# The derivation problem for one type of bimodules over group algebras

Andronick Arutyunov  Alexey Naianzin

###### Abstract

The derivation problem is a familiar one concerning group algebras,
particularly $L_{1}(G)$ and von Neumann algebras. In this paper, we study the
Banach bimodule $\ell_{p}(G)$, which is generated by the $\ell_{p}$ norm over
a specific class of groups with well-organized conjugacy classes. For this
case, we will demonstrate that all $\ell_{p}(G)$ derivations are inner.

##  1 Introduction

This article is devoted to the study of derivations in bimodules over group
rings equipped with various types of norms. The research mainly focuses on
classes of groups that act in some way nicely on their conjugacy graph. By
employing an approach proposed in earlier works, it is demonstrated that for
this class of groups all derivations are inner.

Before providing the historical background and formulating the problem, let us
introduce some notation.

######  Definition 1 ([Dal01], Definition 1.8.1).

Let $A$ be an algebra over $\mathbb{C}$, and $M$ be an $A$-bimodule. A
derivation in the algebra $A$ with values in $M$ is a linear map

$d\colon A\to M,$ (1)

such that for all $a,\,b\in A$ we have

$d(ab)=d(a)b+ad(b).$ (2)

Let us consider an important example of derivations.

######  Definition 2.

Given $x\in M$, the inner derivation $D_{x}\colon A\to M$ is a derivation
defined by the formula

$D_{x}(a)=xa-ax.$ (3)

In appendix A, we provide an example of discontinuous inner derivation in
$l_{2}(G).$ While existence of unbounded inner derivations may be a known
fact, we have not encountered any prior mention of it in the literature.

A classical question known as a derivation problem (also known as a Johnson’s
problem) is whether all continuous derivations are inner. See [Dal01, AM19]
for more details. Typically the case of $A=\mathbb{C}[G]$ is considered. In
terms of Hochschild cohomology, it means the triviality of the first
cohomology group.

This problem is stated in [Dal01] as follows: Let $G$ be a locally compact
group. Is every derivation from $L^{1}(G)$ to $M(G)$ inner? Here, $M(G)$
denotes the space of complex-valued regular Borel measures on $G$.

In many special cases, B. Johnson provided answers to this problem. He
investigated it as a suitable example for the theory of homology in Banach
algebras. For instance, in [Joh01] he showed that for a connected Lie group
$G$ all derivations from $L_{1}\left(G\right)$ to itself have the form:

$Da=a\mu-\mu a,$ (4)

where $\mu\in M(G)$. In the work [JR69], he proved that all derivations of the
algebra $\ell_{1}(G)$ for a discrete group $G$ are inner. The original
formulation of the problem was discussed in [Los08] by V. Losert.

The algebra of outer derivations in group rings without a norm turns out to be
nontrivial. For instance any nontrivial central derivation is not inner (See
[Aru20], Definition 3). The structure of this algebra was studied in [AM19,
AMS16], and its description in terms of the original group $G$ was provided.
These works also developed a technique for describing derivations using
characters.

### Main results

In this article we introduce a class of groups that we will call BC-groups
(see Definition 12), this are groups such that $\sup_{g\in
G}\operatorname{diam}(g^{-1}Kg)<\infty$ for any finite $K\subset G$. That is,
conjugations don’t change the distances between elements very much.

The main goal of this paper is to prove that for BC-groups $G$ all continuous
derivations in $\ell_{p}(G),\,p\geq 1$ are inner. Here is the precise
statement, the proof is given in the subsection 4.2.

######  Theorem 2.

Let $G$ be a BC-group with uniformly bounded finite components. Then every
$G$-bounded derivation $d\colon\hyperlink{finite-
combinations}{f_{p}(G)}\to\hyperlink{finite-combinations}{\ell_{q}(G)}$ is
inner.

All necessary notation is introduced below.

##  2 Key Definitions

Hereinafter, we consider only finitely generated groups $G$, i.e., groups
which can be presented as $G=\langle\mathscr{X}|R\rangle$, where
$\mathscr{X}=\\{x_{1},\dots,x_{n}\\}$ is a finite set of generators, and
${R=\\{r_{i}|i\in I\\}}$ is a set of relations.

Firstly, let us introduce some definitions and notations.

######  Definition 3.

The  $\ell_{p}$ norm on the group ring $\mathbb{C}[G]$ is a norm of the form

$\left\lVert\sum_{g\in G}\alpha(g)g\right\rVert_{p}=\sqrt[p]{\sum_{g\in
G}|\alpha(g)|^{p}}.$ (5)

The supremum or sup norm is a norm of the form

$\left\lVert\sum_{g\in G}\alpha(g)g\right\rVert_{\textnormal{s}}=\sup_{g\in
G}{|\alpha(g)|}.$ (6)

We denote the group ring equipped with the $\ell_{p}$ or $\sup$ norm as
$f_{p}(G)$ or $f_{\infty}(G)$, respectively. Let $\ell_{p}(G)$ and $c_{0}(G)$
be their completions. By $\widehat{\mathbb{C}}[G]$ denote a completion in case
of arbitrary norm.

Note that in both cases of the $\ell_{p}$ and $\sup$ norms, multiplication by
an element of the group $G$ is a continuous operator from $\mathbb{C}[G]$ to
$\mathbb{C}[G]$, so it extends to an operator from $\widehat{\mathbb{C}}[G]$
to $\widehat{\mathbb{C}}[G]$. Therefore $\widehat{\mathbb{C}}[G]$ is a
bimodule over $\mathbb{C}[G].$

Note that a continuous derivation
$d\colon\left(\mathbb{C}[G],\left\lVert\cdot\right\rVert_{1}\right)\to\left(\widehat{\mathbb{C}}[G],\left\lVert\cdot\right\rVert_{2}\right)$
can be extended to a continuous operator
$\hat{d}\colon\left(\widehat{\mathbb{C}}[G],\left\lVert\cdot\right\rVert_{1}\right)\to\left(\widehat{\mathbb{C}}[G],\left\lVert\cdot\right\rVert_{2}\right)$.

The technique described in [AMS16] allows us to assign a character on a
groupoid to each derivation and vice versa. Let us recall the main
definitions. We will mostly follow the notation suggested in [Aru20].

######  Definition 4.

The groupoid of conjugacy action $\Gamma$ is a small category where the
objects are the elements of $G$. The set of morphisms
$\operatorname{Hom}(\Gamma)$ consists of all possible pairs of elements
$(u,v)\in G\times G$, here
$(u,v)\in\operatorname{Hom}_{\Gamma}(v^{-1}u,uv^{-1}).$

Let $\varphi=\left(u_{1},v_{1}\right)$ and $\psi=\left(u_{2},v_{2}\right)$ be
composable, then their composition $\psi\circ\varphi$ is defined by the
formula

$\psi\circ\varphi:=\left(v_{2}u_{1},v_{2}v_{1}\right).$

######  Definition 5.

A map $\chi\colon\operatorname{Hom}(\Gamma)\to\mathbb{C}$ is called a
character if for any two composable arrows $\varphi=\left(u_{1},v_{1}\right)$,
$\psi=\left(u_{2},v_{2}\right)$ we have

$\chi(\psi\circ\varphi)=\chi(\varphi)+\chi(\psi).$

######  Definition 6.

By $\Gamma_{[u]}$ denote the connected component of $\Gamma$ containing an
element $u\in G.$ In other words, $\Gamma_{[u]}$ is a subgroupoid of $\Gamma$
such that

$\operatorname{Obj}(\Gamma_{[u]})=\\{g\in
G\mid\operatorname{Hom}(u,g)\neq\emptyset\\},\quad\operatorname{Hom}(\Gamma_{[u]})=\\{(u,v)\in\operatorname{Hom}(\Gamma)\mid
v^{-1}u,uv^{-1}\in\operatorname{Obj}(\Gamma_{[u]})\\}.$

Let $\delta_{h}\colon\widehat{\mathbb{C}}[G]\to\mathbb{C}$ denote a map
defined by the formula $\delta_{h}\left(\sum_{g\in
G}\alpha(g)g\right)=\alpha(h).$

######  Definition 7.

Given a derivation
$d\colon\left(\mathbb{C}[G],\left\lVert\cdot\right\rVert_{1}\right)\to\left(\widehat{\mathbb{C}}[G],\left\lVert\cdot\right\rVert_{2}\right)$,
the assigned character is a function
$\chi\colon\operatorname{Hom}(\Gamma)\to\mathbb{C}$ defined by the formula

$\chi(h,g)=\delta_{h}\left(d(g)\right).$ (7)

######  Proposition 1.

Assigned character is indeed a character and it satisfies the formula

$d(g)=\sum_{h\in G}\chi(h,g)h=g\left(\sum_{t\in
G}\chi(gt,g)t\right),\quad\forall g\in G.$ (8)

###### Proof.

It is easy to see that

$d(g)=\sum_{h\in G}\delta_{h}\left(d(g)\right)h=\sum_{h\in G}\chi(h,g)h.$ (9)

The equality

$\sum_{h\in G}\chi(h,g)h=g\left(\sum_{t\in G}\chi(gt,g)t\right)$ (10)

is obtained by changing the summation indices $h=gt.$ Verification that $\chi$
is a character is conducted similarly to the proof of Theorem 1 in [AMS16].
For the sake of clarity, it will be repeated here.

If the composition $(h_{2},g_{2})\circ(h_{1},g_{1})$ is defined, then

$\displaystyle h_{1}g_{1}^{-1}=g_{2}^{-1}h_{2},$ (11)
$\displaystyle(h_{2},g_{2})\circ(h_{1},g_{1})=(g_{2}h_{1},g_{2}g_{1}).$ (12)

Taking this into account, we have

$\displaystyle\chi(g_{2}h_{1},g_{2}g_{1})=\delta_{g_{2}h_{1}}\left(d(g_{2}g_{1})\right)=\delta_{g_{2}h_{1}}\left(d(g_{2})g_{1}\right)+\delta_{g_{2}h_{1}}\left(g_{2}d(g_{1})\right)=$
$\displaystyle=\delta_{g_{2}h_{1}g_{1}^{-1}}\left(d(g_{2})\right)+\delta_{h_{1}}\left(d(g_{1})\right)\stackrel{{\scriptstyle(\ref{eq:compose_morphisms})}}{{=}}\delta_{h_{2}}\left(d(g_{2})\right)+\delta_{h_{1}}\left(d(g_{1})\right)=\chi(h_{2},g_{2})+\chi(h_{1},g_{1}).$
(13)

This implies that $\chi$ is a character and completes the proof. ∎

Recall that a derivation is called quasi-inner if the character assigned to
this derivation equals zero on all loops. A morphism $(u,v)$ is called a loop
if $uv=vu.$ One can see that every inner derivation is quasi-inner.

