Outline
B-series for order conditions and
preservation of invariants
Philippe Chartier
INRIA-Rennes, ENS Cachan Bruz
Formation IPR 2008
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Historical panorama
Order conditions for Runge-Kutta methods
B-series formalism Connes and Kreimer Hopf algebra
Outline
1
2
3
4
Historical panorama
From Runge to Butcher
From Hairer and Wanner to Connes and Kreimer
Recent developments
Order conditions for Runge-Kutta methods
Taylor expansions and the necessity of a representation
Trees and elementary differentials
Order conditions
B-series formalism
Formal development and Examples
Composition law
Applications and extensions
Connes and Kreimer Hopf algebra
Origins
Construction of the graded bialgebra
The Hopf algebra structure and the antipode
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S
Historical panorama
Order conditions for Runge-Kutta methods
B-series formalism Connes and Kreimer Hopf algebra
Outline
1
2
3
4
Historical panorama
From Runge to Butcher
From Hairer and Wanner to Connes and Kreimer
Recent developments
Order conditions for Runge-Kutta methods
Taylor expansions and the necessity of a representation
Trees and elementary differentials
Order conditions
B-series formalism
Formal development and Examples
Composition law
Applications and extensions
Connes and Kreimer Hopf algebra
Origins
Construction of the graded bialgebra
The Hopf algebra structure and the antipode
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Order conditions for Runge-Kutta methods
B-series formalism Connes and Kreimer Hopf algebra
From Runge to Butcher
Around 1900, two german mathematicians Runge and
Kutta introduced methods of high-order for the numerical
solution of ordinary differential equations. The problem of
computing the coefficients became notoriously tedious (for
instance, one of the methods published by Runge was not
correct) .
In a series of articles in 1963, 1964, 1969, 1972, Butcher
introduced the use of trees and developped a theory of
order conditions, endind up with the Butcher group of
Runge-Kutta methods.
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From Hairer and Wanner to Connes and Kreimer
Hairer and Wanner introduced in 1974 the formalism of
B-series (B stands for Butcher). These series are formal
expansions which can represent most numerical schemes
for ODEs (not only Runge-Kutta methods). They
developped a composition law for such series.
In 1999, Connes and Kreimer construct a Hopf algebra of
trees and recover the composition law of B-series. A. Dur
had established an implicit link in 1984. Brouder in 2000
helped to make the connection with ODEs more notorious.
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Recent developments
B-series have been extended into several directions: to
partitionned or split systems (P-series with bicolor trees by
Hairer and Wanner and NB-series with multicolour trees
by Murua), to series of differential operators (S-series of
Murua in 1998).
They have been used to study the preservation of
invariants for Hamiltonian systems (Hairer 1994, Murua
1996), general ODEs ( C., Faou and Murua, 2006) or
divergence free systems (Iserles, Quispel and Tse 2007,
C. and Murua, 2007)
They lie at the core of modified equations for backward
error analysis (Reich 1996, Hairer 1994, Murua 1995) and
for modifying vector fields integrators (C., Hairer and
Vilmart 2007). In this context, a new substitution law for
B-series has been introduced by C., Hairer and Vilmart in
2005 which turned out to give rise to a new Hopf algebra of
trees (Calaque, Ebrahimi-Fard and Manchon, 2008)
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Order conditions for Runge-Kutta methods
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Outline
1
2
3
4
Historical panorama
From Runge to Butcher
From Hairer and Wanner to Connes and Kreimer
Recent developments
Order conditions for Runge-Kutta methods
Taylor expansions and the necessity of a representation
Trees and elementary differentials
Order conditions
B-series formalism
Formal development and Examples
Composition law
Applications and extensions
Connes and Kreimer Hopf algebra
Origins
Construction of the graded bialgebra
The Hopf algebra structure and the antipode
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Order conditions for Runge-Kutta methods
B-series formalism Connes and Kreimer Hopf algebra
Taylor expansions and the necessity of a representation
Context
All numerical methods presented in this talk are aimed at
solving an ordinary differential equation of the form
ẏ =
dy
= f (y),
dt
y(0) = y0 ,
(1)
where :
y is a vector of Rn .
f is a smooth vector-field, i.e. a function from Rn into Rn ,
differentiable as many times as necessary.
the conditions for existence and uniqueness of the exact
solution are fullfilled, so that the flow ϕt of (1) is defined in
Rn for all time.
