Resonance-based schemes for dispersive equations via decorated trees

2.1 Decorated trees and frequency interactions

We consider a set of decorated trees following the formalism developed in [Reference Bruned, Hairer and Zambotti13]. These trees will encode the Fourier coefficients of the numerical scheme.

We assume a finite set $ {\mathfrak {L}}$ and frequencies $ k_1,\ldots ,k_n \in {\mathbf {Z}}^{d} $ . The set $ {\mathfrak {L}}$ parametrises a set of differential operators with constant coefficients, whose symbols are given by the polynomials $ (P_{{\mathfrak {t}}})_{{\mathfrak {t}} \in {\mathfrak {L}}} $ . We define the set of decorated trees $ \hat {\mathcal {T}} $ as elements of the form $ T_{{\mathfrak {e}}}^{{\mathfrak {n}}, {\mathfrak {f}}} = (T,{\mathfrak {n}},{\mathfrak {f}},{\mathfrak {e}}) $ where

• • $ T $ is a nonplanar rooted tree with root $ \varrho _T $ , node set $N_T$ and edge set $E_T$ . We denote the leaves of $ T $ by $ L_T $ . $ T $ must also be a planted tree, which means that there is only one edge outgoing the root.

• • The map $ {\mathfrak {e}} : E_T \rightarrow {\mathfrak {L}} \times \lbrace 0,1\rbrace $ are edge decorations.

• • The map $ {\mathfrak {n}} : N_T \setminus \lbrace \varrho _T \rbrace \rightarrow \mathbf {N} $ are node decorations. For every inner node $ v$ , this map encodes a monomial of the form $ \xi ^{{\mathfrak {n}}(v)} $ where $ \xi $ is a time variable.

• • The map $ {\mathfrak {f}} : N_T \setminus \lbrace \varrho _T \rbrace \rightarrow {\mathbf {Z}}^{d}$ are node decorations. These decorations are frequencies that satisfy for every inner node $ u $ :

  (43) $$ \begin{align} (-1)^{\mathfrak{p}(e_u)}{\mathfrak{f}}(u) = \sum_{e=(u,v) \in E_T} (-1)^{\mathfrak{p}(e)} {\mathfrak{f}}(v) \end{align} $$
  where $ {\mathfrak {e}}(e) = ({\mathfrak {t}}(e),\mathfrak {p}(e)) $ and $ e_u $ is the edge outgoing $ u $ of the form $ (v,u) $ . From this definition, one can see that the node decorations $ ({\mathfrak {f}}(u))_{u \in L_T} $ determine the decoration of the inner nodes. We assume that the node decorations at the leaves are linear combinations of the $ k_i $ with coefficients in $ \lbrace -1,0,1 \rbrace $ .

• • We assume that the root of $ T $ has no decoration.

When the node decoration $ {\mathfrak {n}} $ is zero, we will denote the decorated trees $ T_{{\mathfrak {e}}}^{{\mathfrak {n}},{\mathfrak {f}}} $ as $ T_{{\mathfrak {e}}}^{{\mathfrak {f}}} = (T,{\mathfrak {f}},{\mathfrak {e}}) $ . The set of decorated trees satisfying such a condition is denoted by $ \hat {\mathcal {T}}_0 $ . We say that $ \bar T_{\bar {\mathfrak {e}}}^{\bar {\mathfrak {f}}} $ is a decorated subtree of $ T_{{\mathfrak {e}}}^{{\mathfrak {f}}} \in \hat {\mathcal {T}}_0 $ if $ \bar T $ is a subtree of $ T $ and the restriction of the decorations $ {\mathfrak {f}}, {\mathfrak {e}} $ of $ T $ to $ \bar T $ are given by $ \bar {\mathfrak {f}} $ and $ \bar {\mathfrak {e}} $ . Notice that because trees considered in this framework are always planted, we look only at subtrees that are planted. A planted subtree of $ T $ is of the form $ T_e $ where $ e \in E_T $ and $ T_e $ corresponds to the tree above $ e $ . The nodes of $ T_e $ are given by all of the nodes whose path to the root contains $ e $ .

Example 5. Below, we give an example of a decorated tree $ T_{{\mathfrak {e}}}^{{\mathfrak {n}}, {\mathfrak {f}}} $ where the edges are labelled with numbers from $ 1 $ to $ 7 $ and the set $ N_T \setminus \lbrace \varrho _T\rbrace $ is labelled by $\lbrace a,b,c,d,e,f,g \rbrace $ :

We denote by $\hat H $ (respectively $ \hat H_0 $ ) the (unordered) forests composed of trees in $ \hat {\mathcal {T}} $ (respectively $ \hat {\mathcal {T}}_0 $ ; including the empty forest denoted by ${\mathbf {1}}$ ). Their linear spans are denoted by $\hat {\mathcal {H}} $ and $ \hat {\mathcal {H}}_0 $ . We extend the definition of decorated subtrees to forests by saying that $ T $ is a decorated subtree of the decorated forest $ F $ if their exists a decorated tree $ \bar T$ in $ F $ such that $ T $ is a decorated subtree of $ \bar T $ . The forest product is denoted by $ \cdot $ and the counit is $ {\mathbf {1}}^{\star } $ which is nonzero only on the empty forest.

In order to represent these decorated trees, we introduce a symbolic notation. An edge decorated by $ o = ({\mathfrak {t}},{\mathfrak {p}}) $ is denoted by $ {\cal I}_{o} $ . The symbol $ {\cal I}_{o}(\lambda _{k}^{\ell } \cdot ) : \hat {\mathcal {H}} \rightarrow \hat {\mathcal {H}} $ is viewed as the operation that merges all of the roots of the trees composing the forest into one node decorated by $(\ell ,k) \in \mathbf {N} \times {\mathbf {Z}}^{d} $ . We obtain a decorated tree which is then grafted onto a new root with no decoration. If the condition (43) is not satisfied on the argument, then ${\cal I}_{o}( \lambda _{k}^{\ell } \cdot )$ gives zero. If $ \ell = 0 $ , then the term $ \lambda _{k}^{\ell } $ is denoted by $ \lambda _{k} $ as a shorthand notation for $ \lambda _{k}^{0} $ . When $ \ell = 1 $ , it will be denoted by $ \lambda _{k}^{1} $ . The forest product between $ {\cal I}_{o_1}( \lambda ^{\ell _1}_{k_1}F_1) $ and $ {\cal I}_{o_2}( \lambda ^{\ell _2}_{k_2}F_2) $ is given by

$$ \begin{align*} {\cal I}_{o_1}( \lambda^{\ell_1}_{k_1} F_1) {\cal I}_{o_2}( \lambda^{\ell_2}_{k_2} F_2) := {\cal I}_{o_1}( \lambda^{\ell_1}_{k_1} F_1) \cdot {\cal I}_{o_2}( \lambda^{\ell_2}_{k_2} F_2). \end{align*} $$

Example 6. The following symbol

$$ \begin{align*} {\cal I}_{({\mathfrak{t}}(1),{\mathfrak{p}}(1))}( \lambda^{{\mathfrak{n}}(a)}_{{\mathfrak{f}}(a)}{\cal I}_{({\mathfrak{t}}(2),{\mathfrak{p}}(2))}( \lambda^{{\mathfrak{n}}(b)}_{{\mathfrak{f}}(b)}){\cal I}_{({\mathfrak{t}}(3),{\mathfrak{p}}(3))}( \lambda^{{\mathfrak{n}}(c)}_{{\mathfrak{f}}(c)})) \end{align*} $$

encodes the tree

(44)

We will see later (in Example 7) that the above tree with suitably chosen decorations describes the first iterated integral of the Kortweg–de Vries equation (3).

