23 research outputs found
On algebraic structures of numerical integration on vector spaces and manifolds
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 the Butcher-Connes-Kreimer Hopf algebra first appeared
in Butcher's work on composition of integration methods in 1972. In more recent
years, the analysis of structure preserving algorithms, geometric integration
techniques and integration algorithms on manifolds have motivated the
incorporation of other algebraic structures in numerical analysis. In this
paper we will survey structures that have found applications within these
areas. This includes pre-Lie structures for the geometry of flat and torsion
free connections appearing in the analysis of numerical flows on vector spaces.
The much more recent post-Lie and D-algebras appear in the analysis of flows on
manifolds with flat connections with constant torsion. Dynkin and Eulerian
idempotents appear in the analysis of non-autonomous flows and in backward
error analysis. Non-commutative Bell polynomials and a non-commutative Fa\`a di
Bruno Hopf algebra are other examples of structures appearing naturally in the
numerical analysis of integration on manifolds.Comment: 42 pages, final versio
Invariant connections, Lie algebra actions, and foundations of numerical integration on manifolds
Motivated by numerical integration on manifolds, we relate the algebraic
properties of invariant connections to their geometric properties. Using this
perspective, we generalize some classical results of Cartan and Nomizu to
invariant connections on algebroids. This has fundamental consequences for the
theory of numerical integrators, giving a characterization of the spaces on
which Butcher and Lie-Butcher series methods, which generalize Runge-Kutta
methods, may be applied.Comment: 18 page
Symmetric spaces and Lie triple systems in numerical analysis of differential equations
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 differential geometry from the study of spaces of constant
curvature and their tangents. This theory can be used to unify a range of
different topics, such as polar-type matrix decompositions, splitting methods
for computation of the matrix exponential, composition of selfadjoint numerical
integrators and dynamical systems with symmetries and reversing symmetries. The
thread of this paper is the following: involutive automorphisms on groups
induce a factorization at a group level, and a splitting at the algebra level.
In this paper we will give an introduction to the mathematical theory behind
these constructions, and review recent results. Furthermore, we present a new
Yoshida-like technique, for self-adjoint numerical schemes, that allows to
increase the order of preservation of symmetries by two units. Since all the
time-steps are positive, the technique is particularly suited to stiff
problems, where a negative time-step can cause instabilities
Post-Lie Algebras and Isospectral Flows
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 flow is presented. It is based on the solution of a
post-Lie Magnus-type differential equation
The aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators
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-preserving B-series method is the exact flow of the
differential equation. The question was raised whether there exists a
volume-preserving integrator that can be expanded as an aromatic B-series. In
this work, we introduce a new algebraic tool, called the aromatic bicomplex,
similar to the variational bicomplex in variational calculus. We prove the
exactness of this bicomplex and use it to describe explicitly the key object in
the study of volume-preserving integrators: the aromatic forms of vanishing
divergence. The analysis provides us with a handful of new tools to study
aromatic B-series, gives insights on the process of integration by parts of
trees, and allows to describe explicitly the aromatic B-series of a
volume-preserving integrator. In particular, we conclude that an aromatic
Runge-Kutta method cannot preserve volume.Comment: 41 page
On post-Lie algebras, Lie--Butcher series and moving frames
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 operadic points of view and through numerous applications
in numerical analysis, control theory, stochastic differential equations and
renormalization. Butcher series are formal power series founded on pre-Lie
algebras, used in numerical analysis to study geometric properties of flows on
euclidean spaces. Motivated by the analysis of flows on manifolds and
homogeneous spaces, we investigate algebras arising from flat connections with
constant torsion, leading to the definition of post-Lie algebras, a
generalization of pre-Lie algebras. Whereas pre-Lie algebras are intimately
associated with euclidean geometry, post-Lie algebras occur naturally in the
differential geometry of homogeneous spaces, and are also closely related to
Cartan's method of moving frames. Lie--Butcher series combine Butcher series
with Lie series and are used to analyze flows on manifolds. In this paper we
show that Lie--Butcher series are founded on post-Lie algebras. The functorial
relations between post-Lie algebras and their enveloping algebras, called
D-algebras, are explored. Furthermore, we develop new formulas for computations
in free post-Lie algebras and D-algebras, based on recursions in a magma, and
we show that Lie--Butcher series are related to invariants of curves described
by moving frames.Comment: added discussion of post-Lie algebroid