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Examples: A first “guess” of order 1, Euler methods
The most simple method is Euler method: the solution ϕt (y0 )
on (0, h) is approximated by a line with slope f (y0 ) or f (y1 )
ϕh (y0 ) ≈ y1 = y0 + hf (y0 )
= y0 + hf (y1 )
for explicit Euler,
for implicit Euler.
y1
y1
ỹ (t)
D
y0
y0
y(t)
h
t0
Explicit
D
y(t)
h
t1 = t0 + h
t0
Implicit
t1 = t0 + h
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Example: A refined method of order 2, a method of Runge
Euler methods can be improved by evaluating the slope at
midpoint:
y1 = y0 + hf ϕh/2 (y0 ) .
The value of ϕh/2 (y0 ) is itself approximated by
ϕh/2 (y0 ) ≈ y0 +
h
f (y0 ).
2
This new method [Runge (1895)] proceeds in two stages:
Y
y1
h
f (y0 ),
2
= y0 + hf (Y ) .
= y0 +
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Definition
Let b ∈ Rs and A ∈ Rs×s . The formulae
P
Yi = y0 + h sj=1 aij f (Yj ), i = 1, . . . , s,
P
y1 = y0 + h sj=1 bj f (Yj ),
(2)
define one step of the RKM. Vectors Yi are internal stages.
Remarks
If A is strictly lower triangular, then the method is explicit.
RKMs are represented by its Butcher Tableau (c = Ae)
c1
.
.
.
cs
a11
.
.
.
as1
b1
...
...
...
a1s
.
.
.
ass
bs
Explicit
0
Euler
0
1
Implicit
1
Euler
1
1
Runge′ s
0
1/2
Method
0
1/2
0
0
0
1
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Example
In trying to get the Taylor expansion of the implicit Euler solution
y1 = y0 + hf (y1 )
one gets successively (omitting the argument y0 in f , f ′ , ...)
y1 = y0 + h |{z}
f +O(h2 ),
=y ′
f ′ f +O(h3 ),
y1 = y0 + h |{z}
f +h2 |{z}
=y ′
=y ′′
1
f ′ f +h3 f ′ f ′ f + f ′′ f , f
y1 = y0 + h |{z}
f +h2 |{z}
+ O(h4 ).
2
|
{z
}
=y ′′
=y ′
6=y (3) =f ′ f ′ f +f ′′ f ,f
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Trees and elementary differentials
Definition (Rooted trees and forests)
The sets of rooted trees and forests are recursively defined by:
1
2
3
∅∈F
∈ T , σ( ) = 1.
if (t1 , . . . , tn ) ∈ T n are distincts trees, then
Q
u = t1 · · · t1 · · · tn . . . tn ∈ F and σ(u) = ni=1 ri !σ(ti ))ri .
| {z } | {z }
r1
4
rn
if u ∈ F then t = [u] ∈ T and and σ(t) = σ(u).
The order of a tree |t| is its number of vertices.
Example
Tree t
Order |t|
Symetry σ(t)
1
1
2
1
3
2
3
1
4
6
4
1
4
2
4
1
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Trees and elementary differentials
Definition (Elementary differentials)
For each t ∈ T , the elementary differential F (t) associated with
t is the mapping from Rn to Rn , defined recursively by:
1
2
F ( )(y) = f (y),
F ([t1 , . . . , tn ])(y) = f (n) (y) F (t1 )(y), . . . , F (tn )(y) .
Example
F ( )(y)
= f ′ (y)f (y),
F ( )(y)
= f ′ (y)f ′ (y)f
(y),
F(
)(y) = f (3) (y) f (y), f (y), f (y) .
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Theorem (Expansion of the exact solution)
The exact solution of the ODE y ′ = f (y) with initial condition
y(0) = y0 can be expanded as
y(h) = y0 +
X
t∈T
h|t|
F (t)(y0 )
σ(t)γ(t)
where γ is defined by γ( ) = 1 and γ(t) = |t|
Qn
i=1 γ(ti ).
Proof.