We are interested in the following quantity which represents the frequencies associated to this tree:

(45) $$ \begin{align} \mathscr{F}( T_{{\mathfrak{e}}}^{{\mathfrak{f}}} ) = \sum_{u \in N_T} P_{({\mathfrak{t}}(e_u),{\mathfrak{p}}(e_u))}({\mathfrak{f}}(u)) \end{align} $$

where $ e_u $ is the edge outgoing $ u $ of the form $ (v,u) $ and

(46) $$ \begin{align} P_{({\mathfrak{t}}(e_u),{\mathfrak{p}}(e_u))}({\mathfrak{f}}(u)){ \, := \,} (-1)^{{\mathfrak{p}}(e_u)}P_{{\mathfrak{t}}(e_u)}((-1)^{{\mathfrak{p}}(e_u)}{\mathfrak{f}}(u)). \end{align} $$

The term $ \mathscr {F}( T_{{\mathfrak {e}}}^{{\mathfrak {f}}}) $ has to be understood as a polynomial in multiple variables given by the $ k_i $ .

In the numerical scheme, what matters are the terms with maximal degree of frequency, which are here the monomials of higher degree; cf. $\mathcal {L}_{\tiny {\text {dom}}}$ . We compute them using the symbolic notation in the next section. We assume fixed $ {\mathfrak {L}}_{+} \subset {\mathfrak {L}} $ . This subset encodes integrals in time that are of the form $ \int _0^{\tau } e^{s P_{(\mathfrak {t},\mathfrak {p})}(\cdot )} \cdots ds $ for $ (\mathfrak {t},\mathfrak {p}) \in {\mathfrak {L}}_+ \times \lbrace 0,1 \rbrace $ ; see also its interpretation given in (82).

2.2 Dominant parts of trees and physical space maps

Definition 2.2. Let $ P(k_1,\ldots ,k_n) $ a polynomial in the $ k_i $ . If the higher monomials of $ P $ are of the form

$$ \begin{align*} a \sum_{i=1}^{n} (a_i k_i)^{m}, \quad a_i \in { \lbrace 0,1 \rbrace}, \, a \in {\mathbf{Z}}, \end{align*} $$

then we define $ {\mathcal {P}}_{\tiny {\text {dom}}}(P) $ as

(47) $$ \begin{align} {\mathcal{P}}_{\tiny{\text{dom}}}(P) = a \left(\sum_{i=1}^{n} a_i k_i\right)^{m}. \end{align} $$

Otherwise, it is zero.

Remark 2.3. Given a polynomial $ P $ , one can compute its dominant part $ \mathcal {L}_{\text {dom}} $ and its lower part $\mathcal {L}_{\text {low}}$

$$ \begin{align*} \mathcal{L}_{\tiny{\text{dom}}} = {\mathcal{P}}_{\tiny{\text{dom}}}(P), \quad \mathcal{L}_{\tiny{\text{low}}} = \left({\mathrm{id}} - {\mathcal{P}}_{\tiny{\text{dom}}} \right)(P). \end{align*} $$

In our discretisation we will treat the dominant parts of the frequency interactions $\mathcal {L}_{\tiny {\text {dom}}}$ exactly, while approximating the lower-order parts $\mathcal {L}_{\tiny {\text {low}}} $ by Taylor series expansions (cf. also (12)). This will be achieved by applying recursively the operator $ {\mathcal {P}}_{\tiny {\text {dom}}} $ introduced in Definition 2.6.

Note that in the special case that ${\mathcal {P}}_{\tiny {\text {dom}}}(P) = 0$ , we have to expand all frequency interactions into a Taylor series. The latter, for instance, arises in the context of quadratic Schrödinger equations

(48) $$ \begin{align} i \partial_t u = -\Delta u + u^2, \quad u(0,x) = v(x) \end{align} $$

for which we face oscillations of type (cf. (10))

$$ \begin{align*} \int_0^\tau e^{i \xi \Delta } (e^{-i \xi \ \Delta} v)^2 d\xi & = \sum_{k_{1},k_{2}\in {\mathbf{Z}}^d}\hat{v}_{k_1}\hat{v}_{k_2} e^{i (k_1+k_2) x} \int_0^\tau e^{ -i \big(k_1+k_2\big)^2 \xi} e^{ i \big(k_1^2 + k_2^2 \big)\xi} d\xi \\ &=\sum_{k_{1},k_{2}\in {\mathbf{Z}}^d}\hat{v}_{k_1}\hat{v}_{k_2} e^{i (k_1+k_2) x} \int_0^\tau e^{ -2 i k_1 k_2 \xi} d\xi. \end{align*} $$

Here we recall the notation $ k \ell = k_1 \ell _1 + \ldots + k_d \ell _d $ for $k, \ell \in {\mathbf {Z}}^d$ . In contrast to the cubic NLS (21) where we have that (cf. (26))

$$ \begin{align*} { \mathcal{L}_{\text{dom}}(k_1) = 2k_1^2 \quad \text{and}\quad \mathcal{L}_{\text{low}}(k_1,k_2,k_3) = - 2 k_1 (k_2+k_3) + 2 k_2 k_3}, \end{align*} $$

we observe for the quadratic NLS (48) with the map $ {\mathcal {P}}_{\tiny {\text {dom}}} $ given in (47) that

$$ \begin{align*} P(k_1,k_2) & = - (k_1 + k_2)^2 + (k_1^2 + k_2^2) = - 2k_1 k_2, \quad {\mathcal{P}}_{\tiny{\text{dom}}}(P) = 0, \\ \mathcal{L}_{\tiny{\text{dom}}} & = {\mathcal{P}}_{\tiny{\text{dom}}}(P) =0, \quad \mathcal{L}_{\tiny{\text{low}}} = P - {\mathcal{P}}_{\tiny{\text{dom}}}(P) = - 2k_1 k_2. \end{align*} $$

Hence, although $\mathcal {L}= -\Delta $ and $\mathcal {L}_{\text {dom}}= 0 $ (which means that no oscillations are integrated exactly), we ‘only’ lose one derivative in the local approximation error (cf. (16)) as

$$ \begin{align*} \mathcal{O}\left(\tau^2 \mathcal{L}_{\text{low}}v\right)= \mathcal{O}\left(\tau^2 \vert \nabla \vert v\right). \end{align*} $$

Remark 2.4. Terms of type (47) will naturally arise when filtering out the dominant nonlinear frequency interactions in the PDE. We have to embed integrals over their exponentials into our discretisation. For their practical implementation it will therefore be essential to map fractions of (47) back to physical space.

Indeed, if we apply the inverse Fourier transform $ \mathcal {F}^{-1} $ , we get

(49) $$ \begin{align} \mathcal{F}^{-1} & \left( \sum_{\substack{0 \neq k=k_1 +\ldots +k_n\\ k_\ell \neq 0} } \frac{1}{(k_1 +\ldots +k_n)^m} \frac{1}{k_1^{m_1}} \ldots \frac{1}{k_n^{m_n}}v^1_{k_1}\ldots v^n_{k_n} e^{i kx} \right) \end{align} $$

$$\begin{align*}& = (-\Delta)^{- m/2} \prod_{\ell = 1}^n \left( (-\Delta)^{-m_\ell/2} v^\ell(x)\right) \end{align*}$$

where by abuse of notation we define the operator $(-\Delta )^{-1}$ in Fourier space as $(-\Delta )^{-1} f(x) = \sum _{ k \neq 0} \frac { \hat {f}_k }{k^2}e^{i k x}.$

In the next proposition, we elaborate on (49) and give a nice class of functions depending on the $k_i$ that we can map back to the physical space.

