51 research outputs found
On bialgebras and Hopf algebras of oriented graphs
We define two coproducts for cycle-free oriented graphs, thus building up two
commutative con- nected graded Hopf algebras, such that one is a
comodule-coalgebra on the other, thus generalizing the result obtained
previously for Hopf algebras of rooted trees.Comment: 7 pages, error on Proposition 1 corrected, one figure adde
A L\'evy area by Fourier normal ordering for multidimensional fractional Brownian motion with small Hurst index
The main tool for stochastic calculus with respect to a multidimensional
process with small H\"older regularity index is rough path theory. Once
has been lifted to a rough path, a stochastic calculus -- as well as solutions
to stochastic differential equations driven by -- follow by standard
arguments. Although such a lift has been proved to exist by abstract arguments
\cite{LyoVic07}, a first general, explicit construction has been proposed in
\cite{Unt09,Unt09bis} under the name of Fourier normal ordering. The purpose of
this short note is to convey the main ideas of the Fourier normal ordering
method in the particular case of the iterated integrals of lowest order of
fractional Brownian motion with arbitrary Hurst index.Comment: 20 page
Time-ordering and a generalized Magnus expansion
Both the classical time-ordering and the Magnus expansion are well-known in
the context of linear initial value problems. Motivated by the noncommutativity
between time-ordering and time derivation, and related problems raised recently
in statistical physics, we introduce a generalization of the Magnus expansion.
Whereas the classical expansion computes the logarithm of the evolution
operator of a linear differential equation, our generalization addresses the
same problem, including however directly a non-trivial initial condition. As a
by-product we recover a variant of the time ordering operation, known as
T*-ordering. Eventually, placing our results in the general context of
Rota-Baxter algebras permits us to present them in a more natural algebraic
setting. It encompasses, for example, the case where one considers linear
difference equations instead of linear differential equations
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
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
Formal series and numerical integrators: some history and some new techniques
This paper provides a brief history of B-series and the associated Butcher
group and presents the new theory of word series and extended word series.
B-series (Hairer and Wanner 1976) are formal series of functions parameterized
by rooted trees. They greatly simplify the study of Runge-Kutta schemes and
other numerical integrators. We examine the problems that led to the
introduction of B-series and survey a number of more recent developments,
including applications outside numerical mathematics. Word series (series of
functions parameterized by words from an alphabet) provide in some cases a very
convenient alternative to B-series. Associated with word series is a group G of
coefficients with a composition rule simpler than the corresponding rule in the
Butcher group. From a more mathematical point of view, integrators, like
Runge-Kutta schemes, that are affine equivariant are represented by elements of
the Butcher group, integrators that are equivariant with respect to arbitrary
changes of variables are represented by elements of the word group G.Comment: arXiv admin note: text overlap with arXiv:1502.0552
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