ẏ
= f = F ( ),
ÿ
= f ′ ẏ = f ′ f = F ( )
y (3) = f ′′ (f , ẏ ) + f ′ f ′ ẏ = f ′′ (f , f ) + f ′ f ′ f = F (
y
(4)
′′′
′′
′
′ ′′
) + F( )
′ ′ ′
= f (f , f , f ) + 3f (f f , f ) + f f (f , f ) + f f f f
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= F(
) + F(
) + F(
) + F( )
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Order conditions
Expansion of the Runge-Kutta solution
Theorem
Let R = (A, b, c) be a Runge-Kutta method. Then one has
X b T Φ(t) y1 = y0 +
F (t)
σ(t)
t∈T
Proof.
Yi
= y0 + h
X
j
= y0 + h
X
j
= y0 + h
X
j
aij f (Yj ) = y0 + h
X
aij f (y0 ) + O(h2 )
j
X
aij f (y0 +
ajk f (y0 ) + O(h3 )
k
aij f (y0 ) + h2
X
j,k
aij ajk f ′ (y0 )f (y0 ) + O(h3 )
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Definition (Coefficients of Runge-Kutta methods)
The function Φ is defined on the set of trees recursively by:
1
2
Φ( ) = e with e = (1, . . . , 1)T ∈ Rs ,
Q
Φ([t1 , . . . , tn ]) = ni=1 AΦ(ti ),
where A is the matrix of coefficients of the method.
Example
Note that Φ takes its values in Rs , so that the product of vectors
is meant to be componentwise. For instance, we have:
1
t = [ , , ] : Φ(t) = (Ae).(Ae).(Ae).
2
t = [[ , ], ] : Φ(t) = (Ae). (A[(Ae).(Ae)]).
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Theorem (Order conditions for Runge-Kutta methods)
A Runge-Kutta method R = (A, b, c) is of local order p + 1 (i.e.
global order p) iff
∀t ∈ T , |t| ≤ p, b T Φ(t) =
1
.
γ(t)
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Order conditions for Runge-Kutta methods
Order 4 conditions
Graph
t =
γ(t)
γ(t) = 1
Φ(t)
Φ(t) = e
Order Condition
bT e = 1
2
t =
γ(t) = 2
Φ(t) = Ae
bT Ae = 1/2
3
t =
γ(t) = 3
Φ(t) = (Ae).(Ae)
bT (Ae)2 = 1/3
3
t =
γ(t) = 6
Φ(t) = A2 e
bT A2 e = 1/6
γ(t) = 4
Φ(t) = (Ae)3
bT (Ae)3 = 1/4
Order
1
4
t =
4
t =
γ(t) = 8
Φ(t) = (Ae).(A2 e)
bT [(Ae).(A2 e)] = 1/8
4
t =
γ(t) = 12
Φ(t) = A(Ae)2
bT A(Ae)2 = 1/12
4
t =
γ(t) = 24
Φ(t) = A3 e
bT A3 e = 1/24
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Outline
1
2
3
4
Historical panorama
From Runge to Butcher
From Hairer and Wanner to Connes and Kreimer
Recent developments
Order conditions for Runge-Kutta methods
Taylor expansions and the necessity of a representation
Trees and elementary differentials
Order conditions
B-series formalism
Formal development and Examples
Composition law
Applications and extensions
Connes and Kreimer Hopf algebra
Origins
Construction of the graded bialgebra
The Hopf algebra structure and the antipode
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Order conditions for Runge-Kutta methods
B-series formalism Connes and Kreimer Hopf algebra
Formal development and Examples
Definition
A B-series B(a, y) is a formal expression of the form
B(f , a) = idRn +
X h|t|
a(t) F (t)
σ(t)
t ∈T
= idRn + ha( )f (·) + h2 a( )(f ′ f )(·) + · · ·
the index set T = { , ,
, , · · · } is the set of trees,
| · | (order) and σ(·) (symmetry) are maps from T to
positive integers,
F (t) (elementary differential) is a map from Rn to Rn
obtained from f and its derivatives,
a is a function defined on T which characterizes B(a, y).
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Examples of B-series integrators
B-series expansions of some integrators
1
2
h
Exact solution: y(h) = y + hf (y) + 2.1
(f ′ f )(y) +
h3 ′ ′
h3 ′′
3.2 (f (f , f ))(y) + 6.1 (f f f ))(y) + . . . = B(f , 1/γ)(y) with
γ([t1 , . . . , tn ] = |t|γ(t1 ) · · · γ(tn ).
