130 citations · 450 across the 22 of their papers we have counts for
23 papers
Lie Admissible Triple Algebras: The Connection Algebra of Symmetric Spaces
Hans Munthe-Kaas, Jonatan Stava
Associated to a symmetric space there is a canonical connection with zero torsion and parallel curvature. This connection acts as a binary operator on the vector space of smooth se…
The universal equivariance properties of exotic aromatic B-series
Adrien Laurent, Hans Munthe-Kaas
The exotic aromatic Butcher series were originally introduced for the calculation of order conditions for the high order numerical integration of ergodic stochastic differential eq…
Post-Lie Algebra Structure of Manifolds with Constant Curvature and Torsion
Erlend Grong, Hans Z. Munthe-Kaas, Jonatan Stava
For a general affine connection with parallel torsion and curvature, we show that a post-Lie algebra structure exists on its space of vector fields, generalizing previous results f…
The aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators
Adrien Laurent, Robert I. McLachlan, Hans Z. Munthe-Kaas +1
Aromatic B-series were introduced as an extension of standard Butcher-series for the study of volume-preserving integrators. It was proven with their help that the only volume-pres…
The universal pre-Lie-Rinehart algebras of aromatic trees
Gunnar Fløystad, Dominique Manchon, Hans Z. Munthe-Kaas
We organize colored aromatic trees into a pre-Lie-Rinehart algebra (i.e. a flat torsion-free Lie-Rinehart algebra) endowed with a natural trace map, and show the freeness of this o…
Invariant connections, Lie algebra actions, and foundations of numerical integration on manifolds
Hans Z. Munthe-Kaas, Ari Stern, Olivier Verdier
Motivated by numerical integration on manifolds, we relate the algebraic properties of invariant connections to their geometric properties. Using this perspective, we generalize so…
Planarly branched rough paths and rough differential equations on homogeneous spaces
Charles Curry, Kurusch Ebrahimi-Fard, Dominique Manchon +1
The central aim of this work is to understand rough differential equations on homogeneous spaces. We focus on the formal approach, by giving an explicit expansion of the solution a…
Pre- and Post-Lie Algebras: The Algebro-Geometric View
Gunnar Fløystad, Hans Munthe-Kaas
We relate composition and substitution in pre- and post-Lie algebras to algebraic geometry. The Connes-Kreimer Hopf algebras, and MKW Hopf algebras are then coordinate rings of the…
Post-Lie algebras and factorization theorems
Kurusch Ebrahimi-Fard, Igor Mencattini, Hans Munthe-Kaas
In this note we further explore the properties of universal enveloping algebras associated to a post-Lie algebra. Emphasizing the role of the Magnus expansion, we analyze the prope…
Lie-Butcher series, Geometry, Algebra and Computation
Hans Z. Munthe-Kaas, Kristoffer K. Føllesdal
Lie-Butcher (LB) series are formal power series expressed in terms of trees and forests. On the geometric side LB-series generalizes classical B-series from Euclidean spaces to Lie…
Butcher series: A story of rooted trees and numerical methods for evolution equations
Robert I McLachlan, Klas Modin, Hans Munthe-Kaas +1
Butcher series appear when Runge-Kutta methods for ordinary differential equations are expanded in power series of the step size parameter. Each term in a Butcher series consists o…
Post-Lie Algebras and Isospectral Flows
Kurusch Ebrahimi-Fard, Alexander Lundervold, Igor Mencattini +1
In this paper we explore the Lie enveloping algebra of a post-Lie algebra derived from a classical -matrix. An explicit exponential solution of the corresponding Lie bracket flo…
On the Lie enveloping algebra of a post-Lie algebra
Kurusch Ebrahimi-Fard, Alexander Lundervold, Hans Munthe-Kaas
We consider pairs of Lie algebras and , defined over a common vector space, where the Lie brackets of and are related via a post-Lie algebra structure. T…
B-series methods are exactly the affine equivariant methods
Robert I. McLachlan, Klas Modin, Hans Munthe-Kaas +1
Butcher series, also called B-series, are a type of expansion, fundamental in the analysis of numerical integration. Numerical methods that can be expanded in B-series are defined…
Integrators on homogeneous spaces: Isotropy choice and connections
Hans Munthe-Kaas, Olivier Verdier
We consider numerical integrators of ODEs on homogeneous spaces (spheres, affine spaces, hyperbolic spaces). Homogeneous spaces are equipped with a built-in symmetry. A numerical i…
Aromatic Butcher Series
Hans Munthe-Kaas, Olivier Verdier
We show that without other further assumption than affine equivariance and locality, a numerical integrator has an expansion in a generalized form of Butcher series (B-series) whic…
Symmetric spaces and Lie triple systems in numerical analysis of differential equations
Hans Z. Munthe-Kaas, Gilles Reinout W. Quispel, Antonella Zanna
A remarkable number of different numerical algorithms can be understood and analyzed using the concepts of symmetric spaces and Lie triple systems, which are well known in differen…
On post-Lie algebras, Lie--Butcher series and moving frames
Hans Munthe-Kaas, Alexander Lundervold
Pre-Lie (or Vinberg) algebras arise from flat and torsion-free connections on differential manifolds. They have been studied extensively in recent years, both from algebraic operad…
Algebraic structure of stochastic expansions and efficient simulation
Kurusch Ebrahimi-Fard, Alexander Lundervold, Simon J. A. Malham +2
We investigate the algebraic structure underlying the stochastic Taylor solution expansion for stochastic differential systems.Our motivation is to construct efficient integrators.…
On algebraic structures of numerical integration on vector spaces and manifolds
Alexander Lundervold, Hans Z. Munthe-Kaas
Numerical analysis of time-integration algorithms has been applying advanced algebraic techniques for more than fourty years. An explicit description of the group of characters in…
Backward error analysis and the substitution law for Lie group integrators
Alexander Lundervold, Hans Munthe-Kaas
Butcher series are combinatorial devices used in the study of numerical methods for differential equations evolving on vector spaces. More precisely, they are formal series develop…
Finite element systems of differential forms
Snorre Harald Christiansen
We develop the theory of mixed finite elements in terms of special inverse systems of complexes of differential forms, defined over cellular complexes. Inclusion of cells correspon…
Hopf algebras of formal diffeomorphisms and numerical integration on manifolds
Alexander Lundervold, Hans Munthe-Kaas
B-series originated from the work of John Butcher in the 1960s as a tool to analyze numerical integration of differential equations, in particular Runge-Kutta methods. Connections…