69 research outputs found
Integrators on homogeneous spaces: Isotropy choice and connections
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 integrator respects this symmetry if it is
equivariant. One obtains homogeneous space integrators by combining a Lie group
integrator with an isotropy choice. We show that equivariant isotropy choices
combined with equivariant Lie group integrators produce equivariant homogeneous
space integrators. Moreover, we show that the RKMK, Crouch--Grossman or
commutator-free methods are equivariant. To show this, we give a novel
description of Lie group integrators in terms of stage trees and motion maps,
which unifies the known Lie group integrators. We then proceed to study the
equivariant isotropy maps of order zero, which we call connections, and show
that they can be identified with reductive structures and invariant principal
connections. We give concrete formulas for connections in standard homogeneous
spaces of interest, such as Stiefel, Grassmannian, isospectral, and polar
decomposition manifolds. Finally, we show that the space of matrices of fixed
rank possesses no connection
On the Lie enveloping algebra of a post-Lie algebra
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. The latter can be extended to the Lie enveloping
algebra . This permits us to define another associative product on
, which gives rise to a Hopf algebra isomorphism between and
a new Hopf algebra assembled from with the new product.
For the free post-Lie algebra these constructions provide a refined
understanding of a fundamental Hopf algebra appearing in the theory of
numerical integration methods for differential equations on manifolds. In the
pre-Lie setting, the algebraic point of view developed here also provides a
concise way to develop Butcher's order theory for Runge--Kutta methods.Comment: 25 page
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
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
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
B-series methods are exactly the affine equivariant methods
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 in all dimensions, so they correspond to
\emph{sequences of maps}---one map for each dimension. A long-standing problem
has been to characterise those sequences of maps that arise from B-series. This
problem is solved here: we prove that a sequence of smooth maps between vector
fields on affine spaces has a B-series expansion if and only if it is
\emph{affine equivariant}, meaning it respects all affine maps between affine
spaces
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
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