Let a norm on the group algebra be subordinate to the $\operatorname{sup}$
norm, then all continuous derivations are quasi-inner. The proof is given in
[Aru23].

######  Definition 8.

A potential of a character $\xi$ is a function
$\varphi\colon\operatorname{Obj}(G)\to\mathbb{C}$ such that

$\chi(h,g)=\varphi(hg^{-1})-\varphi(g^{-1}h).$ (14)

Conversely, each potential induces a character defined by the formula above.

One can show that a potential of a character $\chi$ exists iff $\chi$ is
quasi-inner. Let $\varphi$, $\varphi^{\prime}$ be two potentials, if
$\psi=\varphi-\varphi^{\prime}$ is constant on connected components function,
then $\varphi$ and $\varphi^{\prime}$ induce the same character.

Rewriting formula (8) in terms of potentials, we get

$d(g)=\sum_{h\in G}\left(\varphi(hg^{-1})-\varphi(g^{-1}h)\right)h=\sum_{t\in
G}\left(\varphi(gtg^{-1})-\varphi(t)\right)gt.$ (15)

Consider a formal sum $a=\sum_{t\in G}\varphi(t)t,$ then the formula can be
rewritten as:

$d(g)=\sum_{t\in G}\varphi(t)\left(tg-gt\right)=[a,g].$ (16)

Let $D_{x}\colon\mathbb{C}[G]\to\ell_{p}(G)$ be an inner derivation, where
$x=\sum_{g\in G}\alpha(g)g\in\ell_{p}(G).$ Then by the last formula we obtain
that $\alpha\colon G\to\mathbb{C}$ is the potential of $D_{x}.$ Conversely, a
quasi-inner derivation $d$ is inner if and only if we can find a potential
$\varphi$ such that $a$ is not only a formal linear combination, but also an
element of
$\left(\widehat{\mathbb{C}}[G],\left\lVert\cdot\right\rVert\right)$, which is
equivalent to $\left\lVert a\right\rVert<\infty.$

######  Definition 9.

Let $G$ be a group and $\mathscr{X}$ be a generating set of $G$. The conjugacy
graph $\operatorname{sk}=\operatorname{sk}(G,\mathscr{X})$ is an edge-labeled
directed graph constructed as follows:

  * •

Each element $g$ of $G$ is assigned a vertex: the vertex set of $\Gamma$ is
identified with $G$.

  * •

For every $g\in G$ and $x\in\mathscr{X}\cup\mathscr{X}^{-1}$ there is a
directed edge with label $x$ from the vertex corresponding to $g$ to the one
corresponding to $xgx^{-1}$.

The distance between vertices is defined as the minimum number of edges in the
paths from one vertex to another. Note that all balls in
$\operatorname{sk}(G)$ contain a finite number of elements, and distance is
infinite if elements lie in distinct connected components.

######  Remark 1.

The graph $\operatorname{sk}(G,\mathscr{X})$ can be embedded in the groupoid
$\Gamma.$ The set of vertices of $\operatorname{sk}(G)$ and objects of
$\Gamma(G)$ just coincide. An edge with label $x$ connecting $g$ and
$xgx^{-1}$ maps to the morphism $(xg,x)\in\operatorname{Hom}(g,xgx^{-1}).$

Let us look at an example of a conjugacy graph.

######  Example 1.

Consider the Heisenberg group

$H_{3}(\mathbb{Z})=\langle
A_{x},A_{p},A_{1}\mid[A_{p},A_{x}]=A_{1},[A_{p},A_{1}]=E,[A_{x},A_{1}]=E\rangle.$

The graph $\operatorname{sk}(A_{p})$ is depicted below. We have
$A_{x}A_{p}A_{1}^{k}A_{x}^{-1}=A_{p}A_{1}^{k-1}$, so the edges connecting
distinct vertices are labeled by $A_{x}.$ Edges labeled by $A_{p},\,A_{1}$ are
loops. Edges labeled by $A_{x}^{-1},\,A_{p}^{-1},\,A_{1}^{-1}$ are not shown.
To avoid cluttering the notation on the figures in the future, we will refrain
from depicting cycles on the graphs.

$A_{p}A_{1}^{3}$$A_{x}$$A_{p},\,A_{1}$$A_{p}A_{1}^{2}$$A_{x}$$A_{p},\,A_{1}$$A_{p}A_{1}^{1}$$A_{x}$$A_{p},\,A_{1}$$A_{p}A_{1}^{0}$$A_{x}$$A_{p},\,A_{1}$$A_{p}A_{1}^{-1}$$A_{x}$$A_{p},\,A_{1}$$A_{p}A_{1}^{-2}$$A_{x}$$A_{p},\,A_{1}$$A_{p}A_{1}^{-3}$$A_{x}$$A_{p},\,A_{1}$
Figure 1: Conjugacy graph $\operatorname{sk}_{[A_{p}]}$

######  Definition 10.

A potential $\varphi$ is called stabilised to the value $a_{0}$ at infinity on
$\Gamma_{[x_{0}]}$ if the following condition is satisfied:

$\forall\varepsilon>0\,\,\exists K:\forall g\in\Gamma_{[u_{0}]}\setminus
K\hookrightarrow|\varphi(g)-a_{0}|<\varepsilon,$ (17)

where $K$ is a finite set. This definition only makes sense if the number of
vertices in $\Gamma_{[u_{0}]}$ is infinite.

We can adjust the value of a potential by a constant, so in cases where a
potential is stabilised, we will consider it to be stabilised to $0.$

In the following sections, it will be shown that for a certain class of
groups, potentials of continuous derivations are stabilised.

The following proposition is obvious:

######  Proposition 2.

Let the elements of $\Gamma_{[u_{0}]}$ be indexed in some way by $\mathbb{N}$.
The potential $\varphi$ is stabilised iff the sequence
$\\{\varphi(g_{k})\\}_{k=1}^{\infty}$ has a finite limit.

First, let us show that a potential of derivation
$d\colon\mathbb{C}[G]\to\ell_{p}(G)$ has no sharp changes in potential from
point to point as it tends to infinity. This is formulated more precisely in
the following proposition.

######  Proposition 3.

Let $\varphi$ be a potential of derivation
${d\colon\mathbb{C}[G]\to\ell_{p}(G)}$, continuity of $d$ is not assumed. Then
for all $\varepsilon>0,$ there exists a finite set $K\subset G$ such that for
all $g_{1},\,g_{2}\in G\setminus K$ with $\rho(g_{1},g_{2})=1,$ we have
$|\varphi(g_{1})-\varphi(g_{2})|<\varepsilon.$

###### Proof.

Otherwise there will be an infinite number of edges within
$\operatorname{sk}{G}$ where the potential difference exceeds $\varepsilon.$
So there exists at least one $x\in\mathscr{X}$ such that the set ${\\{g\in
G:\chi(g,x)>\varepsilon\\}}$ is infinite. Consequently, according to the
formula $d(g)=\sum_{g\in G}\chi(h,g)g,$ it follows that
$d(x_{i})\notin\mathbb{C}[G]_{p}.$ ∎

######  Definition 11.

A derivation $d\colon\mathbb{C}[G]\to\widehat{\mathbb{C}}[G]$ is called
$G$-bounded if $\sup_{g\in G}\left\lVert d(g)\right\rVert$ is finite.

It is obvious that continuity implies $G$-boundedness, and in the case of the
$\ell_{1}$ norm the converse is also true (for an operator $A\colon\ell_{1}\to
X,$ where $X$ is an arbitrary normed space, it holds that $\left\lVert
A\right\rVert=\sup_{i\in N}\left\lVert d(e_{i})\right\rVert$). In the general
case the converse is false, see appendix A.

##  3 Bounded Conjugacy Condition

In contrast to the case of Cayley graphs and the translation action, in the
case of the conjugation action we are interested in, the distance between
vertices can change uncontrollably. For example, in a free group
$F_{2}=\langle a,b|\emptyset\rangle$ we have $\rho(a,bab^{-1})=1,$ but
$\rho(a^{n}aa^{-n},a^{n}bab^{-1}a^{-n})=n+1,$ i.e. tends to infinity. For our
construction we will introduce a class of groups in which conjugations act in
a controllable manner.

######  Definition 12.

A group $G$ is called a group with bounded conjugations (a BC-group for short)
if the following is satisfied:

$\forall h_{1},h_{2}:\rho(h_{1},h_{2})=1\,\,\exists\,C>0:\forall g\in
G\,\hookrightarrow\rho(gh_{1}g^{-1},gh_{2}g^{-1})<C.$ (18)

It is easy to show that $G$ is a BC-group iff for any finite $K\subset G$
holds

$\sup_{g\in G}\operatorname{diam}(g^{-1}Kg)<\infty.$

######  Theorem 1.

The property of a group to be a BC-group is well defined, and invariant under
the choice of finite generating set.

###### Proof.

Consider two generating sets $\mathscr{X}=\\{x_{1},\dots,x_{n}\\}$ and
$\mathscr{Y}=\\{y_{1},\dots,y_{m}\\}.$ Suppose the BC condition holds with
respect to $\mathscr{X}$, i.e., for any $h\in G,$ there exists a constant
$C=C(B_{1}(h))$ such that for every $g\in G$ and $x_{i}\in\mathscr{X},$ we
have $\rho_{\mathscr{X}}(gxhx^{-1}g^{-1},ghg^{-1})<C.$

Now express all generators $\mathscr{Y}$ through generators $\mathscr{X}$:

$\begin{split}&y_{1}=w_{1}(x_{1},\dots,x_{n});\\\ &\dots\\\
&y_{m}=w_{m}(x_{1},\dots,x_{n}).\end{split}$ (19)

Let $L_{y}$ be the maximum length of words $w_{1},\dots,w_{m}.$ Similarly, by
expressing $x_{i}$ through $\mathscr{Y},$ we define $L_{x}.$ Then, for any
$h,g\in G,\,y_{i}\in\mathscr{Y}$ we obtain

$\displaystyle\rho_{\mathscr{Y}}(gyhy^{-1}g^{-1},ghg^{-1})\leqslant
L_{x}\rho_{\mathscr{X}}(gyhy^{-1}g^{-1},ghg^{-1})=$
$\displaystyle=L_{x}\rho_{\mathscr{X}}(gw_{i}(x)hw_{i}(x)^{-1}g^{-1},ghg^{-1})\leqslant
L_{x}C\left(B_{L_{y}}(h)\right),$ (20)

which completes the proof. ∎

###  3.1 Examples

Let us now look at some examples of BC-groups.

######  Example 2.

Any nilpotent group of rank 2 is a BC-group because for such groups, the graph
$\operatorname{sk}_{u}$ is isomorphic to the Cayley graph $G/Z(u)$ (see
[Aru20], lemma 4). The group $G/Z(u)$ is abelian, so the BC condition is
satisfied with the constant $C=1.$

In particular the Heisenberg group

$H_{3}(\mathbb{Z})=\left\\{\begin{pmatrix}1&a&c\\\ 0&1&b\\\
0&0&1\end{pmatrix}:a,\,b,\,c\in\mathbb{Z}\right\\}$

is a BC-group.

