.…”

Section: Introductionmentioning

confidence: 99%

Resonance-based schemes for dispersive equations via decorated trees

Bruned

Schratz

2022

Forum of Mathematics, Pi

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We introduce a numerical framework for dispersive equations embedding their underlying resonance structure into the discretisation. This will allow us to resolve the nonlinear oscillations of the partial differential equation (PDE) and to approximate with high-order accuracy a large class of equations under lower regularity assumptions than classical techniques require. The key idea to control the nonlinear frequency interactions in the system up to arbitrary high order thereby lies in a tailored decorated tree formalism. Our algebraic structures are close to the ones developed for singular stochastic PDEs (SPDEs) with regularity structures. We adapt them to the context of dispersive PDEs by using a novel class of decorations which encode the dominant frequencies. The structure proposed in this article is new and gives a variant of the Butcher–Connes–Kreimer Hopf algebra on decorated trees. We observe a similar Birkhoff type factorisation as in SPDEs and perturbative quantum field theory. This factorisation allows us to single out oscillations and to optimise the local error by mapping it to the particular regularity of the solution. This use of the Birkhoff factorisation seems new in comparison to the literature. The field of singular SPDEs took advantage of numerical methods and renormalisation in perturbative quantum field theory by extending their structures via the adjunction of decorations and Taylor expansions. Now, through this work, numerical analysis is taking advantage of these extended structures and provides a new perspective on them.

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Section: Introductionmentioning

confidence: 99%