Proposition 2.5. Assume that we have polynomials $ Q $ in $ k_1, \ldots , k_n $ and $ k $ is a linear combination of the $k_i$ such that

$$ \begin{align*} Q & = \prod_j (\sum_{u \in V_j} a_{u,V_j} k_u )^{m_i}, \quad V_j \subset \lbrace 1,\ldots ,n \rbrace, \quad a_{u,V_j} \in \lbrace-1,1\rbrace, \\ k & = \sum_{u=1}^{n} a_u k_u, \quad a_u \in \lbrace -1,1 \rbrace, \end{align*} $$

where the $ V_i $ are either disjoint or if $ V_j \subset V_i $ we assume that there exist $ p_{i,j} $ such that

$$ \begin{align*} a_{u,V_i} = (-1)^{p_{i,j}} a_{u,V_j}, \quad u \in V_j. \end{align*} $$

We also suppose that the $ V_i $ are included in $ k $ in the sense that there exist $ p_{V_i} $ such that

$$ \begin{align*} a_u = (-1)^{p_{V_i}} a_{u,V_i}, \quad u \in V_i. \end{align*} $$

Then, one gets

$$ \begin{align*} \mathcal{F}^{-1} & \left( \sum_{ \substack{0 \neq k= a_1 k_1 +\ldots + a_n k_n\\ Q(k_1,\ldots ,k_n) \neq 0}} \frac{1}{Q} v^{1,a_1}_{k_1}\ldots v^{n,a_n}_{k_n} e^{i kx} \right)\\ & = \left(\prod_{V_i \subset V_j} (-1)^{p_{V_i}} (-\Delta)^{- m_i/2}_{V_i} \right) v^{1,a_1}\ldots v^{n,a_n} \end{align*} $$

where $ v^{i,1} = v^{i} $ and $ v^{i,-1} = \overline {v^{i}} $ . The operator $ (-\Delta )^{- m_i/2}_{V_i} $ acts only on the functions $\prod _{u \in V_i} v^{u,a_u} $ and the product starts by the smaller elements for the inclusion order.

Proof

We proceed by induction on the number of $ V_i $ . Let $ V_{\max } $ be an element among the $ V_i $ maximum for the inclusion order. Then, we get

$$ \begin{align*} \sum_{ \substack{0 \neq k= a_1 k_1 +\ldots + a_n k_n\\ Q(k_1,\ldots ,k_n) \neq 0}} & \frac{1}{Q} v^{1,a_1}_{k_1}\ldots v^{n,a_n}_{k_n} e^{i kx} = \sum_{\substack{0 \neq k= r + \ell \\ \ell \neq 0}} \frac{(-1)^{p_{V_{\max}}}}{\ell^{m_{\max}}}\sum_{ \substack{0 \neq r= \sum_{u \notin V_{\max}} a_u k_u \\ R \neq 0}} \frac{1}{R} \\ & \left( \prod_{j \notin V_{\max}} v^{j,a_j}_{k_j} \right) e^{i r x} \times \sum_{ \substack{0 \neq \ell= \sum_{u \in V_{\max}} a_u k_u \\ S \neq 0}} \frac{1}{S} \left( \prod_{j \in V_{\max}} v^{j,a_j}_{k_j} \right) e^{i \ell x} \end{align*} $$

where

$$ \begin{align*} S = \prod_{V_j \varsubsetneq V_{\max}} (\sum_{u \in V_j} a_{u,V_j} k_u )^{m_i}, \quad R = \prod_{V_j \cap V_{\max} = \emptyset} (\sum_{u \in V_j} a_{u,V_j} k_u )^{m_i}, \quad Q = R S \ell^{m_{max}}. \end{align*} $$

Thus, by applying the inverse Fourier transform, we get the term $ (-1)^{p_{V_{\max }}} (-\Delta )^{- m_{\max }/2}_{V_{\max }} $ from $ \frac {(-1)^{p_{V_{\max }}}}{\ell ^{m_{\max }}} $ . We conclude from the induction hypothesis on the two remaining sums.

The next definition will allow us to compute the dominant part of the frequency interactions of a given decorated forest in $ \hat H_0 $ . The idea is to filter out the dominant part using the operator $ {\mathcal {P}}_{\tiny {\text {dom}}} $ which selects the frequencies of highest order. The operator $ {\mathcal {P}}_{\tiny {\text {dom}}} $ only appears if we face an edge in $ {\mathfrak {L}}_+ $ which corresponds to an integral in time that we have to approximate.

Definition 2.6. We recursively define $\mathscr {F}_{\tiny {\text {dom}}}, \mathscr {F}_{\tiny {\text {low}}} : \hat H_{0} \rightarrow \mathbb {R}[{\mathbf {Z}}^d]$ as

$$ \begin{align*} \mathscr{F}_{\tiny{\text{dom}}}({\mathbf{1}}) = 0 \quad \mathscr{F}_{\tiny{\text{dom}}}(F \cdot \bar F) & =\mathscr{F}_{\tiny{\text{dom}}}(F) + \mathscr{F}_{\tiny{\text{dom}}}(\bar F) \\ \mathscr{F}_{\tiny{\text{dom}}}\left( {\cal I}_{({\mathfrak{t}},{\mathfrak{p}})}( \lambda_{k}F) \right) & = \left\{ \begin{array}{l} \displaystyle {\mathcal{P}}_{\tiny{\text{dom}}}\left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) +\mathscr{F}_{\tiny{\text{dom}}}(F) \right), \, \text{if } {\mathfrak{t}} \in {\mathfrak{L}}_+ , \\ P_{({\mathfrak{t}},{\mathfrak{p}})}(k) +\mathscr{F}_{\tiny{\text{dom}}}(F), \quad \text{otherwise} \\ \end{array} \right. \\ \mathscr{F}_{\tiny{\text{low}}} \left( {\cal I}_{({\mathfrak{t}},{\mathfrak{p}})}( \lambda_{k}F) \right) & = \left( {\mathrm{id}} - {\mathcal{P}}_{\tiny{\text{dom}}} \right) \left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) +\mathscr{F}_{\tiny{\text{dom}}}(F) \right). \end{align*} $$

We extend these two maps to $ \hat H $ by ignoring the node decorations $ {\mathfrak {n}} $ .

Remark 2.7. The definition of $ {\mathcal {P}}_{\tiny {\text {dom}}} $ can be adapted depending on what is considered to be the dominant part. For example, if for $ {\mathfrak {t}}_2 \in {\mathfrak {L}}_+$ , one has (cf. (9))

$$ \begin{align*} P_{({\mathfrak{t}}_2,p)}( \lambda) = \frac{1}{\varepsilon^{\sigma}} + F_{({\mathfrak{t}}_2,p)}( \lambda). \end{align*} $$

Then we can define the dominant part only depending on $ \varepsilon $ (see Example 5.3):

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}}\left( P_{({\mathfrak{t}}_2,p)}(k) \right)= \frac{1}{\varepsilon^{\sigma}}. \end{align*} $$

The definition considers only the case where one does not have terms of the form $ 1/\varepsilon ^{\sigma } $ . It is sufficient for covering many interesting examples.

In the following we compute the dominant part $ \mathscr {F}_{\text {dom}}$ for the underlying trees of the cubic Schrödinger (2) and KdV (3) equation.