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Examples of B-series integrators
B-series expansions of some integrators
1
2
2
h
Exact solution: y(h) = y + hf () + 2.1
(f ′ f )(y) +
h3 ′ ′
h3 ′′
3.2 (f (f , f ))(y) + 6.1 (f f f ))(y) + . . . = B(f , 1/γ)(y) with
γ([t1 , . . . , tn ] = |t|γ(t1 ) · · · γ(tn ).
Explicit Euler: y + hf (y) = B(f , a)(y) with a( ) = 1 and
a(t) = 0 for all t 6= .
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Examples of B-series integrators
B-series expansions of some integrators
1
2
h
Exact solution: y(h) = y + hf () + 2.1
(f ′ f )(y) +
h3 ′ ′
h3 ′′
3.2 (f (f , f ))(y) + 6.1 (f f f ))(y) + . . . = B(f , 1/γ)(y) with
γ([t1 , . . . , tn ] = |t|γ(t1 ) · · · γ(tn ).
2
Explicit Euler: y + hf (y) = B(f , a)(y) with a( ) = 1 and
a(t) = 0 for all t 6= .
3
Implicit Euler: Y = y + hf (Y ) and y + hf (Y ) = B(f , a)(y)
with a(t) = 1 for all t ∈ T .
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Examples of B-series integrators
B-series expansions of some integrators
1
2
h
Exact solution: y(h) = y + hf () + 2.1
(f ′ f )(y) +
h3 ′ ′
h3 ′′
3.2 (f (f , f ))(y) + 6.1 (f f f ))(y) + . . . = B(1/γ, y0 ) with
γ([t1 , . . . , tn ] = |t|γ(t1 ) · · · γ(tn ).
2
Explicit Euler: y + hf (y) = B(f , a)(y) with a( ) = 1 and
a(t) = 0 for all t 6= .
3
Implicit Euler: Y = y + hf (Y ) and y + hf (Y ) = B(f , a)(y)
with a(t) = 1 for all t ∈ T .
4
Midpoint rule: Y = y + h2 f (Y ) and y + hf (Y ) = B(f , a)(y)
with a(t) = (1/2)|t|−1 for all t ∈ T .
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A partition p of a tree t is obtained by cutting some of its
edges.
The resulting forest (unordered collection of trees) is
denoted up and the set of partitions p of t is denoted P(t).
We distinguish rp as the tree of vp whose root coincides
with the root of t and we denote by vp∗ the forest obtained
by removing rp from vp .
An admissible partition is a partition with at most one cut
along any path from the root to any terminal vertex.
We denote AP(t) the set of admissible partitions of t and
by convention, we consider that ∅ ∈ AP(t) and that r∅ = ∅,
v∅ = v∅∗ = t.
Remark
We observe that a tree t ∈ T has exactly 2|t|−1 partitions
p ∈ P(t), and that different partitions p may lead to the same
forest up .
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Table: The 8 partitions of a tree of order 4 with associated forest
p
Admis.?
YES
YES
YES
YES
YES
NO
NO
NO
vp
rp
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Theorem (Hairer & Wanner 1974)
Let a, b : T → R be two mappings, with a(∅) = 1. Then the
B-series B(f , a)(y) inserted into B(f , b)(·) is still a B-series
B(f , b) B(f , a)(y) = B(f , a · b)(y),
and a · b : T ∪ {∅} → R is defined by
(a · b)(∅) = b(∅),
∀t ∈ T , (a · b)(t) =
X
b(rp )a(vp∗ ), (3)
p∈AP(t)
where a is extented to F as follows:
∀u = t1 . . . tn ∈ F, a(u) =
n
Y
a(ti ).
(4)
i=1
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Product and inverse for trees of order ≤ 3
a · b( )
= b(∅)a( ) + b( ),
a · b( )
= b(∅)a( ) + b( )a( ) + b( ),
a · b(
= b(∅)a(
)
) + b( )a( )2 + 2b( )a( ) + b(
a · b( )
= b(∅)a( ) + b( )a( ) + b( )a( ) + b( ),
a−1 ( )
= −a( ),
a
−1
( )
a−1 (
)
a−1 ( )
),
= −a( ) + a( )2 ,
= −a(
) + 2a( )a( ) − a( )3 ,
= −a( ) + 2a( )a( ) − a( )3 .