######  Example 3.

Each FC-group is a BC-group, since by definition, all groupoid components in
an FC-group are finite. It is known that a finitely generated group $G$ is an
FC-group iff the derived subgroup $|G^{\prime}|$ is finite (see [Neu54], (3.1)
Theorem).

######  Example 4.

Two previous examples can be generalised as follows: Let $G$ be a group such
that $|G^{\prime}/(Z(G)\cap G^{\prime})|<\infty,$ then $G$ is a BC-group. Let
us show it. Let $\\{[a_{1}],\dots,[a_{k}]\\}$ be the elements of
$G^{\prime}/(Z(G)\cap G^{\prime}),$ let $A=\\{a_{1},\dots,a_{k}\\}$ be the set
of their representatives. For any $h\in\mathscr{X},$ $g\in G$ we have
$gh=zahg,$ where $a\in A,$ $z\in Z(G).$ Then we obtain

$\rho(gug^{-1},gxux^{-1}g^{-1})=\rho(gug^{-1},axgu(axg)^{-1})\leq|a|+1\leq\max_{a^{\prime}\in
A}|a^{\prime}|+1.$

Here $|a|$ is a length of $a\in G$ with respect to the generating set
$\mathscr{X},$ that is, the minimum number $n$ such that $a$ can be
represented as the product $x_{i_{1}}\dots x_{i_{n}}.$

######  Example 5.

The infinite dihedral group $D_{\infty}=\mathbb{Z}_{2}*\mathbb{Z}_{2}=\langle
a,b\mid a^{2},b^{2}\rangle$ is a BC-group. Let us describe the conjugacy
classes. If a word $w$ starts and ends with the same letter, then it belongs
either to $[a]$ or to $[b].$ If a word $w$ starts and ends with different
letters, then it has the form $(ab)^{n}$ or $(ba)^{n}.$ For words of this
form, we have $[(ab)^{n}]=[(ba)^{n}]=\\{(ab)^{n},(ba)^{n}\\}.$ We are
interested in the infinite conjugacy classes. For definiteness, let us work
with $[a].$

Let’s check the BC condition. Consider an element of the form $(ba)^{k}b.$ The
adjacent elements are $a(ba)^{k}ba$ and $a(ba)^{k-1}.$ Let’s see how the
distance between the elements $h_{1}=(ba)^{k}b$ and $h_{2}=a(ba)^{k}ba$
changes when conjugated by $g.$ Write $g$ as a reduced word and conjugate
letter by letter. If $a$ is the rightmost letter in $g,$ then $h_{1}$ will
move 1 step to the right (see Figure 2), and $h_{2}$ will move 1 step to the
left. Then, when conjugating by $b,$ the element $ah_{1}a$ will move right
again, and $ah_{2}a$ will move left, and so on, until one of them reaches the
beginning of the ray. If this happens, they will start moving in the same
direction. Therefore, $B_{1}(h_{1})=2\rho(h_{1},a)+1.$ The additional 1 comes
from the fact that when we reach the end of the segment, i.e., $a,$ we will
need to conjugate again with $a,$ so the rightward movement will not start at
this step.

$\tiny{a}$$\tiny{bab}$$\tiny{ababa}$ Figure 2: Conjugacy graph
$\operatorname{sk}_{[a]}$

######  Proposition 4.

a) Let $G$ and $H$ be a BC-groups, then $G\times H$ is also a BC-group.

b) Let $H$ be a normal subgroup of a BC-group $G.$ Then $G/H$ is a BC-group.

###### Proof.

a) Suppose $G$ is generated by $x_{1},\dots,x_{n},$ and the group $H$ is
generated by $y_{1},\dots,y_{n}.$ Then, $G\times H$ is generated by the union
of the images of these elements under the embeddings. The BC condition follows
immediately from the following relation on metrics:

$\rho_{G\times
H}(a_{1},a_{2})\leqslant\rho_{G}(\pi_{G}(a_{1}),\pi_{G}(a_{2}))+\rho_{H}(\pi_{H}(a_{1}),\pi_{H}(a_{2})),\text{
where }a_{1},\,a_{2}\in G\times H.$

The sum of the constants could be taken as the boundedness constant.

b) Let $x_{i}$ be the generators of the group $G$. Take $\left[x_{i}\right]$
as the generators of $G/H$. From
$\rho\left(\left[g_{1}\right],\left[g_{2}\right]\right)=1,$ it follows
$\left[xg_{1}x^{-1}\right]=\left[g_{2}\right]$ for some $x\in\mathscr{X}$.
Since $G$ is a BC-group, we have $\forall g\in
G\hookrightarrow\rho\left(gxg_{1}x^{-1}g^{-1},gg_{1}g^{-1}\right)<C$.
Therefore,
$\rho\left([g]\left[g_{1}\right]\left[g^{-1}\right],[g]\left[g_{2}\right]\left[g^{-1}\right]\right)<C$.
∎

Consequently the finite products of the groups mentioned above and finite
groups will satisfy the bounded conjugacy condition.

######  Example 6.

Consider $G=D_{\infty}\rtimes_{\varphi}\mathbb{Z}_{2},$ where
$\varphi\colon\mathbb{Z}_{2}\to\operatorname{Aut}(D_{\infty})$ such that
$\varphi(a)=b,\,\varphi(b)=a.$ This group can be presented using generators
and relations as $G=\langle a,b,c\mid a^{2}=b^{2}=c^{2}=e,cac=b\rangle.$

Any word in this group can be represented as $u=w(a,b)c^{\varepsilon},$ where
$\varepsilon=0,1.$ If $\varepsilon=0,$ then $u$ belongs either to the
conjugacy class $[a]=[b]$ or to a finite conjugacy class of the form
$[(ab)^{n}].$

In the case of $\varepsilon=1,$ the infinite class is
$[c]=\\{(ab)^{n}c:n\in\mathbb{Z}\\}.$ Finite classes have the form
$[(ab)^{n}ac]=\\{(ab)^{n}ac,(ba)^{n}bc\\}.$

Analogously to Example 5, it can be shown that the BC condition is satisfied
for infinite conjugacy classes. These ones are depicted in the figures below.

$\tiny{a}$$\tiny{bab}$$\tiny{ababa}$$\tiny{b}$$\tiny{aba}$$\tiny{babab}$$\tiny{c}$$\tiny{c}$$\tiny{c}$$\tiny{a}$$\tiny{a}$$\tiny{b}$$\tiny{b}$
Figure 3:  $\operatorname{sk_{a}}(D_{\infty}\rtimes_{\varphi}\mathbb{Z}_{2}).$
$\tiny{c}$$\tiny{bac}$$\tiny{(ab)^{2}c}$$\tiny{(ba)^{3}c}$$\tiny{abc}$$\tiny{(ba)^{2}c}$$\tiny{(ab)^{3}c}$$\tiny{a}$$\tiny{b}$$\tiny{c}$$\tiny{c}$$\tiny{c}$$\tiny{b}$$\tiny{b}$$\tiny{a}$$\tiny{a}$
Figure 4: $\operatorname{sk}_{c}(D_{\infty}\rtimes_{\varphi}\mathbb{Z}_{2}.)$

######  Example 7 (Not a BC-group).

Consider $G=H_{3}\rtimes_{\varphi}\mathbb{Z}_{2}$, where
$\varphi\colon\mathbb{Z}_{2}\to\operatorname{Aut}(H_{3})$ is defined by
$\varphi(A_{p})=A_{x},\,\varphi(A_{x})=A_{p},\,\varphi(A_{1})=A_{1}^{-1}.$
This indeed defines a homomorphism, as the relations go to relations. Indeed,
$[\varphi(A_{p}),\varphi(A_{x})]=[A_{x},A_{p}]=[A_{p},A_{x}]^{-1}=A_{1}^{-1}=\varphi(A_{1}),$
and the remaining two are obvious.

Now notice that $cA_{x}c=A_{p},$ but
$\rho(A_{x}^{k}A_{p}A_{x}^{-k},A_{p})\to\infty,$ whereas
$\rho(A_{x}^{k}A_{x}A_{x}^{-k},A_{x})=0.$ So BC condition isn’t satisfied.

$\tiny{A_{p}}$$\tiny{A_{p}A_{1}^{-1}}$$\tiny{A_{p}A_{1}^{-2}}$$\tiny{A_{x}}$$\tiny{A_{x}A_{1}}$$\tiny{A_{x}A_{1}^{2}}$$\tiny{c}$$\tiny{c}$$\tiny{c}$$\tiny{A_{p}}$$\tiny{A_{p}}$$\tiny{A_{x}}$$\tiny{A_{x}}$$A_{x}A_{1}^{-1}$$A_{p}A_{1}$$c$
Figure 5: $\operatorname{sk}_{A_{x}}(H_{3}\rtimes_{\varphi}\mathbb{Z}_{2})$

This example also demonstrates that not all quasi-isometries preserve the BC
condition, so the property of a group to be a BC group is not a coarse
invariant. Also it shows that on conjugacy classes with two ends the BC-
condition can be not satisfied.

##  4 Derivations in BC-groups

In this section we will prove that in BC-groups with uniformly bounded finite
conjugacy classes all continuous derivations are inner. First we will show
that on each infinite component a potential corresponding to the derivation is
stabilised (see Definition 10). Then we will prove the theorem for derivations
with support in one infinite component, and finally we will prove the initial
statement.

######  Lemma 1.

Let $G=\langle x_{1},\dots,x_{n}|r_{i}\rangle$ satisfy the BC condition. Let
$\varphi$ be a potential corresponding to $G$-bounded derivation ${d\colon
f_{p}(G)\to\ell_{q}(G)}$. Then $\varphi$ is stabilised.

###### Proof.

Consider an arbitrary infinite connected component
$\operatorname{sk}_{g_{0}}$. Define numbers

$a:=\inf_{K}\left(\sup\limits_{g\in\Gamma[g_{0}]\setminus
K}\varphi\left(g\right)\right),\quad
b:=\sup_{K}\left(\inf\limits_{g\in\Gamma[g_{0}]\setminus
K}\varphi\left(g\right)\right),\quad\delta:=a-b.$ (21)

Clearly, if the supremum or infimum equals infinity, the derivation is not
continuous.