Example 7 (KdV). We consider the decorated tree $ T $ given in Example 44, where we fix the following decorations:

$$ \begin{align*} {\mathfrak{p}}(1) & = {\mathfrak{p}}(2) = {\mathfrak{p}}(3) = 0, \quad {\mathfrak{t}}(2) = {\mathfrak{t}}(3) = {\mathfrak{t}}_1, \quad {\mathfrak{t}}(1) = {\mathfrak{t}}_2, \\[-3pt] {\mathfrak{f}}(b) & = k_1, \quad {\mathfrak{f}}(c) = k_2, \quad {\mathfrak{f}}(a) = k_1 + k_2, \quad P_{{\mathfrak{t}}_2}( \lambda) = \lambda^3 , \quad P_{{\mathfrak{t}}_1}( \lambda) = - \lambda^3. \end{align*} $$

Now, we suppose $ {\mathfrak {L}}_+ = \lbrace {\mathfrak {t}}_2 \rbrace $ and $ {\mathfrak {L}} = \lbrace {\mathfrak {t}}_1, {\mathfrak {t}}_2 \rbrace $ . Then the tree (44) takes the form

(50)

This tree corresponds to the first iterated integral for the KdV equation (3). In more formal notation, we denote this tree by

(51)

where a blue edge encodes $ (\mathfrak {t}_2,0) $ and a brown edge is used for $(\mathfrak {t}_1,0)$ . The frequencies are given on the leaves. The ones on the inner nodes are determined by those on the leaves. On the left-hand side, we have given the symbolic notation. Together with Definition 2.6, one gets

$$ \begin{align*} \mathscr{F}_{\tiny{\text{dom}}}(T) & = {\mathcal{P}}_{\tiny{\text{dom}}} \left( (k_1+k_2)^{3} - k_1^{3} - k_2^{3} \right) = 0\\ \mathscr{F}_{\tiny{\text{low}}} (T) & = (k_1+k_2)^{3} - k_1^{3} - k_2^{3} = 3k_1 k_2 (k_1+k_2). \end{align*} $$

The fact that $ \mathscr {F}_{\tiny {\text {dom}}}(T) $ is zero comes from the fundamental choice of definition of the operator $ {\mathcal {P}}_{\tiny {\text {dom}}} $ in Definition 2.2.

Example 8 (Cubic Schrödinger). Next we consider the symbol

$$ \begin{align*} {\cal I}_{({\mathfrak{t}}_2,0)}( \lambda_{-k_1+k_2+k_3}{\cal I}_{({\mathfrak{t}}_2,1)}( \lambda_{k_1}){\cal I}_{({\mathfrak{t}}_2,0)}( \lambda_{k_2}){\cal I}_{({\mathfrak{t}}_2,0)}( \lambda_{k_3})) \end{align*} $$

with $P_{{\mathfrak {t}}_2}( \lambda )= \lambda ^2$ , $P_{{\mathfrak {t}}_1}( \lambda ) = - \lambda ^2$ , $ {\mathfrak {L}}_+ = \lbrace {\mathfrak {t}}_2 \rbrace $ and $ {\mathfrak {L}} = \lbrace {\mathfrak {t}}_1, {\mathfrak {t}}_2 \rbrace $ which encodes the tree

(52)

This tree corresponds to the frequency interaction of the first iterated integral for the cubic Schrödinger equation (2). In a more formal notation, we denote this tree by

(53)

where a blue edge encodes $ (\mathfrak {t}_2,0) $ , a brown edge is used for $ (\mathfrak {t}_1,0) $ and a dashed brown edge is for $ (\mathfrak {t}_2,1) $ . With Definition 2.6, we get

$$ \begin{align*} \mathscr{F}_{\tiny{\text{dom}}}(T) & = {\mathcal{P}}_{\tiny{\text{dom}}} \left( (-k_1+k_2+k_3)^{2} + (-k_1)^{2} - k_2^{2} - k_3^2 \right) \\& = {\mathcal{P}}_{\tiny{\text{dom}}} \left( 2k_1^2 - 2k_1(k_2+k_3) + 2 k_2 k_3\right) = 2 k_1^2\\ \mathscr{F}_{\tiny{\text{low}}} (T)& = (-k_1+k_2+k_3)^{2} + (-k_1)^{2} - k_2^{2} - k_3^2 - 2k_1^2\\ & = - 2k_1(k_2+k_3) + 2 k_2 k_3. \end{align*} $$

The map $ \mathscr {F}_{\tiny {\text {dom}}}$ has a nice property regarding the tree inclusions given in the next proposition. This inclusive property will be important in practical computations; see also Remark 1.3 and the examples in Section 5.

For Proposition 2.8, we need an additional assumption.

With this assumption, for a decorated forest $ F = \prod _i T_i $ such that $ {\mathcal {P}}_{\tiny {\text {dom}}}\left (\mathscr {F}_{\tiny {\text {dom}}}(F) \right ) \neq 0 $ , one has the nice identity

(54) $$ \begin{align} {\mathcal{P}}_{\tiny{\text{dom}}}\left(\mathscr{F}_{\tiny{\text{dom}}}(F) \right) = {\mathcal{P}}_{\tiny{\text{dom}}}\left(\sum_i {\mathcal{P}}_{\tiny{\text{dom}}}\left(\mathscr{F}_{\tiny{\text{dom}}}(T_i) \right) \right). \end{align} $$

We will illustrate this property and give a counterexample in an example below.

Example 9. We give a counterexample in the setting of the Schrödinger equation to (54) when ${\mathcal {P}}_{\tiny {\text {dom}}}\left (\mathscr {F}_{\tiny {\text {dom}}}(F) \right ) = 0 $ . We consider the following forest $ F = \prod _{i=1}^3 T_i$ where the decorated trees $ T_i $ are given by

Then, we can check Assumption 1 for $ F $ . One has

$$ \begin{align*} \mathscr{F}_{\tiny{\text{dom}}}(T_1) & = - k_4^2, \quad\mathscr{F}_{\tiny{\text{dom}}}(T_2) = - k_5^2, \quad\mathscr{F}_{\tiny{\text{dom}}}(T_3) = - (-k_1 + k_2 + k_3)^2 + 2 k_1^2, \\ \mathscr{F}_{\tiny{\text{dom}}}(F) & = - k_4^2 - k_5^2 - (-k_1 + k_2 + k_3)^2 + 2 k_1^2 \end{align*} $$

and

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left( \mathscr{F}_{\tiny{\text{dom}}}(T_1) \right) = - k_4^2, \quad {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T_2) \right) = - k_5^2, \quad {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T_3) \right) = 0. \end{align*} $$

Therefore, one obtains

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left( \sum_{i=1}^3 {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T_i) \right) \right) = - (k_4 + k_5)^2. \end{align*} $$

But, on the other hand,

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(F) \right) = 0. \end{align*} $$

Proposition 2.8. Let $ T_{{\mathfrak {e}}}^{{\mathfrak {f}}} $ be a decorated tree in $ \hat H_0 $ and $ e \in E_T $ . We recall that $ T_e $ corresponds to the subtree of $ T $ above $ e $ . The nodes of $ T_e $ are given by all of the nodes whose path to the root contains $ e $ . Under Assumption 1 one has

(55) $$ \begin{align} \begin{split} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T_{{\mathfrak{e}}}^{{\mathfrak{f}}}) \right) & = a \left( \sum_{u \in V } a_u k_u \right)^{m}, \quad m \in \mathbf{N}, a \in {\mathbf{Z}}, \, V \subset L_T, \, a_u \in \lbrace -1,1 \rbrace \\ {\mathcal{P}}_{\tiny{\text{dom}}} \left( \mathscr{F}_{\tiny{\text{dom}}}((T_e)_{{\mathfrak{e}}}^{{\mathfrak{f}}}) \right) & = b \left( \sum_{u \in \bar V } b_u k_u \right)^{m_e}, \quad m_e \in \mathbf{N}, b \in {\mathbf{Z}}, \, \bar V \subset L_T, \, b_u \in \lbrace -1,1 \rbrace \end{split} \end{align} $$

and $\bar V \subset V \text { or } \bar V \cap V = \emptyset $ . If $ \bar V \subset V $ , then there exists $ \bar p \in \lbrace 0,1 \rbrace $ such that $ a_u = (-1)^{\bar p} b_u $ for every $ u \in \bar V $ .