The set of maps a : T ∪ {∅} → R with a(∅) = 1, forms a group
called the Butcher group. Its unit element is δ∅ , defined as
δ∅ (∅) = 1, δ∅ (t) = 0 for t ∈ T .
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The composition law is used for the
study of the order of composed methods φ1h ◦ φ2h ; in
particular, this supports the idea of effective order
(φ1h ◦ φ2h ◦ (φ1h )−1 )
study of order conditions for many classes of methods (hf
is a special B-series)
It can be extended to
P-series or NB-series of coloured trees where each colour
corresponds to a vector field (for systems of the form
y ′ = f1 (y) + f2 (y) + . . . + fN (y));
S-series of differential operators, where elementary
differentials are replaced by operators acting on smooth
functions
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Outline
1
2
3
4
Historical panorama
From Runge to Butcher
From Hairer and Wanner to Connes and Kreimer
Recent developments
Order conditions for Runge-Kutta methods
Taylor expansions and the necessity of a representation
Trees and elementary differentials
Order conditions
B-series formalism
Formal development and Examples
Composition law
Applications and extensions
Connes and Kreimer Hopf algebra
Origins
Construction of the graded bialgebra
The Hopf algebra structure and the antipode
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Construction of the graded bialgebra
Definition
The set F can be naturally endorsed in an algebra HCK as
follows:
∀ (u, v) ∈ F 2 , ∀ (λ, µ) ∈ R2 , λu + µv ∈ HCK ,
∀ (u, v) ∈ F 2 , u v ∈ HCK , where u v is the (commutative)
juxtaposition of the two forests u and v,
∀ u ∈ F, u e = e u = u, where e = ∅ is the unity element.
Maps a from T ∪ {∅} onto R with a(∅) = 1 can be uniquely
extended to F as algebra morphisms and to HCK by
linearity. Hence, unital algebra maps from Alg(HCK , R) are
in one-to-one correspondance with integrator B-series
B(f , a) satisfying a(∅) = 1.
General linear maps a from HCK to R with a(∅) = 1
correspond to S-series S(f , a) satisfying a(∅) = 1.
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Definition
The tensor product of HCK with itself is the set of elements of
the form u ⊗ v with u and v in H, such that:
3 , (u + v) ⊗ w = u ⊗ w + v ⊗ w ,
∀ (u, v, w ) ∈ HCK
3 , u ⊗ (v + w ) = u ⊗ v + u ⊗ w ,
∀ (u, v, w ) ∈ HCK
2 , ∀ λ ∈ R, λ · u ⊗ v = (λ · u) ⊗ v = u ⊗ (λ · v).
∀ (u, v) ∈ HCK
Remark
Using the definition of tensor products, the juxtaposition of
forests may also be written as a multiplication µ defined on
HCK ⊗ HCK :
2
∀ (u, v) ∈ HCK
, u v = µ(u ⊗ v).
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The algebra HCK (over R) gives rise to a bialgebra (over R) if
there exist two linear maps:
∆CK : HCK
ǫ : HCK
→ HCK ⊗ HCK
→ R
Coproduct
Counit
such that
(idHCK ⊗ ∆CK ) ◦ ∆CK
(idHCK ⊗ ǫ) ◦ ∆CK
= (∆CK ⊗ idHCK ) ◦ ∆CK ,
(5)
= (ǫ ⊗ idHCK ) ◦ ∆CK .
(6)
It then becomes a Hopf algebra if in addition, there exists a
linear map S (called the antipode) such that
µ ◦ (idHCK ⊗ SCK ) ◦ ∆ = µ ◦ (SCK ⊗ idHCK ) ◦ ∆ = ǫ. (7)
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Construction of the graded bialgebra
Define the maps B − and B + as follows
B+ :
F
→
T
B− :
T
→
F
u = t1 . . . tn 7→ [t1 , . . . , tn ]
t = [t1 , . . . , tn ] 7→ t1 . . . tn
Example
B+(
)=
h
i
=
and B − (
)=
Definition
The coproduct of Connes and Kreimer ∆CK is defined
recursively as the only algebra map which satisfies
∆CK (∅) = ∅ ⊗ ∅,
∀t ∈ T ,
(8)
+
−
∆CK (t) = ∅ ⊗ t + (B ⊗ idHCK ) ◦ ∆CK ◦ B (t).