Suppose $\varphi$ isn’t stabilised; then $\delta$ is positive. Fix an
arbitrary natural number $n$. Consider unbounded sets

$V_{a}=\\{g\mid|\varphi(g)-a|<\frac{\delta}{4}\\},\quad
V_{b}=\\{g\mid|\varphi(g)-b|<\frac{\delta}{4}\\}.$

Take an arbitrary $n$-element subset $V_{n}\subset V_{a}.$ Applying the BC
condition we find a constant $C$ such that for all $h_{1},h_{2}\in V_{n}$ and
all $g\in G,$ the inequality $\rho(gh_{1}g^{-1},gh_{2}g^{-1})<C$ is satisfied.
According to Proposition 3, there exists $h\in V_{b}$ such that
$B_{C}(h)\subset V_{b}.$ Consider an arbitrary $u\in V_{n}.$ Since $u,\,h$
belong to the same connected component, we have $h=gug^{-1}.$ Thus,
$gV_{n}g^{-1}\subset B_{C}(h)\subset V_{b}.$

Using the formula 15, we obtain

$\displaystyle\left\lVert d(g)\right\rVert=\left\lVert\sum_{t\in
G}\left(\varphi(gtg^{-1})-\varphi(t)\right)gt\right\rVert=\geqslant\sqrt[q]{\sum_{t\in
V_{n}}\left|\varphi(gtg^{-1})-\varphi(t)\right|^{q}}\geqslant\sqrt[q]{n\frac{\delta}{2}}.$
(22)

Since $n$ was chosen arbitrarily, we conclude that $d$ is unbounded. ∎

###  4.1 Case of a Single Component

######  Lemma 2.

Let $G$ be a BC-group. Then any $G$-bounded derivation with support in one
infinite component $\Gamma_{[u_{0}]}$ is inner.

###### Proof.

Due to equation (16), it is sufficient to prove that a potential from
$\ell_{q}$ can be chosen for the given derivation. Suppose the opposite:

$\sum_{g\in\Gamma[u_{0}]}|\varphi(g)|^{q}=\infty.$ (23)

For each $x_{0}>0$, there exist
$g_{1},\dots,g_{n}\in\operatorname{Obj}(\Gamma_{[u_{0}]})$ such that

${|\varphi(g_{1})|^{q}+\dots+|\varphi(g_{n})|^{q}>x_{0}^{q}},$

and each term in the sum is nonzero. Set
$m=\frac{1}{2}\min\\{|\varphi(g_{1})|,\dots|\varphi(g_{n})|\\}.$ According to
Lemma 1, the potential converges to $0$ at infinity. So we can find $R>0$ such
that $|\varphi(g)|<m$ for all $g\in\Gamma[u_{0}]\setminus B_{R}(u_{0})$. By
the BC condition there exists a constant $C=C(B_{R}(u_{0}))>0$ such that for
all $g\in G$ and all $h_{1},h_{2}\in B_{R}(u_{0})$, we have
$\rho(gh_{1}g^{-1},gh_{2}g^{-1})<C.$

Consider an element $g\in G$ such that $\rho(u_{0},gu_{0}g^{-1})>R+C$. Then

$\forall h\in B_{R}(u_{0})\hookrightarrow|\varphi(ghg^{-1})|<m,$ (24)

because $\rho(ghg^{-1},gu_{0}g^{-1})<C,$ and therefore $ghg^{-1}\notin
B_{R}(u_{0}).$

From (15) we obtain lower bound of $\left\lVert d(g)\right\rVert$:

$\displaystyle\left\lVert d(g)\right\rVert=\left\lVert\sum_{t\in
G}\left(\varphi(gtg^{-1})-\varphi(t)\right)gt\right\rVert=\sqrt[q]{\sum_{t\in
G}\left|\varphi(gtg^{-1})-\varphi(t)\right|^{q}}\geqslant$
$\displaystyle\geqslant\sqrt[q]{\sum_{t\in\\{g_{1},\dots,g_{n}\\}}\left|\varphi(gtg^{-1})-\varphi(t)\right|^{q}}=\sqrt[q]{\sum_{i=1}^{n}\left|\varphi(g_{i})-\varphi(gg_{i}g^{-1})\right|^{q}}\geqslant$
$\displaystyle\geqslant\sqrt[q]{\sum_{i=1}^{m}\left|\left|\varphi(g_{i})\right|-m\right|}\geqslant\frac{1}{2}\sqrt[q]{\sum_{i=1}^{n}|\varphi(g_{i})|^{q}}=\frac{x_{0}}{2}.$
(25)

The first inequality holds because we just omitted all summands with
$t\notin\\{g_{1},\dots,g_{n}\\},$ after that we applied the triangle
inequality and took into account that
$\varphi(gg_{i}g^{-1})<m<\min_{i=1,\dots,n}|\varphi(g_{i})|.$

Thus, for any chosen $x_{0}$, we have found an element $g\in G$ for which
$\left\lVert d(g)\right\rVert\geqslant\frac{x_{0}}{2}.$ This implies that $d$
is not $G$-bounded. ∎

Therefore, for BC-groups, it is shown that a derivation with support in one
component can be associated with a potential from $\ell_{q}.$

###  4.2 Case of Multiple Components

Let us examine how the distances $\rho(h,ghg^{-1})$ and $\rho(h,g^{-1}hg)$ are
related. Consider $g=x^{k}y,\,h=x\in F_{2}=\langle x,y\rangle.$ Then

$\rho(h,ghg^{-1})=\rho(x,x^{k}yxy^{-1}x^{-k})=k+1,$
$\rho(h,g^{-1}hg)=\rho(x,y^{-1}xy)=1,$

so in general case one of the distances can increase largely, whereas other is
bounded. But if $G$ is a BC-group, it is impossible.

######  Proposition 5.

Let $G$ be a BC-group, and let $\\{a_{k}\\}_{k=1}^{\infty}$ be a sequence of
elements of $G$ such that ${\rho(u,a_{k}ua_{k}^{-1})\to\infty.}$ Then
${\rho(u,a_{k}^{-1}ua_{k})\to\infty.}$

###### Proof.

Otherwise, there exists a constant $L$ and a subsequence $a_{n_{k}}$ such that
$\rho(u,a_{n_{k}}^{-1}ua_{n_{k}})\leqslant L,$ meaning
$a_{n_{k}}^{-1}ua_{n_{k}}\in B_{L}(u).$ Conjugating by $a_{n_{k}}$ and using
the boundedness of conjugations, we obtain
$\rho(a_{n_{k}}ua_{n_{k}}^{-1},u)<C(B_{L}(u))$, which contradicts the
hypothesis that ${\rho(u,a_{k}ua_{k}^{-1})\to\infty.}$ ∎

######  Lemma 3.

Let $G$ be a BC-group. There exists a sequence $\\{a_{k}\\}_{k=1}^{\infty},$
$a_{k}\in G$ such that for every $h$ from any infinite connected component we
have $\rho(h,a_{k}ha_{k}^{-1})\to\infty.$

###### Proof.

First, consider the finite number of components. Let us enumerate all infinite
connected components and proceed by induction on the number of components. In
each component, we arbitrarily choose an origin $u_{i}.$

The base case is evident; the existence of such a sequence for a single
component follows from the fact that we are dealing with infinite components.

Suppose $a_{n}$ is a sequence such that
$\rho(u_{i},a_{n}u_{i}a_{n}^{-1})\to\infty$ for $i\leqslant k.$ Consider the
component $\Gamma_{[u_{k+1}]}.$ If we can extract a subsequence from the
sequence $\rho(u_{k+1},a_{n}u_{k+1}a_{n}^{-1})$ that converges to infinity,
this completes the induction step.

Otherwise, $\\{a_{n}ha_{n}^{-1}|n\in\mathbb{Z}\\}$ will be bounded for each
$h\in\Gamma_{[u_{k+1}]}.$ Take an arbitrary sequence $b_{n}$ such that
$\rho(u_{k+1},b_{n}u_{k+1}b_{n}^{-1})\to\infty.$ Extract a subsequence
$a_{n}^{\prime}$ from the sequence $a_{n}$ such that $\min_{i\leqslant
k}\rho(u_{i},a_{n}^{\prime}u_{i}{a_{n}^{\prime}}^{-1})>2l(b_{n}).$ Consider
the sequence $c_{n}=a_{n}^{\prime}b_{n}.$ Then, for each $i\leqslant k+1,$ we
have $\rho(u_{i},c_{n}u_{i}c_{n}^{-1})\to\infty.$ For $i\leqslant k,$ this
follows from the triangle inequality. For $i=k+1,$ it follows from the fact
that $\\{a_{n}u_{k+1}a_{n}^{-1}|n\in\mathbb{Z}\\}$ is a bounded set, $G$ is a
BC-group and $\rho(u_{k+1},b_{n}u_{k+1}b_{n}^{-1})\to\infty.$

In case of an arbitrary number of components, applying the result above, we
can choose an element $a_{k}\in G$ such that
$\min_{i\in\mathbb{N}}\rho(u_{i},a_{k}u_{i}a_{k}^{-1})>k.$ Then, for every
$i\in\mathbb{N},$ it holds that $\rho(u_{i},a_{k}u_{i}a_{k}^{-1})\to\infty,$
and thus, for every $h\in\Gamma_{[u_{i}]},$
$\rho(u_{i},a_{k}ha_{k}^{-1})\to\infty.$

∎

Let $\Gamma_{\text{inf}}$ and $\Gamma_{\operatorname{f}}$ denote the union of
all infinite and all finite components, respectively.

The result of Lemma 2 can be generalised, it is shown in Lemma 4.

Define the $\ell_{q}$ norm of the potential $\varphi$ by the formula

$\left\lVert\varphi\right\rVert_{q}=\sqrt[q]{\sum_{g\in G}|\varphi(g)|^{q}}.$

######  Lemma 4.

Let the support of the derivation $d$ be contained in
$\Gamma_{\operatorname{inf}}$. Suppose a potential $\varphi$ of $d$ stabilises
to zero. Let $\\{a_{k}\\}_{k=1}^{\infty}$ be a sequence such that for each
$u\in\operatorname{\Gamma_{\operatorname{inf}}},$
$\rho(u,a_{k}ua_{k}^{-1})\to\infty.$ Then ${\lim_{k\to\infty}\left\lVert
d(a_{k})\right\rVert=\sqrt[q]{2}\left\lVert\varphi\right\rVert_{q}.}$

###### Proof.

In each infinite connected component choose origin $u_{i}$. Pick
$0<\varepsilon<1.$

Case 1: $0<\left\lVert\varphi\right\rVert_{q}<\infty$ (the case of zero norm
is trivial).

Choose $x_{0}=\left\lVert\varphi\right\rVert_{q}-\varepsilon/2$. Then, there
exists a finite set
$\mathcal{S}=\\{g_{1},\dots,g_{n}\\}\subset\operatorname{Obj}(\Gamma_{\operatorname{inf}}),$
such that

$\sum_{g\in\mathcal{S}}|\varphi(g)|^{q}>x_{0}^{q},\quad\sum_{g\notin\mathcal{S}}|\varphi(g)|^{q}<\min\left(\frac{\varepsilon}{2},\,\,2^{-q}\varepsilon\left(\left\lVert\varphi\right\rVert_{q}\right)^{q}\right),$

and $\varphi(g)\neq 0$ for any $g\in\mathcal{S}$. We will need the first part
of the minimum for the lower bound estimate and the second part for the upper
bound estimate.