Proof

We proceed by induction over the size of the decorated trees. We consider $T= {\cal I}_{({\mathfrak {t}},{\mathfrak {p}})}( \lambda _{k} F) $ where $ F = \prod _{i=1}^m T_i $ .

(i) If $F ={\mathbf {1}}$ , then one has

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T) \right) = {\mathcal{P}}_{\tiny{\text{dom}}} \left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) \right). \end{align*} $$

We conclude from the definition of $ {\mathcal {P}}_{\tiny {\text {dom}}} $ .

(ii) If $ m> 2 $ , then one gets

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T) \right) = {\mathcal{P}}_{\tiny{\text{dom}}} \left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) + \sum_{i=1}^m \mathscr{F}_{\tiny{\text{dom}}}(T_i) \right). \end{align*} $$

Using Assumption 1, one obtains

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T) \right) = {\mathcal{P}}_{\tiny{\text{dom}}} \left( \sum_{i=1}^m (P_{({\mathfrak{t}},{\mathfrak{p}})}(k^{(i)}) +\mathscr{F}_{\tiny{\text{dom}}}(T_i)) \right) \end{align*} $$

where $ k^{(i)} $ corresponds to the frequency attached to the node connected to the root of $ T_i $ . If $ {\mathcal {P}}_{\tiny {\text {dom}}} \left (\mathscr {F}_{\tiny {\text {dom}}}(T) \right ) \neq 0 $ , then from (54) we have

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T) \right) = {\mathcal{P}}_{\tiny{\text{dom}}} \left( \sum_{i=1}^m {\mathcal{P}}_{\tiny{\text{dom}}} \left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k^{(i)}) +\mathscr{F}_{\tiny{\text{dom}}}(T_i) \right) \right). \end{align*} $$

We apply the induction hypothesis to each decorated tree $ \tilde T_i = {\cal I}_{({\mathfrak {t}},{\mathfrak {p}})}( \lambda _{k^{(i)}} T_i) $ and we recombine the various terms in order to conclude.

(iii) If $ m=1 $ , then $ F = {\cal I}_{({\mathfrak {t}}_1,{\mathfrak {p}}_1)}( \lambda _{ \bar k} T_1) $ where $ \bar k $ is equal to $ k $ up to a minus sign. We can assume without loss of generality that

(56) $$ \begin{align} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(F) \right) = \mathscr{F}_{\tiny{\text{dom}}}(F). \end{align} $$

Indeed, otherwise,

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left( \mathscr{F}_{\tiny{\text{dom}}}(T)\right) = {\mathcal{P}}_{\tiny{\text{dom}}} \left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) + P_{({\mathfrak{t}}_1,{\mathfrak{p}}_1)}(\bar k) + \mathscr{F}_{\tiny{\text{dom}}}(T_1) \right). \end{align*} $$

We can see $ P_{({\mathfrak {t}},{\mathfrak {p}})}(k) + P_{({\mathfrak {t}}_1,{\mathfrak {p}}_1)}(\bar k) $ as a polynomial and then apply the induction hypothesis. We are down to the case (56) and we consider

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}}\left(\mathscr{F}_{\tiny{\text{dom}}}(T) \right) & = {\mathcal{P}}_{\tiny{\text{dom}}}\left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) + {\mathcal{P}}_{\tiny{\text{dom}}}\left(\mathscr{F}_{\tiny{\text{dom}}}(\bar T) \right) \right) \end{align*} $$

where now $ F = \bar T$ is just a decorated tree. We apply the induction hypothesis on $ \bar T $ and we get

(57) $$ \begin{align} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(\bar T) \right) & = a \left( \sum_{u \in V } a_u k_u \right)^{m}, \quad a \in {\mathbf{Z}}, \, V \subset L_T, \, a_u \in \lbrace -1,1 \rbrace \\k & = \sum_{u \in L_T } c_u k_u, \quad c_{ u} = (-1)^{ p} a_{u} , \quad u \in V. \notag \end{align} $$

If the degree of $P_{({\mathfrak {t}},{\mathfrak {p}})}(k)$ is higher than the degree of $\mathscr {F}_{\tiny {\text {dom}}}(\bar T)$ , we obtain that

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) +\mathscr{F}_{\tiny{\text{dom}}}(\bar T) \right) = {\mathcal{P}}_{\tiny{\text{dom}}} \left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) \right). \end{align*} $$

On the other hand, if the degree of $P_{({\mathfrak {t}},{\mathfrak {p}})}(k)$ is lower than the degree of $\mathscr {F}_{\tiny {\text {dom}}}(\bar T)$ ,

$$ \begin{align*} {\mathcal{P}}_{\tiny{\text{dom}}} \left( P_{({\mathfrak{t}},{\mathfrak{p}})}(k) +\mathscr{F}_{\tiny{\text{dom}}}(\bar T) \right) = {\mathcal{P}}_{\tiny{\text{dom}}} \left( \mathscr{F}_{\tiny{\text{dom}}}(\bar T) \right). \end{align*} $$

If $ P_{({\mathfrak {t}},{\mathfrak {p}})}(k)$ and $\mathscr {F}_{\tiny {\text {dom}}}(\bar T)$ have the same degree $ m $ , we get using the definition of $ P_{({\mathfrak {t}},{\mathfrak {p}})}(k)$ in (46) as well as the induction hypothesis on $\bar T$ given in (57) that

$$ \begin{align*} P_{({\mathfrak{t}},{\mathfrak{p}})}(k) + {\mathcal{P}}_{\tiny{\text{dom}}}\left(\mathscr{F}_{\tiny{\text{dom}}}(\bar T) \right) & = \sum_{u \in V } \left( a (-1)^{p + m + {\mathfrak{p}}} + (- 1)^{{\mathfrak{p}}} \right)( (-1)^{{\mathfrak{p}}}c_u k_u)^{m} \\ & \quad{}+ \sum_{u \in L_T \setminus V } (- 1)^{{\mathfrak{p}}} ((-1)^{{\mathfrak{p}}} c_u k_u)^{m} + R \end{align*} $$

where $ R $ terms of lower orders.

By applying the map $ {\mathcal {P}}_{\tiny {\text {dom}}} $ defined in (47), we thus obtain an expression of the form

(58) $$ \begin{align} {\mathcal{P}}_{\tiny{\text{dom}}} \left(\mathscr{F}_{\tiny{\text{dom}}}(T) \right) = b \left( \sum_{u \in \tilde{V} } (-1)^{{\mathfrak{p}}} c_u k_u \right)^{m} \,\text{for some } b \in {\mathbf{Z}} \end{align} $$

where $ \tilde V $ could be either $ L_T $ or $ L_T \setminus V $ .

Let $ e \in E_T $ , $ T_e \neq T $ , then $ T_e $ is a subtree of $ \bar T $ . By the induction hypothesis, one obtains (55), meaning that if we denote by $ V $ (respectively $ \bar V $ ) the set associated to $ \bar T $ (respectively $ T_e $ ), we get $ \bar V \subset V $ or $ \bar V \cap V = \emptyset $ .