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The following equivalent formula can be shown by induction
X
∀t ∈ T , ∆CK (t) =
rp ⊗ vp∗ .
p∈AP(t)
For the first trees, this yields (compare with formulae for the
composition law) :
∆CK (∅)
∆CK ( )
∆CK ( )
∆CK (
= ∅⊗∅
= ∅⊗ + ⊗∅
= ∅⊗ + ⊗ +
) = ∅⊗
+
⊗
= ∅⊗
+
⊗
∆CK ( )
⊗∅
+2
+
⊗
(9)
⊗
+
+
⊗∅
⊗ ∅.
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Defining the counit ǫ as ǫ(u) = 1 iff u = ∅, one can check
coassociativity. For instance, ((∆CK ⊗ idHCK ) ◦ ∆CK )( ) gives
∆CK (∅) ⊗
+ ∆CK ( ) ⊗
+ 2∆CK ( ) ⊗
+ ∆CK (
=∅⊗∅⊗
+∅⊗ ⊗
+ ⊗∅⊗
+ 2∅ ⊗
+2 ⊗ ∅ ⊗ + ∅ ⊗
⊗∅+ ⊗
⊗∅+2 ⊗
and ((idHCK ⊗ ∆CK ) ◦ ∆CK )(
) + 2 ⊗ ∆CK ( ) +
+ 2∅ ⊗ ⊗ + ∅ ⊗
+ ⊗
+
⊗
⊗ +2 ⊗ ⊗
⊗∅+
⊗ ∅ ⊗ ∅,
)
∅ ⊗ ∆CK ( ) + ⊗ ∆CK (
=∅⊗∅⊗
+∅⊗ ⊗
⊗∅+2 ⊗
)⊗∅
⊗∅⊗
+2
⊗ ∆CK (∅)
⊗∅+
⊗∅⊗∅
⊗∅⊗
+2
⊗
⊗ ∅.
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The Hopf algebra structure and the antipode
n = Span{u ∈ F; |u| = n}, then
Denote HCK
M
n
HCK =
HCK
,
n≥0
0 ≡ R), bialgebra.
so that HCK is a graded, connected (i.e. HCK
It is thus a Hopf algebra and hence has an antipode SCK , which
can be obtained recursively. For the first trees, this yields :
SCK (∅) = ∅,
SCK ( ) = − ,
SCK ( ) = −
+
2
,
) = −
+2
−
SCK ( ) = −
+2
−
SCK (
3
3
,
.
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An explicit formula for SCK exists in terms of partitions.
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Now, the main observation is that the composition product of
B-series or S-series with coefficients a and b can be seen as
the convolution product
(b · a) = µR ◦ (a ⊗ b) ◦ ∆CK
(10)
Of course a · b 6= b · a, i.e. the Hopf algebra in not
cocommutative.
Group of characters = Butcher group
The convolution product a · b makes the set of unital algebra
mappings from HCK to R a group: the Butcher group, or the
group of characters from HCK to R. It is then standard that the
inverse for the convolution product of a unital algebra morphism
a : HCK → R is given by the antipode:
∀u ∈ HCK , a−1 (u) = a ◦ SCK (u).
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Outline
1
2
3
4
Historical panorama
From Runge to Butcher
From Hairer and Wanner to Connes and Kreimer
Recent developments
Order conditions for Runge-Kutta methods
Taylor expansions and the necessity of a representation
Trees and elementary differentials
Order conditions
B-series formalism
Formal development and Examples
Composition law
Applications and extensions
Connes and Kreimer Hopf algebra
Origins
Construction of the graded bialgebra
The Hopf algebra structure and the antipode
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The idea of backward error analysis for ODEs comes to
computing the coefficients of a modified equation. For instance,
considering the trapezoidal rule with stepsize h,
h
f (y0 ) + f (y1 )
(11)
y1 = y0 +
2
h2
= y0 + hf (y0 ) + f ′ (y0 )f (y0 )
2
1
1
+h3 f ′ (y0 )f ′ (y0 )f (y0 ) + f ′′ (y0 )(f (y0 ), f (y0 )) + . . .
4
8
we search for a modified field efh (y) such that the trapezoidal
rule formally yields the exact solution of a modified differential
equation,
yė = ef (ye) = f (ye) +
h2 ′′
h2 ′
f (ye)f ′ (ye)f (ye) −
f (ye)(f (ye), f (ye)) + . . . ,
12
24
i.e. yn = ye(nh) for n = 0, 1, 2, . . .