Since $\mathcal{S}$ is a finite set it is contained in a finite union of
infinite connected components, that is, $\mathcal{S}\subset\cup_{i\in
I}\Gamma_{[u_{i}]},\,|I|<\infty.$ So, there exists $r>0$ such that
$\\{g_{1},\dots,g_{n}\\}\subset\cup_{i\in I}B_{r}(u_{i})$.

By hypothesis, the potential stabilises to zero at infinity on each component.
Therefore, for any $m>0$ there exists $R>r>0$ such that for each
$g\in\cup_{i\in I}\left(\Gamma[u_{i}]\setminus B_{R}(u_{i})\right),$ we have
$|\varphi(g)|<m$. Set
$m=\frac{\varepsilon}{2}\min\\{|\varphi(g_{1})|,\dots|\varphi(g_{n})|\\}$.
Since $G$ is a BC-group, the numbers $C_{i}=\sup_{g\in
G}{\operatorname{diam}(gB_{r}(u_{i})g^{-1})}$ are finite. So the constant
${C=\max_{i\in I}C(B_{r}(u_{i}))}$ is finite and satisfies

$\forall g\in G,\,\,\forall i\in I,\forall h_{1},h_{2}\in
B_{r}(u_{i})\hookrightarrow\rho(gh_{1}g^{-1},gh_{2}g^{-1})<C.$

By hypothesis and Proposition 5, there exists a number $N$ such that for each
$k\geqslant N$, for each $i\in I,$ we have
$\rho(u_{i},a_{k}u_{i}a_{k}^{-1})>R+C$ and
$\rho(u_{i},a_{k}^{-1}u_{i}a_{k})>R+C,$ remembering that $I$ is finite. Then
for all $h\in B_{r}(u_{i}),$ $k\geqslant N$ it follows that

$|\varphi(a_{k}ha_{k}^{-1})|<m,\quad|\varphi(a_{k}^{-1}ha_{k})|<m.$ (26)

Note that $\mathcal{S}\cap\left(a_{k}^{-1}\mathcal{S}a_{k}\right)=\emptyset$
by the triangle inequality.

Now from (15), we obtain lower bound:

$\displaystyle\left(\left\lVert
d(a_{k})\right\rVert\right)^{q}=\left\lVert\sum_{t\in
G}\left(\varphi(a_{k}ta_{k}^{-1})-\varphi(t)\right)a_{k}t\right\rVert^{q}={\sum_{t\in
G}\left|\varphi(a_{k}ta_{k}^{-1})-\varphi(t)\right|^{q}}\geqslant$
$\displaystyle\geqslant{\sum_{t\in\mathcal{S}}\left|\varphi(a_{k}ta_{k}^{-1})-\varphi(t)\right|^{q}}+{\sum_{t\in
a_{k}^{-1}\mathcal{S}a_{k}}\left|\varphi(a_{k}ta_{k}^{-1})-\varphi(t)\right|^{q}}=$
$\displaystyle={\sum_{i=1}^{n}\left|\varphi(a_{k}g_{i}a_{k}^{-1})-\varphi(g_{i})\right|^{q}}+{\sum_{i=1}^{n}\left|\varphi(g_{i})-\varphi(a_{k}^{-1}g_{i}a_{k})\right|^{q}}\geqslant$
$\displaystyle\geqslant
2{\sum_{i=1}^{m}\left|\left|\varphi(g_{i})\right|-m\right|^{q}}\geqslant
2(1-\frac{\varepsilon}{2})^{q}\sum_{i=1}^{n}|\varphi(g_{i})|^{q}\geqslant
2\left(\left\lVert\varphi\right\rVert_{q}\right)^{q}\left(1-\frac{\varepsilon}{2}\right)^{q}\left(1-\frac{\varepsilon}{2}\right).$
(27)

This means that

$\left\lVert
d(a_{k})\right\rVert\geqslant\sqrt[q]{2}\left\lVert\varphi\right\rVert_{q}(1-\frac{\varepsilon}{2})\sqrt[q]{1-\frac{\varepsilon}{2}}\geqslant(1-\varepsilon)\sqrt[q]{2}\left\lVert\varphi\right\rVert_{q}.$

Now, let us obtain upper bound:

$\displaystyle\left(\left\lVert d(a_{k})\right\rVert\right)^{q}={\sum_{t\in
G}\left|\varphi(a_{k}ta_{k}^{-1})-\varphi(t)\right|^{q}}\leqslant$
$\displaystyle{\sum_{t\in\mathcal{S}}\left|\varphi(a_{k}ta_{k}^{-1})-\varphi(t)\right|^{q}}+{\sum_{t\in
a_{k}^{-1}\mathcal{S}a_{k}}\left|\varphi(a_{k}ta_{k}^{-1})-\varphi(t)\right|^{q}}+{\sum_{t\notin\mathcal{S}\cup
a_{k}^{-1}\mathcal{S}a_{k}}\left|\varphi(a_{k}ta_{k}^{-1})-\varphi(t)\right|^{q}}\leqslant$
$\displaystyle
2\left(1+\frac{\varepsilon}{2}\right)^{q}\left(\left\lVert\varphi\right\rVert_{q}\right)^{q}+{\sum_{t\notin\mathcal{S}\cup
a_{k}^{-1}\mathcal{S}a_{k}}2^{q-1}\left(\left|\varphi(a_{k}ta_{k}^{-1})\right|^{q}+\left|\varphi(t)\right|^{q}\right)}\leqslant
2(1+\frac{\varepsilon}{2})^{q}\left\lVert\varphi\right\rVert_{q}^{q}+\frac{\varepsilon}{2}\left\lVert\varphi\right\rVert_{q}^{q}.$
(28)

To estimate the term ${\sum_{t\notin\mathcal{S}\cup
a_{k}^{-1}\mathcal{S}a_{k}}\left|\varphi(a_{k}ta_{k}^{-1})-\varphi(t)\right|^{q}}$
we used an inequality

$(|a|+|b|)^{q}\leqslant 2^{q-1}(|a|^{q}+|b|^{q}),$

and after that applied
$\sum_{g\notin\mathcal{S}}|\varphi(g)|^{q}<2^{-q}\varepsilon\left(\left\lVert\varphi\right\rVert_{q}\right)^{q}.$

Thus,

$\left\lVert
d(a_{k})\right\rVert\leqslant\left(1+\frac{\varepsilon}{2}\right)\sqrt[q]{2}\left\lVert\varphi\right\rVert\sqrt[q]{1+\frac{\varepsilon}{4}}\leqslant(1+\varepsilon)\sqrt[q]{2}\left\lVert\varphi\right\rVert.$

This gives us the desired result:

$\lim_{k\to\infty}\left\lVert
d(a_{k})\right\rVert=\sqrt[q]{2}\left\lVert\varphi\right\rVert_{q}.$ (29)

Case 2: $\left\lVert\varphi\right\rVert_{q}=\infty$.

Choose an arbitrary $x_{0}>0$. Then, there exist
$g_{1},\dots,g_{n}\in\cup_{i\in I}\operatorname{Obj}(\Gamma_{[u_{i}]})$ such
that

${|\varphi(g_{1})|^{q}+\dots+|\varphi(g_{n})|^{q}>x_{0}^{q}}.$

Analogously to how we calculate the lower bound, for large enough $k$ we
obtain

$\left(\left\lVert d(a_{k})\right\rVert\right)^{q}\geqslant
2{\sum_{i=1}^{m}\left|\left|\varphi(g_{i})\right|-m\right|^{q}}\geqslant
2\left(1-\frac{\varepsilon}{2}\right)x_{0}.$

Since $x_{0}$ could be chosen arbitrarily large, we conclude that sequence
converges to infinity. ∎

######  Corollary 1.

Let $G$ be a BC-group, and let ${d\colon f_{p}(G)\to\ell_{q}(G)}$ be a bounded
derivation with support in $\Gamma_{\text{inf}}$. Then $d$ has a potential
from $\ell_{q}.$

###### Proof.

Any potential stabilises by lemma 1. Choose such $\varphi$ that stabilises to
$0$. Consider a sequence $a_{k}$ from lemma 3. By lemma 4 we get

$\left\lVert\varphi\right\rVert_{q}=\lim_{i\to\infty}\left\lVert
d(a_{i})\right\rVert\leqslant\left\lVert d\right\rVert_{el},$

where $\left\lVert d\right\rVert_{el}=\sup_{g\in G}\left\lVert
d(g)\right\rVert.$ ∎

These results allow us to understand what happens on infinite components. It
turns out that if the sizes of finite components are uniformly bounded, then a
similar result holds for them.

######  Proposition 6.

Let the diameters of all finite connected components of
$\operatorname{sk}_{G}$ be uniformly bounded by some constant $N$. Then a
$G$-bounded derivation $d\colon f_{p}(G)\to\ell_{q}(G)$, whose support is
contained in $\Gamma_{\operatorname{f}}$, has a potential from $\ell_{q}.$

###### Proof.

In each finite component, pick a vertex $u_{i}$ and set the potential at that
vertex to zero. Consider a new generating system for the group $G$, containing
all words of length at most $N$. Denote this system by $\mathcal{A}$.

Then for each $g\in\Gamma_{\operatorname{f}}$, there exists a vertex $u_{i}$
(a vertex of the component containing $g$) and a generating element $a$ such
that $g=a^{-1}u_{i}a.$

Therefore,

$\sum_{g\in\Gamma_{\operatorname{f}}}|\varphi(g)|^{q}\leqslant\sum_{a\in\mathcal{A}}\sum_{u_{i}\in\Gamma_{\operatorname{f}}}|\varphi(a^{-1}u_{i}a)|^{q}\leqslant\sum_{a\in\mathcal{A}}\left\lVert
d(a)\right\rVert\leqslant|\mathcal{A}|\left\lVert d\right\rVert_{el}.$

So the potential belongs to $\ell_{q}.$ ∎

######  Theorem 2.

Let $G$ be a BC-group with uniformly bounded finite components. Then every
derivation $G$-bounded derivation $d\colon f_{p}(G)\to\ell_{q}(G)$ is inner.