In the first case, $\bar V \subset V $ , the assertion follows as $ V \subset \tilde {V} $ or $ V \cap \tilde {V} = \emptyset $ such that necessarily $\bar V \subset \tilde V$ or $\bar V \cap \tilde V = \emptyset $ .

In the second case, $ \bar V \cap V = \emptyset $ , we apply the induction hypothesis on $ T_e $ . Then, for $ v $ of $ T_e $ , there exists $ p $ such that the decoration $ {\mathfrak {f}}(v) $ is given by

$$ \begin{align*} {\mathfrak{f}}(v) = \sum_{u \in L_{T_v} } d_u k_u, \quad d_{ u} = (-1)^{ p} b_{u} , \quad u \in \bar V. \end{align*} $$

As $ {\mathfrak {f}}(v) $ appears as a subfactor in $ k $ , one has $ \bar V \subset L_T $ . Then, $ \bar {V} \cap V = \emptyset $ also gives that $ \bar V \subset L_T \setminus V $ . Therefore, we have $ \bar V \subset \tilde {V} $ , which concludes the proof.

Corollary 2.9. Let $ T_{{\mathfrak {e}}}^{{\mathfrak {f}}} $ a decorated tree in $ \hat {\mathcal {T}}_0 $ . We assume that Assumption 1 holds true for a set $ A $ of decorated subtrees of $ T_{{\mathfrak {e}}}^{{\mathfrak {f}}} $ such that ${\mathcal {P}}_{\tiny {\text {dom}}} \left (\mathscr {F}_{\tiny {\text {dom}}}(\bar T) \right ) =\mathscr {F}_{\tiny {\text {dom}}}(\bar T) \neq 0$ for $ \bar T \in A $ . Moreover, we assume that the $ \bar T $ are of the form $ (T_e)_{{\mathfrak {e}}}^{{\mathfrak {f}}} $ where $ e \in E_T $ . Then, the following product

$$ \begin{align*} \prod_{\bar T\kern-1pt \in A} \frac{1}{\left(\mathscr{F}_{\tiny{\text{dom}}}(\bar T) \right)^{m_T}} \end{align*} $$

can be mapped back to physical space using operators of the form $(- \Delta )^{-m/2}_V $ as defined in Proposition 2.5.

Example 10. We illustrate Corollary 2.9 via an example extracted from the cubic Schrödinger equation (2). We consider the following decorated trees:

(59)

We observe that these trees satisfy Assumption 1 and that $ T_1 $ is a subtree of $ T_2 $ . One has that

$$ \begin{align*} \mathscr{F}_{\tiny{\text{dom}}}( T_1) = 2 k_1^2 , \quad\mathscr{F}_{\tiny{\text{dom}}}( T_2) = 2 (k_1 + k_4)^2 \end{align*} $$

and the following quantity can be mapped back into physical space:

$$ \begin{align*} &\sum_{\substack{k= -k_1-k_4+k_2+k_3+k_5 \\ k_1\neq 0, k_1+k_4\neq 0 \\k_1,k_2,k_3,k_4,k_5\in {\mathbf{Z}}^d}} \frac{1}{ \mathscr{F}_{\tiny{\text{dom}}}( T_1)} \frac{1}{ \mathscr{F}_{\tiny{\text{dom}}}( T_2)} \overline{\hat{v}_{k_1}} \overline{\hat{v}_{k_4}}\hat{v}_{k_2}\hat{v}_{k_3}\hat{v}_{k_5} e^{i k x}\\ &\quad= \sum_{\substack{k= -k_1-k_4+k_2+k_3+k_5\\ k_1\neq 0, k_1+k_4\neq 0 \\k_1,k_2,k_3,k_4,k_5\in {\mathbf{Z}}^d}} \frac{1}{4 (k_1^2) (k_1 + k_4)^2}\overline{\hat{v}_{k_1}} \overline{\hat{v}_{k_4}}\hat{v}_{k_2}\hat{v}_{k_3}\hat{v}_{k_5} e^{i k x} \\ &\quad = \frac14 v(x)^3(-\Delta)^{-1}\left(\overline{v}(x) (-\Delta)^{-1} \overline{v}(x)\right). \end{align*} $$

2.3 Approximated decorated trees

We denote by $ {\mathcal {T}} $ the set of decorated trees $ T_{{\mathfrak {e}},r}^{{\mathfrak {n}},{\mathfrak {f}}} = (T,{\mathfrak {n}},{\mathfrak {f}},{\mathfrak {e}},r) $ where

• • $ T_{{\mathfrak {e}}}^{{\mathfrak {n}},{\mathfrak {f}}} \in \hat {\mathcal {T}} $ .

• • The decoration of the root is given by $ r \in {\mathbf {Z}} $ , $ r \geq -1 $ such that

  (60) $$ \begin{align} r +1 \geq \deg(T_{{\mathfrak{e}}}^{{\mathfrak{n}},{\mathfrak{f}}}) \end{align} $$
  where $ \deg $ is defined recursively by
  $$ \begin{align*} \deg({\mathbf{1}}) & = 0, \quad \deg(T_1 \cdot T_2 ) = \max(\deg(T_1),\deg(T_2)), \\ \deg({\cal I}_{({\mathfrak{t}},{\mathfrak{p}})}( \lambda^{\ell}_{k}T_1) ) & = \ell + {\mathbf{1}}_{\lbrace{\mathfrak{t}} \in {\cal L}_+\rbrace} + \deg(T_1) \end{align*} $$
  where $ {\mathbf {1}} $ is the empty forest and $ T_1, T_2 $ are forests composed of trees in $ {\mathcal {T}} $ . The quantity $\deg (T_{{\mathfrak {e}}}^{{\mathfrak {n}},{\mathfrak {f}}})$ is  the maximum number of edges with type in $ {\mathfrak {L}}_+ $ and node decorations $ {\mathfrak {n}} $ lying on the same path from one leaf to the root.

We call decorated trees in $ {\mathcal {T}} $ approximated decorated trees. The main difference with decorated trees introduced before is the adjunction of the decoration $ r $ at the root. The idea is that these trees correspond to different analytical objects. We summarise this below:

$$ \begin{align*} T_{{\mathfrak{e}}}^{{\mathfrak{n}},{\mathfrak{f}}} & \in \hat{{\mathcal{T}}} \equiv \text{Iterated integral} \\ T_{{\mathfrak{e}},r}^{{\mathfrak{n}},{\mathfrak{f}}} & \in \hat{{\mathcal{T}}} \equiv \text{Approximation of an iterated integral of order } r. \end{align*} $$

The interpretation (83) of approximated decorated trees gives the numerical scheme; see Definition 4.4.