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In C., Hairer and Vilmart, we introduce the idea of modifying
(or preprocessed) vector integrators. For instance, applying the
trapezoidal rule to a suitable modified ODE
ẏ = f (y) −
h2 ′′
h2 ′
f (y)f ′ (y)f (y) +
f (y)(f (y), f (y)) + . . . ,
12
24
formally yields the exact solution of the original ODE).
Truncating after the second-order terms then provides a
fourth-order approximation.
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B ACKWARD
ẏ = f (y )
ż = fh (z)
num
er
met ical
hod q
t
exac n
i
to
solu
M ODIFYING
ẏ = f (y )
ż = f (z)
ERROR ANALYSIS
y0 , y1 , y2 , y3 , . . .
=
z(0),
z(h),
z(2h), . . .
1
NUMERICAL METHOD
exa
solu ct
tion
q y (0), y (h), y (2h), . . .
=
z0 , z1 , z2 , z3 , . . .
al
eric 1
num hod
met
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In terms of B-series, the two procedure read:
A B-series integrator B(f , a)(y0 ) can be seen as the exact
solution B(·, 1/γ) of a modified field h1 B(f , b):
B(f , a)(y0 ) = B
1
h
B(f , b), 1/γ (y0 ) := B(f , b ⋆ 1/γ)(y0 ).
A B-series integrator y1 = B(·, a)(y0 ) applied to a modified
vector field h1 B(f , b)(y) can be regarded as a new
integrator of the form
B
1
h
B(f , b), a (y0 ) = B(f , b ⋆ a)(y0 ).
In this situation, one substitutes a B-series for the vector field
used to integrate. The corresponding law, denoted ⋆, has been
called substitution law.
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From partitions and skeletons to the formula
Definition
Given a partition p of t, the corresponding skeleton χp is the
tree obtained by contracting each tree of p to a single vertex
and by re-establishing the cut edges.
Table: The 8 partitions of a tree of order 4 with associated skeleton
and forest
p
χp
vp
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Theorem
For b(∅) = 0, the vector field h−1 B(f , b) inserted into B(·, a)
gives a B-series
B h−1 B(f , b), a = B(f , b ⋆ a).
We have (b ⋆ a)(∅) = a(∅) and for all t ∈ T ,
X
(b ⋆ a)(t) =
a(χp )b(vp )
(12)
p∈P(t)
where b is extented to F as follows:
∀u = t1 . . . tn ∈ F, b(u) =
n
Y
b(ti ).
(13)
i=1
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Table: Substitution law ⋆ for the first trees.
(b ⋆ a)(∅) = a(∅)
(b ⋆ a)( ) = a( )b( )
(b ⋆ a)( ) = a( )b( ) + a( )b( )2
(b ⋆ a)(
) = a( )b(
) + 2a( )b( )b( ) + a(
)b( )3
(b ⋆ a)( ) = a( )b( ) + 2a( )b( )b( ) + a( )b( )3
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Theorem
e c, e
Let a, b, b,
c : T ∪ {∅} → R be mappings satisfying a(∅) = 1
e
and b(∅) = b(∅) = 0. Then, for all λ and µ in R:
b ⋆ δ∅ = δ∅ ,
(14)
b⋆δ
= δ ⋆ b = b,
b ⋆ (λc + µe
c ) = λ(b ⋆ c) + µ(b ⋆ e
c ),
e
e
(b ⋆ b) ⋆ c = b ⋆ (b ⋆ c),
(15)
(16)
(17)
b ⋆ (a · c) = (b ⋆ a) · (b ⋆ c),
·−1
(b ⋆ a)
a
−1
= b⋆a
−1
(18)
,
(19)
−1
= (a − δ∅ ) ⋆ (δ∅ + δ )
,
(20)
Remark
Given a tree t, we have (δ∅ + δ )−1 (t) = (−1)|t| , and for all
p ∈ P(t), |χ(p)| = #(p).
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The character ω and its role
The coefficients b for backward error analysis can be computed
in terms of a: let ω denote the inverse element of γ1 − δ∅ for ⋆,
then
Backward error character ω
∀t ∈ T , b(t) = ((a − δ∅ ) ⋆ ω)(t).
Remarks
The coefficients ω can thus be interpreted as the
coefficients of the modified field obtained by backward
error analysis for the Euler explicit method.