###### Proof.

Let’s show that for such $d$, we can find a potential from $\ell_{q}.$ The
derivation $d$ can be represented as the sum
$d=d_{\text{inf}}+d_{\operatorname{f}}.$ The support of $d_{\text{inf}}$ is
contained in the infinite components, and the support of
$d_{\operatorname{f}}$ is contained in the finite components. Note that for
every $g\in G$, it holds $\left\lVert
d_{\text{inf}}(g)\right\rVert\leqslant\left\lVert
d(g)\right\rVert\leqslant\left\lVert d\right\rVert_{el},$ similarly for the
second term. This means that $d_{\operatorname{f}}$ and $d_{\text{inf}}$ are
$G$-bounded, and therefore, by Proposition 6 and Corollary 1 their potentials
lie in $\ell_{q}.$ Thus, we obtain

$\sum_{g\in
G}|\varphi(g)|^{q}=\sum_{g\in\Gamma_{\text{inf}}}|\varphi(g)|^{q}+\sum_{g\in\Gamma_{\operatorname{f}}}|\varphi(g)|^{q}=\sum_{g\in
G}|\varphi_{\text{inf}}(g)|^{q}+\sum_{g\in
G}|\varphi_{\operatorname{f}}(g)|^{q}<\infty.$ (30)

Thus, we have shown $\varphi$ lies in $\ell_{q}.$ ∎

######  Example 8.

Nilpotent groups of rank 2 are BC-groups with uniformly bounded finite
components.

Recall that a group $G$ is called a nilpotent group of rank 2 if its factor
group modulo the center $Z$ is commutative.

Choose a presentation of $G$ of the following form $\langle a_{1},a_{2},\dots
a_{m},b_{ij}\dots|R\rangle,$ where $b_{ij}=[a_{i}^{-1},a_{j}^{-1}]$.
Additionally, assume that if $a_{i}$ is a generator, then $a_{i}^{-1}$ is also
a generator, i.e., there exists $j$ such that $a_{j}={a_{i}}^{-1}.$ We
consider the group to be finitely generated, as always. We denote the center
of the group by $Z,$ and the subgroup generated by the commutators of
generators by $B=\langle b_{ij}\rangle\subset Z.$

######  Proposition 7.

In a nilpotent group of rank 2, the bounded connected components are uniformly
bounded by some constant $N.$

###### Proof.

The group $G^{\prime}=[G,G]$ is finitely generated. Indeed, for a nilpotent
group of rank 2, it holds that
$a_{i}a_{j}=a_{j}a_{i}a_{i}^{-1}a_{j}^{-1}a_{i}a_{j}=a_{j}a_{i}[a_{i}^{-1},a_{j}^{-1}]=a_{j}a_{i}b_{ij},$
where $b_{ij}\in Z.$ Consequently, the commutator of any two elements is equal
to the product of elements $b_{ij},$ and there is a finite number of $b_{ij}$.

In a finitely generated abelian group, the torsion subgroup is finite.
Consider a finite component $\operatorname{sk}_{u_{0}}$. Notice that
$gu_{0}g^{-1}=u_{0}b,$ where $b\in\operatorname{Tor}(G^{\prime}).$ Therefore,
the number of elements in each component is bounded by
$N=|\operatorname{Tor}(G^{\prime})|.$ ∎

######  Corollary 2.

In a nilpotent group of rank 2, every $G$-bounded ${d\colon
f_{p}(G)\to\ell_{q}(G)}$, has a potential from $\ell_{q},$ so $d$ is inner.

##  Appendix A Unbounded inner derivation

Consider the following matrices

$A_{p}=\begin{pmatrix}1&1&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix},\quad
A_{x}=\begin{pmatrix}1&0&0\\\ 0&1&1\\\ 0&0&1\end{pmatrix},\quad
A_{1}=\begin{pmatrix}1&0&1\\\ 0&1&0\\\ 0&0&1\end{pmatrix}.$ (31)

A matrix with elements $(a,b,c)$ can be represented as the product
$A_{x}^{b}A_{p}^{a}A_{1}^{c}$. Taking into account the relations, we have
$H_{3}(\mathbb{Z})=\langle
A_{x},A_{p},A_{1}\mid[A_{p},A_{x}]=A_{1},[A_{p},A_{1}]=E,[A_{x},A_{1}]=E\rangle.$
Let’s consider the connected component containing
$A_{x}^{b}A_{p}^{a}A_{1}^{c}.$ Evidently $a$ and $b$ are invariant under
conjugation and $c$ can change to values multiples of $a$ and $b.$ The
distance between elements is determined only by the difference in $c$, so it
does not change during conjugation.

The property of derivation $d$ to be $G$-bounded is weaker than to be
continuous. So unbounded inner derivations can exist.

Indeed, consider the Heisenberg group $G=H_{3}(\mathbb{Z}).$ We will examine
derivations of the form $d\colon\mathbb{C}[G]_{2}\to\mathbb{C}[G]_{2}.$

Let’s take $h=A_{p}$ and define a derivation in such a way that there is a
non-zero coefficient before $h$ in $d(A_{x}^{k})$. To achieve this, define the
potential as $\frac{1}{k}$ on the vertices
$A_{p}A_{x}^{-k}=A_{x}^{-k}A_{p}A_{1}^{-k}$ for $k\geqslant 1$ and set it to
zero on the remaining vertices.

Derivation $d$ defined by $\varphi$ is obviously inner and $G$-bounded since
$\varphi$ lies in $\ell_{2}.$ Now let us calculate the images of some
elements.

$\displaystyle d(A_{x})$ $\displaystyle=\sum_{t\in
G}\left(\varphi(A_{x}tA_{x}^{-1})-\varphi(t)\right)A_{x}t=\sum_{t\in
G}\varphi(t)tA_{x}-\sum_{t\in G}\varphi(t)A_{x}t=$
$\displaystyle=\sum_{k=1}^{\infty}\frac{1}{k}\left(A_{p}A_{x}^{-k+1}-A_{x}A_{p}A_{x}^{-k}\right)=\sum_{k=1}^{\infty}\frac{1}{k}\left(A_{x}^{-k+1}A_{p}A_{1}^{-k+1}-A_{x}^{-k+1}A_{p}A_{1}^{-k}\right)$
(32) $\displaystyle d(A_{x}^{m})$
$\displaystyle=\sum_{k=1}^{\infty}\frac{1}{k}\left(A_{x}^{-k+m}A_{p}A_{1}^{-k+m}-A_{x}^{-k+m}A_{p}A_{1}^{-k}\right))$
(33)

Consider the elements $a_{m}=\sum_{k=-m}^{m}A_{x}^{k}.$ The coefficient in
$d(a_{m})$ before $A_{x}^{-n}A_{p}A_{1}^{-n}$ is:

$\delta_{A_{x}^{-n}A_{p}A_{1}^{-n}}(d(a_{m}))=\sum_{k=-m}^{m}\delta_{A_{x}^{-n}A_{p}A_{1}^{-n}}(d(A_{x}^{k}))=\sum_{k=\max\\{-n+1,-m\\},\,k\neq
0}^{m}\frac{1}{k+n}=\sum_{j=1,\,j\neq
n}^{m+n}\frac{1}{j}\geqslant\sum_{j=2}^{m}\frac{1}{j},$ (34)

the last equality holds only when $n\leqslant m+1.$

Now we can estimate the norm of the image:

$\displaystyle\left\lVert
d(a_{m})\right\rVert\geqslant\left(\sum_{n=0}^{m}|\delta_{A_{x}^{-n}A_{p}A_{1}^{-n}}(d(a_{m}))|^{2}\right)^{\frac{1}{2}}\geqslant\sqrt{m}\left(\sum_{j=2}^{m}\frac{1}{j}\right),$

where $\left\lVert a_{m}\right\rVert=\sqrt{2m+1}.$ Thus, $\frac{\left\lVert
d(a_{m})\right\rVert}{\left\lVert a_{m}\right\rVert}\to\infty,$ so the
operator is unbounded.

## References

  * [Neu54] B.. Neumann  ‘‘Groups Covered By Permutable Subsets’’  In _Journal of the London Mathematical Society_ s1-29 Oxford University Press, 1954, pp. 236–248  DOI: 10.1112/jlms/s1-29.2.236
  * [JR69] B.. Johnson and J.. Ringrose  ‘‘Derivations of Operator Algebras and Discrete Group Algebras’’  In _Bulletin of the London Mathematical Society 1969-mar vol. 1 iss. 1_ 1, 1969  DOI: 10.1112/blms/1.1.70
  * [Dal01] H.. Dales  ‘‘Banach Algebras and Automatic Continuity’’, London Mathematical Society Monographs New Series  Oxford University Press, USA, 2001
  * [Joh01] B.. Johnson  ‘‘The Derivation Problem for Group Algebras of Connected Locally Compact Groups’’  In _Journal of the London Mathematical Society 2001-apr vol. 63 iss. 2_ 63, 2001  DOI: 10.1112/S002461070000185X
  * [Los08] V. Losert  ‘‘The Derivation Problem for Group Algebras’’  In _Annals of Mathematics 2008-jul vol. 168 iss. 1_ 168, 2008  DOI: 10.4007/annals.2008.168.221
  * [AMS16] A.. Arutyunov, A.. Mishchenko and A.. Shtern  ‘‘Derivations of group algebras’’  In _Фундамент. и прикл. матем._ 21, 2016  URL: https://www.mathnet.ru/eng/fpm1768
  * [AM19] A.. Arutyunov and A.. Mishchenko  ‘‘A smooth version of Johnson’s problem on derivations of group algebras’’  In _Mat. Sb._ 210, 2019  DOI: 10.1070/SM9119
  * [Aru20] A.. Arutyunov  ‘‘Derivation Algebra in Noncommutative Group Algebras’’  In _Proc. Steklov Inst. Math._ 308, 2020  DOI: 10.4213/tm4048
  * [Aru23] A. Arutyunov  ‘‘A combinatorial view on derivations in bimodules’’  In _Advances in Systems Science and Applications_ , 2023  DOI: 10.25728/assa.2023.23.2.1408


# Effective Preconditioners for Mixed-Dimensional Scalar Elliptic Problems

Xiaozhe Hu  Department of Mathematics, Tufts University, Medford, MA 02155,
USA  Center for Modeling of Coupled Subsurface Dynamics, Department of
Mathematics, University of Bergen, 5020, Bergen, Norway  Eirik Keilegavlen
e-mail<EMAIL_ADDRESS>Center for Modeling of Coupled Subsurface
Dynamics, Department of Mathematics, University of Bergen, 5020, Bergen,
Norway  Center for Modeling of Coupled Subsurface Dynamics, Department of
Mathematics, University of Bergen, 5020, Bergen, Norway  Jan M. Nordbotten
Center for Modeling of Coupled Subsurface Dynamics, Department of Mathematics,
University of Bergen, 5020, Bergen, Norway

(June 2022)

###### Abstract

Discretization of flow in fractured porous media commonly lead to large
systems of linear equations that require dedicated solvers. In this work, we
develop an efficient linear solver and its practical implementation for mixed-
dimensional scalar elliptic problems. We design an effective preconditioner
based on approximate block factorization and algebraic multigrid techniques.
Numerical results on benchmarks with complex fracture structures demonstrate
the effectiveness of the proposed linear solver and its robustness with
respect to different physical and discretization parameters.