Example 11. We continue Example 6 with the decorated tree $ T_{{\mathfrak {e}}}^{{\mathfrak {n}},\mathfrak {f}} $ given in (44). We suppose that $ {\mathfrak {t}}(1) $ is in $ \mathfrak {L}_+ $ but not $ {\mathfrak {t}}(2) $ and ${\mathfrak {t}}(3) $ . We are now in the context of Example 7. Then, one has

$$ \begin{align*} \deg(T_{{\mathfrak{e}}}^{{\mathfrak{n}},{\mathfrak{f}}}) = {\mathfrak{n}}(a) + 1+ \max({\mathfrak{n}}(b),{\mathfrak{n}}(c)). \end{align*} $$

We denote by $ {\mathcal {H}} $ the vector space spanned by forests composed of trees in $ {\mathcal {T}}$ and $ \lambda ^n $ , $ n \in \mathbf {N} $ where $ \lambda ^n $ is the tree with one node decorated by $ n $ . When the decoration $ n $ is equal to zero, we identify this tree with the empty forest: $ \lambda ^{0} ={\mathbf {1}} $ . Using the symbolic notation, one has

$$ \begin{align*} {\mathcal{H}} = \langle \lbrace \prod_j \lambda^{m_j} \prod_i {\cal I}^{r_i}_{o_i}( \lambda_{k_i}^{\ell_i} F_i), \, {\cal I}_{o_i}( \lambda_{k_i}^{\ell_i} F_i) \in \hat {\mathcal{T}} \rbrace \rangle \end{align*} $$

where the product used is the forest product. We call decorated forests in $ {\mathcal {H}} $ approximated decorated forests. The map $ {\cal I}^{r}_{o}( \lambda _{k}^{\ell } \cdot ) : \hat {\mathcal {H}} \rightarrow {\mathcal {H}} $ is defined the same as for $ {\cal I}_{o}( \lambda _{k}^{\ell } \cdot ) $ except now the root is decorated by $ r $ and it could be zero if the inequality (60) is not satisfied. We extend this map to $ {\mathcal {H}} $ by

$$ \begin{align*} {\cal I}^{r}_{o}( \lambda_{k}^{\ell} (\prod_j \lambda^{m_j} \prod_i {\cal I}^{r_i}_{o_i}( \lambda_{k_i}^{\ell_i} F_i))) { \, := }{\cal I}^{r}_{o}( \lambda_{k}^{\ell+\sum_j m_j} (\prod_i {\cal I}_{o_i}( \lambda_{k_i}^{\ell_i} F_i))). \end{align*} $$

In the extension, we remove the decorations $ r_i $ and we add up the decorations $ m_j $ with $ \ell $ . In the sequel, we will use a recursive formulation and move from $ \hat {\mathcal {H}} $ to $ {\mathcal {H}} $ . Therefore, we define the map $ {\cal D}^{r} : \hat {\mathcal {H}} \rightarrow {\mathcal {H}} $ which replaces the root decoration of a decorated tree by $ r $ and performs the projection along the identity (60). It is given by

(61) $$ \begin{align} {\cal D}^{r}({\mathbf{1}})= {\mathbf{1}}_{\lbrace 0 \leq r+1\rbrace} , \quad {\cal D}^r\left( {\cal I}_{o}( \lambda_{k}^{\ell} F) \right) = {\cal I}^{r}_{o}( \lambda_{k}^{\ell} F) \end{align} $$

and we extend it multiplicatively to any forest in $ \hat {\mathcal {H}} $ . The map $ {\cal D}^r $ projects according to the order of the scheme $ r $ . We disregard decorated trees having a degree bigger than $ r $ .

We denote by $ {\mathcal {T}}_{+} $ the set of decorated forests composed of trees of the form $ (T,{\mathfrak {n}},{\mathfrak {f}},{\mathfrak {e}},(r,m)) $ where

• • $ T_{{\mathfrak {e}},r}^{{\mathfrak {n}},{\mathfrak {f}}} \in {\mathcal {T}} $ .

• • The edge connecting the root has a decoration of the form $ ({\mathfrak {t}},{\mathfrak {p}}) $ where $ {\mathfrak {t}} \in {\mathfrak {L}}_+ $ .

• • The decoration $ (r,m) $ is at the root of $ T $ and $ m \in \mathbf {N} $ is such that $ m \leq r +1 $ .

The linear span of $ {\mathcal {T}}_+ $ is denoted by $ {\mathcal {H}}_+ $ . One can observe that the main difference with ${\mathcal {H}}$ is that $ \lambda ^m \notin {\mathcal {T}}_+ $ . We can define the same grafting operator as before $ {\cal I}^{(r,m)}_{o}( \lambda _{k}^{\ell } \cdot ) : {\mathcal {H}} \rightarrow {\mathcal {H}}_+ $ the same as $ {\cal I}^{r}_{o}( \lambda _{k}^{\ell } \cdot ) : {\mathcal {H}} \rightarrow {\mathcal {H}} $ but now we add the decoration $ (r,m) $ at the root where $ m \leq r+1 $ . We also define $ \hat {\cal D}^{(r,m)}: \hat {\mathcal {H}} \rightarrow {\mathcal {H}}_+ $ the same as $ {\cal D}^{r} $ . It is given by

(62) $$ \begin{align} \hat {\cal D}^{(r,m)}({\mathbf{1}})= {\mathbf{1}}, \quad \hat {\cal D}^{(r,m)}\left( {\cal I}_{o}( \lambda_{k}^{\ell} F) \right) = {\cal I}^{(r,m)}_{o}( \lambda_{k}^{\ell} F). \end{align} $$

Example 12. For the tree (44) we obtain when applying $ {\cal D}^r $ and $ \hat {\cal D}^{(r,m)} $

(63)

For the decorated tree in (53), one obtains

(64)

2.4 Operators on approximated decorated forests

In the subsection, we introduce two maps $ \Delta $ and $ \Delta ^{\!+} $ that will act on approximated decorated forests, splitting them into two parts by the use of the tensor product. They act at two levels. First, on the shapes of the trees, they extract a subtree at the root. Then, they induce subtle changes in the decorations that could be interpreted as abstract Taylor expansions.

We define a map $\Delta : {\mathcal {H}} \rightarrow {\mathcal {H}} \otimes {\mathcal {H}}_+$ for a given $T^{{\mathfrak {n}},{\mathfrak {f}}}_{{\mathfrak {e}},r} \in {{\mathcal T}}$ by

(65) $$ \begin{align} \Delta T^{{\mathfrak{n}},{\mathfrak{f}}}_{{\mathfrak{e}},r} & = \sum_{A \in {\mathfrak{A}}(T) } \sum_{{\mathfrak{e}}_A} \frac1{{\mathfrak{e}}_A!} (A,{\mathfrak{n}} + \pi{\mathfrak{e}}_A, {\mathfrak{f}}, {\mathfrak{e}},r)\\ & \otimes \prod_{e \in \partial(A,T)}( T_{e}, {\mathfrak{n}} , {\mathfrak{f}}, {\mathfrak{e}}, (r-\deg(e),{\mathfrak{e}}_A(e))), \notag \\ & = \sum_{A \in {\mathfrak{A}}(T) } \sum_{{\mathfrak{e}}_A} \frac1{{\mathfrak{e}}_A!} A^{{\mathfrak{n}} +\pi{\mathfrak{e}}_A, {\mathfrak{f}} }_{{\mathfrak{e}},r} \otimes \prod_{e \in \partial(A,T)} (T_{e})^{{\mathfrak{n}} , {\mathfrak{f}}}_{{\mathfrak{e}}, (r-\deg(e),{\mathfrak{e}}_A(e))} \notag\end{align} $$

where we use the following notation:

• • We write $T_e $ as the planted tree above the edge $ e $ in $ T $ . For $g : E_T \rightarrow \mathbf {N}$ , we define for every $x \in N_T$ , $(\pi g)(x) = \sum _{e=(x,y) \in E_T} g(e)$ .

• • In $ A^{{\mathfrak {n}} +\pi {\mathfrak {e}}_A, {\mathfrak {f}} }_{{\mathfrak {e}},r} $ , the maps $ {\mathfrak {n}}, {\mathfrak {f}} $ and $ {\mathfrak {e}} $ are restricted to $ N_A $ and $ E_A $ . The same is valid for $ (T_{e})^{{\mathfrak {n}} , {\mathfrak {f}}}_{{\mathfrak {e}}, (r-\deg (e),{\mathfrak {e}}_A(e))} $ where the restriction is on $ N_{T_e} \setminus \lbrace \varrho _{T_e}\rbrace $ and $ E_{T_e} $ , $ \varrho _{T_e} $ is the root of $ T_e $ . When $ A $ is reduced to a single node, we set $ A^{{\mathfrak {n}} +\pi {\mathfrak {e}}_A, {\mathfrak {f}} }_{{\mathfrak {e}},r} = \lambda ^{{\mathfrak {n}} +\pi {\mathfrak {e}}_A} $ .

