167 research outputs found
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
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
Backward error analysis and the substitution law for Lie group integrators
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 developments of differential operators indexed over
rooted trees, and can be used to represent a large class of numerical methods.
The theory of backward error analysis for differential equations has a
particularly nice description when applied to methods represented by Butcher
series. For the study of differential equations evolving on more general
manifolds, a generalization of Butcher series has been introduced, called
Lie--Butcher series. This paper presents the theory of backward error analysis
for methods based on Lie--Butcher series.Comment: Minor corrections and additions. Final versio
One-site density matrix renormalization group and alternating minimum energy algorithm
Given in the title are two algorithms to compute the extreme eigenstate of a
high-dimensional Hermitian matrix using the tensor train (TT) / matrix product
states (MPS) representation. Both methods empower the traditional alternating
direction scheme with the auxiliary (e.g. gradient) information, which
substantially improves the convergence in many difficult cases. Being
conceptually close, these methods have different derivation, implementation,
theoretical and practical properties. We emphasize the differences, and
reproduce the numerical example to compare the performance of two algorithms.Comment: Submitted to the proceedings of ENUMATH 201
Gaussian cubature arising from hybrid characters of simple Lie groups
Lie groups with two different root lengths allow two mixed sign homomorphisms
on their corresponding Weyl groups, which in turn give rise to two families of
hybrid Weyl group orbit functions and characters. In this paper we extend the
ideas leading to the Gaussian cubature formulas for families of polynomials
arising from the characters of irreducible representations of any simple Lie
group, to new cubature formulas based on the corresponding hybrid characters.
These formulas are new forms of Gaussian cubature in the short root length case
and new forms of Radau cubature in the long root case. The nodes for the
cubature arise quite naturally from the (computationally efficient) elements of
finite order of the Lie group.Comment: 23 pages, 3 figure
Continuous and discrete Clebsch variational principles
The Clebsch method provides a unifying approach for deriving variational
principles for continuous and discrete dynamical systems where elements of a
vector space are used to control dynamics on the cotangent bundle of a Lie
group \emph{via} a velocity map. This paper proves a reduction theorem which
states that the canonical variables on the Lie group can be eliminated, if and
only if the velocity map is a Lie algebra action, thereby producing the
Euler-Poincar\'e (EP) equation for the vector space variables. In this case,
the map from the canonical variables on the Lie group to the vector space is
the standard momentum map defined using the diamond operator. We apply the
Clebsch method in examples of the rotating rigid body and the incompressible
Euler equations. Along the way, we explain how singular solutions of the EP
equation for the diffeomorphism group (EPDiff) arise as momentum maps in the
Clebsch approach. In the case of finite dimensional Lie groups, the Clebsch
variational principle is discretised to produce a variational integrator for
the dynamical system. We obtain a discrete map from which the variables on the
cotangent bundle of a Lie group may be eliminated to produce a discrete EP
equation for elements of the vector space. We give an integrator for the
rotating rigid body as an example. We also briefly discuss how to discretise
infinite-dimensional Clebsch systems, so as to produce conservative numerical
methods for fluid dynamics
Shape analysis on homogeneous spaces: a generalised SRVT framework
Shape analysis is ubiquitous in problems of pattern and object recognition
and has developed considerably in the last decade. The use of shapes is natural
in applications where one wants to compare curves independently of their
parametrisation. One computationally efficient approach to shape analysis is
based on the Square Root Velocity Transform (SRVT). In this paper we propose a
generalised SRVT framework for shapes on homogeneous manifolds. The method
opens up for a variety of possibilities based on different choices of Lie group
action and giving rise to different Riemannian metrics.Comment: 28 pages; 4 figures, 30 subfigures; notes for proceedings of the Abel
Symposium 2016: "Computation and Combinatorics in Dynamics, Stochastics and
Control". v3: amended the text to improve readability and clarify some
points; updated and added some references; added pseudocode for the dynamic
programming algorithm used. The main results remain unchange
Simplicial gauge theory on spacetime
We define a discrete gauge-invariant Yang-Mills-Higgs action on spacetime
simplicial meshes. The formulation is a generalization of classical lattice
gauge theory, and we prove consistency of the action in the sense of
approximation theory. In addition, we perform numerical tests of convergence
towards exact continuum results for several choices of gauge fields in pure
gauge theory.Comment: 18 pages, 2 figure
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