##  1 Introduction

Mixed-dimensional scalar elliptic equations are the backbone of models across
many applications wherein potentially intersecting thin layers are embedded
into a material. This application is of particular relevance in subsurface
flows, where the thin layers correspond to fracture networks embedded in a
porous material.

Flow in fractured porous media can be modeled in several ways. However, three
of the main modeling approaches result in essentially mixed-dimensional
elliptic equations. In sequence, these are 1) Direct mixed-dimensional
modeling [13, 5, 4], 2) Discrete fracture networks [11, 20], and 3) Low-order
numerical methods for equidimensional fracture models [21, 28]. These modeling
approaches are presented in detail in the recent review paper [2], and two
community benchmark studies have recently been conducted [12, 3]. The relative
merits of the three modeling approaches have been extensively analyzed in the
literature. However, for our purpose, it suffices to point out that after
discretization by standard numerical methods, the resulting algebraic linear
systems have a similar structure.

Despite the importance of modeling flow in fractured porous media, the
computational cost of linear solvers has not received much attention, although
linear solvers for discrete fracture networks recently were studied in [17].
Indeed, in recent benchmark studies, the computational cost is simply
represented by proxy in terms of the condition number of the system matrix.
The construction, or even existence, of efficient iterative solvers can not be
taken for granted. The material contrasts between porous rock and fracture may
be significant, which is known to cause a challenge for many iterative
algorithms, see [10, 24, 25, 27, 16, 29] and references therein.

This technical note presents an efficient iterative linear solver for the
prototypical mixed-dimensional scalar elliptic problem, which serves as a
building block for solving other mixed-dimensional problems, e.g., [9] and
[8]. We explore the block structure of the results linear systems and design
effective block preconditioners based on approximate block factorization and
algebraic multigrid (AMG) methods. The successful application requires several
essential steps. Firstly, because the blocks corresponding to fractures are
usually smaller in size and lower in dimension, we build an approximation
Schur complement on the domain via a simple diagonal approximation to the
fracture block. Secondly, for the diagonal blocks in the block
preconditioners, we employ existing AMG methods to approximately invert the
diagonal blocks via several steps of V-cycles. Finally, the block
preconditioners are used to accelerate the Krylov iterative methods, which
results in an effective and robust linear solver.

This combination of stable discretization and efficient linear solvers has not
been reported in the literature to the best of our knowledge. The novelty of
this technical note is identifying an efficient and robust iterative solver
for mixed-dimensional scalar elliptic equations. We support this claim by
applying the solver to one 2D benchmark problem with complex fracture networks
[12], and to two 3D benchmark problems [3] with structured and complex
fracture networks, respectively, and show robustness (in terms of iteration
count until convergence) across a range of material parameters and grid
resolutions.

##  2 Preliminaries

###  2.1 Mixed-dimensional Scalar Elliptic Problems

Our model for flow in fractured porous media is the same as used in the
benchmark for 3d flow [3] and several other works, e.g., [5, 15]. We model the
fractures as $(N-1)$-dimensional objects embedded in the N-dimensional host
medium $\Omega\subset\mathbb{R}^{N}$, with fracture intersections forming
lines and points of dimensions $(N-2)$ and $(N-3)$. We refer to these
geometric objects as subdomains and let $\Omega_{i}$ denote an arbitrary
subdomain, so that the computational domain
$\Omega=\bigcup_{i=1}^{m}\Omega_{i}$. The subdomains are connected via
interfaces, denoted $\Gamma_{j},j=\\{1,...,M\\}$. If a $D$-dimensional
interface $\Gamma_{j}$ connects subdomains $\Omega_{\hat{j}}$ and
$\Omega_{\check{j}}$, of dimension $D+1$ and $D$, respectively, we have that
geometrically $\Gamma_{j}=\Omega_{\check{j}}$, and furthermore denote the part
of the boundary of $\Omega_{\hat{j}}$ coinciding with $\Gamma_{j}$ as
$\Gamma_{j}=\partial_{j}\Omega_{\hat{j}}$. Finally, for $\Omega_{i}$, let
$\hat{S}_{i}$ be the set of interfaces towards neighboring subdomains of
higher dimension, such that $j\in\hat{S}_{i}$ if and only if $\check{j}=i$.
The geometry is illustrated in Figure 1, where we note that a combination of
fracture geometry and boundary condition can lead to subdomains (of dimension
$N$ or lower) being disconnected from Dirichlet boundary conditions.

Figure 1: Left: A mixed-dimensional geometry consisting of three fractures,
one intersection point, and with the top-dimensional domain split into two.
Possible boundary conditions are indicated. Right: Geometry of the interfaces;
$\Gamma_{4}$ is included as a placeholder for a no-flux condition on the tip
of $\Omega_{2}$.

With this notion of geometry, the model for single-phase fluid flow in
$\Omega_{i}$ is given in strong form as,

$\nabla_{\parallel}\cdot(-K_{i,\parallel}\nabla_{\parallel}p_{i})-\sum\limits_{j\in\hat{S}_{i}}\lambda_{j}=f_{i}.$
(1)

Here, $K_{i}$, $p_{i}$, and $f_{i}$ represent permeability, pressure, and
source terms in subdomain $\Omega_{i}$, and subscript $\parallel$ indicates
that operators are tangential to $\Omega_{i}$. The flux over $\Gamma_{j}$,
denoted $\lambda_{j}$, is governed by a Darcy-type law on the form

$\lambda_{j}=-\kappa_{j}(p_{i}-p_{\hat{j}}),$ (2)

The constant $\kappa_{j}$ can be considered related to the normal
transmissivity across $\Gamma_{j}$. Thus the terms $\lambda_{j}$ represent
flow through $\Gamma_{j}$ from $\Omega_{\hat{j}}$ to $\Omega_{i}$. This same
flow through $\Gamma_{j}$ then appears as a Neumann boundary term on
$\partial_{j}\Omega_{\hat{j}}$,

$(-K_{\hat{j},\parallel}\nabla_{\parallel}p_{\hat{j}})\cdot\mathbf{n}_{\hat{j}}=\lambda_{j}$
(3)

where $\mathbf{n}_{\hat{j}}$ is the outer normal vector on the boundary of
$\Omega_{\hat{j}}$.

Equations (1) remain valid also for the highest-dimensional domains (when
$\hat{S}_{i}=\emptyset$, and as such the summation is void), and also for the
lowest-dimensional point intersections (when there is no parallel directions,
and as such the differential operators are void).

###  2.2 Discretization

We discretize the flow model following the unified framework developed in
[26], wherein the subdomains are discretized independently with the interface
fluxes treated as interior boundary terms. The framework is compatible with
various discretization methods for elliptic equations; herein, we apply the
finite volume multipoint flux approximation method [1] to discretize subdomain
problems, but mixed and virtual finite elements have also been used [26].

##  3 Preconditioners for Mixed-dimensional Problems

In this section, we introduce the multigrid-based solver for solving the
discretized mixed-dimensional problem (1) and (2). Our solver takes advantage
of the block structure of the linear systems after discretization. More
precisely, we use the block structure to perform a block factorization and
develop preconditioners based on the block factorization via appropriate
approximated Schur complement. Finally, the multigrid method is used to obtain
an efficient and practical preconditioner.

###  3.1 Block Structure of the Linear Systems of Equations

Since (1) is defined in the domain and (2) is defined on the interface, our
the discretized linear systems naturally inherits the following two-by-two
block structure,

$\mathcal{A}p=f\quad\Leftrightarrow\begin{pmatrix}A_{\Omega\Omega}&A_{\Omega\Gamma}\\\
A_{\Gamma\Omega}&A_{\Gamma\Gamma}\\\ \end{pmatrix}\begin{pmatrix}p_{\Omega}\\\
p_{\Gamma}\end{pmatrix}=\begin{pmatrix}f_{\Omega}\\\ f_{\Gamma}\end{pmatrix},$
(4)

where the subscripts $\Omega$ and $\Gamma$ denote the blocks related to the
domain and the interface, respectively. Correspondingly $p_{\Omega}$ and
$p_{\Gamma}$ denote the unknowns in all the subdomain $\Omega_{i}$ and all the
interfaces $\Gamma_{i}$, respectively. The off-diagonal blocks
$A_{\Omega\Gamma}$ and $A_{\Gamma\Omega}$ denote the interaction and coupling
between the domain and the interfaces. As our implementation follows the
unified discretization approach for fracture [26], we are guaranteed that
$A_{\Omega\Gamma}=A_{\Gamma\Omega}^{T}$. However, for the sake of generality,
we will not exploit this property in the following discussion.

A block factorization-based approach is a natural choice for solving the
linear systems with block structure, such as (4). There are two types of block
factorization for a two-by-two block system which give different Schur
complements. Here, since $A_{\Gamma\Gamma}$ is defined on lower dimension
interfaces, it is relatively smaller and easier to invert compared with
$A_{\Omega\Omega}$. Hence, we use the following form of block factorization.

$\mathcal{A}=\begin{pmatrix}A_{\Omega\Omega}&A_{\Omega\Gamma}\\\
A_{\Gamma\Omega}&A_{\Gamma\Gamma}\\\
\end{pmatrix}=\begin{pmatrix}I&A_{\Omega\Gamma}A_{\Gamma\Gamma}^{-1}\\\
0&I\end{pmatrix}\begin{pmatrix}S_{\Omega\Omega}&0\\\
0&A_{\Gamma\Gamma}\end{pmatrix}\begin{pmatrix}I&0\\\
A_{\Gamma\Gamma}^{-1}A_{\Gamma\Omega}&I\end{pmatrix}=:\mathcal{U}\mathcal{D}\mathcal{L},$
(5)

where $S_{\Omega\Omega}$ is the well-known Schur complement defined as

$S_{\Omega\Omega}:=A_{\Omega\Omega}-A_{\Omega\Gamma}A_{\Gamma\Gamma}^{-1}A_{\Gamma\Omega}.$
(6)

Such a block factorization (5) serves the building block of our proposed block
preconditioner.

###  3.2 Factorization-based Block Preconditioner

Based on the block factorization (5), we can immediately propose several block
preconditioners, e.g., $\mathcal{B}_{D}=\mathcal{D}^{-1}$ can be used as a
block diagonal preconditioner, $\mathcal{B}_{U}:=(\mathcal{UD})^{-1}$ and
$\mathcal{B}_{L}:=(\mathcal{DL})^{-1}$ can be used as block upper and lower
triangular preconditioners, respectively. For the sake of simplicity, in this
subsection, we focus our discussion on $\mathcal{B}_{L}$ (and its variants).

If we use $\mathcal{B}_{L}$ as a right preconditioner for Krylov iterative
methods, such as general minimal residual (GMRes) method, we need to look at
the spectrum of the preconditioned linear system $\mathcal{A}\mathcal{B}_{L}$.
From (5), we have
$\mathcal{A}\mathcal{B}_{L}=\mathcal{UDL}(\mathcal{DL})^{-1}=\mathcal{U}$.
This immediately implies that the eigenvalues of $\mathcal{A}\mathcal{B}_{L}$
are $\lambda(\mathcal{A}\mathcal{B}_{L})=\lambda(\mathcal{U})=1$. Therefore,
$\mathcal{B}_{L}$ is an efficient and robust preconditioner for solving (4).