• • The first sum runs over ${\mathfrak {A}}(T)$ , the set of all subtrees A of T containing the root $ \varrho $ of $ T $ . The second sum runs ${\mathfrak {e}}_{A} : \partial (A,T) \rightarrow \mathbf {N}$ where $\partial (A,T)$ denotes the edges in $E_T \setminus E_A$ of type in $ {\mathfrak {L}}_+ $ that are adjacent to $N_A$ .

• • Factorial coefficients are understood in multi-index notation.

• • We define $ \deg (e) $ for $ e \in E_T $ as the number of edges having $ {\mathfrak {t}}(e) \in {\mathfrak {L}}_+ $ lying on the path from $ e $ to the root in the decorated tree $ T_{{\mathfrak {e}}}^{{\mathfrak {n}}, {\mathfrak {f}}} $ . We also add up the decoration $ {\mathfrak {n}} $ on this path.

Let us comment briefly that the play on decorations can be interpreted as asbtract Taylor expansions. We can use the following dictionary:

$$ \begin{align*} (T_e)_{{\mathfrak{e}}}^{{\mathfrak{n}},{\mathfrak{f}}} \equiv \int_{0}^{\tau} e^{i\xi P(k)} f(k_1,\ldots ,k_n,\xi) d\xi \end{align*} $$

where $ k_1,\ldots ,k_n $ are the frequencies appearing on the leaves of $ (T_e)_{{\mathfrak {e}}}^{{\mathfrak {n}},{\mathfrak {f}}}$ , $ P $ is the polynomial associated to the decoration of the edge $ e $ and $ k $ is the frequency on the node connected to the root. This iterated integral appears inside the iterated integral associated to $ T_{{\mathfrak {e}}}^{{\mathfrak {n}},{\mathfrak {f}}} $ . For our numerical approximation, we need to approximate this integral by giving a scheme of the form

$$ \begin{align*} (T_e)_{{\mathfrak{e}}, \tilde{r}}^{{\mathfrak{n}},{\mathfrak{f}}} \equiv \sum_{\ell \leq \tilde{r}} \frac{\tau^{\ell}}{\ell!} f_{\tilde{r},\ell}(k_1,\ldots ,k_n), \quad (T_{e})^{{\mathfrak{n}} , {\mathfrak{f}}}_{{\mathfrak{e}}, (\tilde{r},\ell)} \equiv f_{\tilde{r},\ell}(k_1,\ldots ,k_n) \end{align*} $$

where the order of the expansion is given by $ \tilde {r} = r - \deg (e) $ . Then, the $ \tau ^{\ell } $ are part of the original iterated integrals and cannot be detached as the $ f_{\tilde {r},\ell } $ . That is why we increase the polynomial decorations where the tree $ T_e $ was originally attached via the term $ {\mathfrak {n}} +\pi {\mathfrak {e}}_A $ . The choice for $ \tilde {r} $ is motivated by the fact that we do not need to go too far for the approximation of $ (T_e)_{{\mathfrak {e}}}^{{\mathfrak {n}},{\mathfrak {f}}} $ . We take into account all of the time integrals and polynomials which lie on the path connecting the root of $ T_e $ to the root of $ T $ .

The map $ \Delta $ is compatible with the projection induced by the decoration $ r $ . Indeed, one has

$$ \begin{align*} \deg(T_{{\mathfrak{e}}}^{{\mathfrak{n}}, {\mathfrak{f}}}) = \max_{e \in E_T} \left( \deg( (T_e)_{{\mathfrak{e}}}^{{\mathfrak{n}}, {\mathfrak{f}}} ) + \deg(e) \right). \end{align*} $$

Therefore, if $ \deg (T_{{\mathfrak {e}}}^{{\mathfrak {n}}, {\mathfrak {f}}}) < r $ , then for every $ e \in E_T $

$$ \begin{align*} \deg( (T_e)_{{\mathfrak{e}}}^{{\mathfrak{n}}, {\mathfrak{f}}} ) + \deg(e) < r. \end{align*} $$

We deduce that if $ T^{{\mathfrak {n}},{\mathfrak {f}}}_{{\mathfrak {e}},r} $ is zero, then the $ (T_e)_{{\mathfrak {e}},(r-\deg (e),{\mathfrak {e}}_A(e))}^{{\mathfrak {n}}, {\mathfrak {f}}} $ are zero, too.

We define a map ${\Delta ^{\!+}} : {\mathcal {H}}_+ \rightarrow {\mathcal {H}}_+ \otimes {\mathcal {H}}_+$ given for $T^{{\mathfrak {n}}, {\mathfrak {f}}}_{{\mathfrak {e}},(r,m)} \in {{\mathcal T}}_+$ by

(66) $$ \begin{align} {\Delta^{\!+}} T^{{\mathfrak{n}},{\mathfrak{f}}}_{{\mathfrak{e}},(r,m)} = \sum_{A \in {\mathfrak{A}}(T) } \sum_{{\mathfrak{e}}_A} \frac1{{\mathfrak{e}}_A!} A^{{\mathfrak{n}} +\pi{\mathfrak{e}}_A, {\mathfrak{f}} }_{{\mathfrak{e}},(r,m)} \otimes \prod_{e \in \partial(A,T)} (T_{e})^{{\mathfrak{n}} , {\mathfrak{f}}}_{{\mathfrak{e}}, (r-\deg(e),{\mathfrak{e}}_A(e))}. \end{align} $$

We require that $ A^{{\mathfrak {n}} +\pi {\mathfrak {e}}_A, {\mathfrak {f}} }_{{\mathfrak {e}},(r,m)} \in {\mathcal {H}}_+ $ , so we implicitly have a projection on zero when $ A $ happens to be a single node with $ {\mathfrak {n}} +\pi {\mathfrak {e}}_A \neq 0 $ . We illustrate this coproduct on a well-chosen example.

Example 13. We continue with the tree in Example 5. We suppose that $ {\mathfrak {L}}_+ = \lbrace {\mathfrak {t}}(2), {\mathfrak {t}}(3), {\mathfrak {t}}(4), {\mathfrak {t}}(5) \rbrace $ . Below, the subtree $ A \in {\mathfrak {A}}(T)$ is coloured in blue. We have $ N_A = \lbrace \varrho , a,b\rbrace $ , $ E_A = \lbrace 1,2 \rbrace $ and $ \partial (A,T) = \lbrace 3, 4,5 \rbrace $ .

We also get

$$ \begin{align*} \deg(4) = \deg(5) = 1 + {\mathfrak{n}}(a) + {\mathfrak{n}}(b), \quad \deg(3) = {\mathfrak{n}}(a). \end{align*} $$

We have for a fixed $ {\mathfrak {e}}_A : \partial (A,T) \rightarrow \mathbf {N} $ :

where $ \bar r = r - {\mathfrak {n}}(a) - {\mathfrak {n}}(b) -1 $ . Now if $ A $ is just equal to the root of $ T $ , then one gets $ N_A = \lbrace \varrho \rbrace $ , $ E_A = \emptyset $ and $ \partial (A,T) = \lbrace 1\rbrace $ as illustrated below:

(67)

The map $ {\Delta ^{\!+}} $ behaves the same way except that we start with a tree decorated by $ (r,m) $ at the root and we exclude the case described in (67).