11 research outputs found
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
Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs
We introduce a new algebraic framework based on a modification (called
exotic) of aromatic Butcher-series for the systematic study of the accuracy of
numerical integrators for the invariant measure of a class of ergodic
stochastic differential equations (SDEs) with additive noise. The proposed
analysis covers Runge-Kutta type schemes including the cases of partitioned
methods and postprocessed methods. We also show that the introduced exotic
aromatic B-series satisfy an isometric equivariance property.Comment: 33 page
A minimal-variable symplectic integrator on spheres
We construct a symplectic, globally defined, minimal-coordinate, equivariant
integrator on products of 2-spheres. Examples of corresponding Hamiltonian
systems, called spin systems, include the reduced free rigid body, the motion
of point vortices on a sphere, and the classical Heisenberg spin chain, a
spatial discretisation of the Landau-Lifschitz equation. The existence of such
an integrator is remarkable, as the sphere is neither a vector space, nor a
cotangent bundle, has no global coordinate chart, and its symplectic form is
not even exact. Moreover, the formulation of the integrator is very simple, and
resembles the geodesic midpoint method, although the latter is not symplectic
The geometry of characters of Hopf algebras
Character groups of Hopf algebras appear in a variety of mathematical
contexts such as non-commutative geometry, renormalisation of quantum field
theory, numerical analysis and the theory of regularity structures for
stochastic partial differential equations. In these applications, several
species of "series expansions" can then be described as characters from a Hopf
algebra to a commutative algebra. Examples include ordinary Taylor series,
B-series, Chen-Fliess series from control theory and rough paths. In this note
we explain and review the constructions for Lie group and topological
structures for character groups. The main novel result of the present article
is a Lie group structure for characters of graded and not necessarily connected
Hopf algebras (under the assumption that the degree zero subalgebra is
finite-dimensional). Further, we establish regularity (in the sense of Milnor)
for these Lie groups.Comment: 25 pages, notes for the Abelsymposium 2016: "Computation and
Combinatorics in Dynamics, Stochastics and Control", v4: corrected typos and
mistakes, main results remains valid, updated reference
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
Constructing general rough differential equations through flow approximations
The non-linear sewing lemma constructs flows of rough differential equations
from a braod class of approximations called almost flows. We consider a class
of almost flows that could be approximated by solutions of ordinary
differential equations, in the spirit of the backward error analysis. Mixing
algebra and analysis, a Taylor formula with remainder and a composition formula
are central in the expansion analysis. With a suitable algebraic structure on
the non-smooth vector fields to be integrated, we recover in a single framework
several results regarding high-order expansions for various kind of driving
paths. We also extend the notion of driving rough path. We also introduce as an
example a new family of branched rough paths, called aromatic rough paths
modeled after aromatic Butcher series.Comment: version R0 (august 4, 2020): bibliography updat
Order conditions for sampling the invariant measure of ergodic stochastic differential equations on manifolds
We derive a new methodology for the construction of high order integrators
for sampling the invariant measure of ergodic stochastic differential equations
with dynamics constrained on a manifold. We obtain the order conditions for
sampling the invariant measure for a class of Runge-Kutta methods applied to
the constrained overdamped Langevin equation. The analysis is valid for
arbitrarily high order and relies on an extension of the exotic aromatic
Butcher-series formalism. To illustrate the methodology, a method of order two
is introduced, and numerical experiments on the sphere, the torus and the
special linear group confirm the theoretical findings.Comment: 40 page