They may be computed by induction. In particular,
ω( ) = B1 = −1/2, ω(
) = B2 = 1/6, ω(
where the Bi ’s are the Bernoulli numbers.
) = B3 = 0, . . .
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Theorem
The logarithmic map log : a 7→ (a − δ∅ ) ⋆ ω establishes a
one-to-one correspondence between:
the subgroup of symplectic B-series with the subgroup of
Hamiltonian B-series vector fields;
the subgroup of symmetric B-series with the subgroup of
B-series vector fields in even powers of h;
the subgroup of volume-preserving B-series with the
subgroup of divergence free B-series vector fields;
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Outline
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2
3
4
Historical panorama
From Runge to Butcher
From Hairer and Wanner to Connes and Kreimer
Recent developments
Order conditions for Runge-Kutta methods
Taylor expansions and the necessity of a representation
Trees and elementary differentials
Order conditions
B-series formalism
Formal development and Examples
Composition law
Applications and extensions
Connes and Kreimer Hopf algebra
Origins
Construction of the graded bialgebra
The Hopf algebra structure and the antipode
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Let us denote F∗ = F/{∅}:
Definition
The set F∗ can be naturally endorsed in an algebra HCEM as
follows:
∀ (u, v) ∈ F∗2 , ∀ (λ, µ) ∈ R2 , λu + µv ∈ HCEM ,
∀ (u, v) ∈ F∗2 , u v ∈ HCEM , where u v is the (commutative)
juxtaposition of the two forests u and v,
∀ u ∈ F∗ , u e = e u = u, where e =
is the unity element.
Maps b from T onto R with b( ) = 1 can be uniquely extended
to F∗ by requiring that they are algebra morphisms, so that
unital algebra maps from Alg(HCEM , R) are in one-to-one
correspondance with vector fields B-series B(f , b) satisfying
b(∅) = 0 and b( ) = 1.
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Definition
The tensor product of HCK with itself is the set of elements of
the form u ⊗ v with u and v in H, such that:
3
∀ (u, v, w ) ∈ HCEM
, (u + v) ⊗ w = u ⊗ w + v ⊗ w ,
3
∀ (u, v, w ) ∈ HCEM
, u ⊗ (v + w ) = u ⊗ v + u ⊗ w ,
2
∀ (u, v) ∈ HCEM
, ∀ λ ∈ R, λ · u ⊗ v = (λ · u) ⊗ v = u ⊗ (λ · v).
Remark
Using the definition of tensor products, the juxtaposition of
forests may also be written as a multiplication µ defined on
HCEM ⊗ HCEM :
2
∀ (u, v) ∈ HCEM
, u v = µ(u ⊗ v).
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The coproduct ∆CEM of Calaque, Ebrahimi-Fard and
Manchon is the only algebra map from HCEM to HCEM ⊗ HCEM
such that
X
∀t ∈ T , ∆CEM (t) =
χp ⊗ vp .
p∈P(t)
and the counit is defined by ǫ( ) = 1, ǫ(t) = 0 if t 6= .
∆CEM ( ) =
⊗
) =
⊗
+2
∆CEM ( ) =
⊗
+
∆CEM ( ) =
⊗
+2
∆CEM (
⊗
+
⊗
+
⊗
⊗
Table: Coproduct ∆CEM .
⊗
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The bialgebra HCEM is then graded with the number of edges
e(·) so that HCEM is a Hopf algebra and we have
Antipode
∀t ∈ T , S(t) = −t +
X
(−1)ν(p) upχp
p∈P(t)
where
p is an edge-coloured partition of t, i.e. ...
P(t) is the set of all edge-coloured partitions of t;
up is the forest made up of the trees formed by vertices in p
which are connected with edges of the same colour...
ν(p) is the number of colours in p;
χp is the skeleton of p...
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The substitution law ⋆ for B-series B(f , a) and B(f , b) (with
b(∅) = 0 and b( ) = 1) corresponds to
convolution product associated to ∆CEM
b ⋆ a = µR ◦ (a ⊗ b) ◦ ∆CEM .
In this situation, one can identify the group of B-series
satisfying b(∅) = 0 and b( ) = 1 as the group of characters of
the Hopf algebra HCEM , and hence one has the following
fundamental relation
b inv = b ◦ SCEM .
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THIS IS THE END
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