Note that the action of

$\mathcal{B}_{L}=\begin{pmatrix}S_{\Omega\Omega}&0\\\
A_{\Gamma\Omega}&A_{\Gamma\Gamma}\end{pmatrix}^{-1}$

requires computing the Schur complement $S_{\Omega\Omega}$, the inverse of the
Schur complement $S_{\Omega\Omega}^{-1}$, and $A_{\Gamma\Gamma}^{-1}$. All
these steps could be quite expensive and we instead approximate them one by
one.

For approximating the Schur complement, we use the fact that
$A_{\Gamma\Gamma}$ in general is diagonally dominated and replace
$A_{\Gamma\Gamma}$ in (6) by its diagonal
$\operatorname{diag}(A_{\Gamma\Gamma})$. Therefore, the approximate Schur
complement is defined as

$\widetilde{S}_{\Omega\Omega}:=A_{\Omega\Omega}-A_{\Omega\Gamma}\left(\operatorname{diag}(A_{\Gamma\Gamma})\right)^{-1}A_{\Gamma\Omega}.$
(7)

This approximation leads to the following approximate block lower triangular
preconditioner

$\widetilde{\mathcal{B}}_{L}=\begin{pmatrix}\widetilde{S}_{\Omega\Omega}&0\\\
A_{\Gamma\Omega}&A_{\Gamma\Gamma}\end{pmatrix}^{-1}.$

Unfortunately, exactly inverting $\widetilde{\mathcal{B}}_{L}$ is still
expensive and, therefore, we approximately invert the two blocks on the main
diagonal, and obtain the following block lower triangular preconditioner which
we use in our numerical experiments,

$\mathcal{M}_{L}=\begin{pmatrix}Q^{-1}_{\Omega\Omega}&0\\\
A_{\Gamma\Omega}&Q^{-1}_{\Gamma\Gamma}\end{pmatrix}^{-1},$ (8)

where $Q_{\Omega\Omega}$ and $Q_{\Gamma\Gamma}$ approximate
$\widetilde{S}_{\Omega\Omega}^{-1}$ and $A_{\Gamma\Gamma}^{-1}$, respectively.

###  3.3 Practical Implementation

In practice, given a right hand side $r=(r_{\Omega},r_{\Gamma})^{T}$, an
algorithm for the action of the precondtioner $\mathcal{M}_{L}$ is shown in
Algorithm 1.

Algorithm 1 Action of the precontioner $\mathcal{M}_{L}$:
$z\leftarrow\mathcal{M}_{L}r$

1:Solve in the domain: $z_{\Omega}\leftarrow Q_{\Omega\Omega}r_{\Omega}$

2:Update the interface residual: $r_{\Gamma}\leftarrow
r_{\Gamma}-A_{\Gamma\Omega}z_{\Omega}$

3:Solve on the interface: $z_{\Gamma}\leftarrow Q_{\Gamma\Gamma}r_{\Gamma}$

4:Output the update: $z\leftarrow(z_{\Omega},z_{\Gamma})^{T}$

The algorithm requires properly choosen $Q_{\Omega\Omega}$ and
$Q_{\Gamma\Gamma}$. Since we are solving the scalar elliptic problem, we do
this by applying one V-cycle of an algebraic multigrid (AMG) method to
$\widetilde{S}_{\Omega\Omega}$ and $A_{\Gamma\Gamma}$, respectively.
Specifically, our implementation uses smoothed aggregation AMG (SA-AMG)
methods to balance the computational complexity and convergence behavior [30,
6, 7, 19]. In general, other variants of AMG methods can be applied and
similarly the number of V-cycle steps can be modified. However, in our
numerical experiments, it seems that one V-cycle is sufficient to provide a
good approximation and, as a result, leads to an effective preconditioner.

We comment that $\widetilde{S}_{\Omega\Omega}$ and $A_{\Gamma\Gamma}$ have
block structures themselves since we put subdomains $\Omega_{i}$ and
interfaces $\Gamma_{i}$ of different dimensions together. Therefore, it is
possible to design special geometric and algebraic MG methods for
approximating $\widetilde{S}_{\Omega\Omega}$ and $A_{\Gamma\Gamma}$. For
example, in the setup phase of SA-AMG, one could carefully design coarsening
strategies so that aggregations will be constructed within the subdomains and
interfaces of the same dimension and then form aggregations that possibly
cross different dimensions. In addition, if the subdomains $\Omega_{i}$ and
$\Gamma_{i}$ are in the one-dimensional space, the cost of directly inverting
the corresponding block is relatively negligible. In this work, our choice,
SA-AMG implemented in the HAZmath package [18], already provides a good
performance, and the specially tailored strategies suggested above do not
appear to be necessary.

##  4 Numerical Results

In this section, we present numerical results to demonstrate the effectiveness
of the proposed block preconditioner for solving the linear systems of
equations after discretizing the mixed-dimensional scalar elliptic problem (1)
and (2). We use the preconditioner $\mathcal{M}_{L}$ (8) to accelerate the
GMRes method. One V-cycle SA-AMG with one step of Gauss-Seidel iteration as
both pre-and post-smoothing steps is used to define $Q_{\Omega\Omega}$ and
$Q_{\Gamma\Gamma}$, respectively. In all our numerical experiments, we use a
zero initial guess, and the GMRes method terminates when the relative residual
is smaller than $10^{-6}$. Numerical experiments are conducted on a
workstation with an 8-core 3GHz Intel Xeon “Sandy Bridge” CPU and 256 GB of
RAM. The software packages used are PorePy [23] (for the discretization of the
mixed-dimensional scalar elliptic problems) and HAZmath [18] (for the
preconditioners and iterative solvers for solving the linear systems of
equations). The meshes are generated by Gmsh [14], using PorePy’s interface to
Gmsh to control the mesh size. The runscripts used to produce the results
presented below are available at [22].

Figure 2: Left: Fracture geometry (black lines) and grid for the complex 2d case. Middle: Fracture geometry and grid for the regular 3d case; 2d objects are colored by the pressure solution. Right: Fracture network (shown as bold lines) and grid for the 3d field case, top view. | 2D Complex | 3D Regular | 3D Field
---|---|---|---
$K_{\parallel}$ | $\kappa$ |  $h$=20 |  $h$=10 |  $h$=5 |  $h$=0.2 |  $h$=0.1 |  $h$=0.05 |  $h_{f}$=80 |  $h_{f}$=40 |  $h_{f}$=20
$10^{-4}$ | $10^{-4}$ | 12 | 12 | 18 | 7 | 7 | 8 | 18 | 16 | 34
$10^{-4}$ | $1$ | 13 | 12 | 14 | 19 | 20 | 25 | 36 | 51 | 44
$10^{-4}$ | $10^{4}$ | 16 | 16 | 19 | 14 | 14 | 16 | 68 | 68 | 44
$1$ | $10^{-4}$ | 10 | 10 | 17 | 8 | 10 | 11 | 13 | 15 | 32
$1$ | $1$ | 13 | 12 | 13 | 8 | 7 | 8 | 43 | 64 | 50
$1$ | $10^{4}$ | 16 | 16 | 19 | 13 | 13 | 13 | 62 | 73 | 47
$10^{4}$ | $10^{-4}$ | 8 | 7 | 10 | 8 | 8 | 8 | 12 | 17 | 28
$10^{4}$ | $1$ | 28 | 28 | 28 | 8 | 10 | 10 | 32 | 40 | 45
$10^{4}$ | $10^{4}$ | 23 | 23 | 27 | 10 | 9 | 9 | 181 | 74 | 55
Table 1: Number of iterations of GMRes with block preconditioner
$\mathcal{M}_{L}$ when varying mesh size $h$ (or mesh size of the fracture
$h_{f}$), tangential fracture permeability $K_{\parallel}$, and normal
fracture permeability $\kappa$. Left: 2D complex fracture example. Middle: 3D
Regular fracture example. Right: 3D field example.

###  4.1 2D Complex Fracture Example

For the first example, we choose a 2D example with a complex fracture
configuration [12] to demonstrate the robustness of the block preconditioner
on a realistic fracture network. Such a complex fracture structure often
occurs in geological rock simulations, where the geometrical and physical
properties of the fracture network can significantly influence the stability
of the linear solvers. In particular, as we can see from Figure 2 (Left), the
fractured porous medium domain where tips and very acute intersections may
decrease the shape regularity of the mesh. Hence, this test case is well
suited to show the robustness of the block preconditioner with respect to
challenging geometric configurations that are common in realistic fracture
geometries.

In the numerical experiments, we set the permeability of the matrix to be $1$.
The tangential and normal permeability of the fractures, denoted
$K_{\parallel}$ and $\kappa$, respectively, are constants throughout the whole
network. To show the robustness of the block preconditioner with respect to
the discretization parameter $h$ and physical parameters $K_{\parallel}$ and
$\kappa$, we perform a set of tests in which we vary the values of those
parameters.

Table 1(Left) summarizes the numerical results. Each row in the table
represents a set of tests where the tangential and normal fracture
permeability values are fixed and the mesh size $h$ is changing. It should be
mentioned that these are mesh sizes prescribed to PorePy, but in practice,
local geometric details combined with Gmsh’s internal functionality to
generate high-quality meshes may lead to further local grid refinement. As we
can see, the numbers of GMRes iterations grow slightly but remain under $30$
iterations, which demonstrates the robustness of the preconditioner with
respect to the mesh size $h$. On the other hand, each column of the table
presents a set of tests where the mesh size is fixed, and the tangential and
normal fracture permeability values vary. Again, the numbers of GMRes
iteration remain stable as expected, which demonstrate the robustness of the
block preconditioner with respect to the physical parameters. Overall, Table
1(Left) illustrates the robustness of the block preconditioner for this 2D
complex fracture benchmark.

###  4.2 3D Regular Fracture Example

Now we consider a 3D problem from a benchmark study [3]. The 3D geometry is a
unit cube as shown in Figure 2 (Middle). The fracture network consists of $9$
fracture planes, $69$ intersection lines, and $27$ intersection points. Again,
to show the robustness of the block preconditioner with respect to the
discretization and physical parameters, we vary the values of the tangential
and normal fracture permeability and the mesh size, keeping the permeability
of the porous medium unitary.

Table 1(Middle) shows the number of GMRes iterations for this 3D regular
fracture example. We can see that the numerical results are consistent with
our 2D example, i.e., the block preconditioners show robustness with respect
to the mesh size $h$, the tangential fracture permeability $K_{\parallel}$,
and normal fracture permeability $\kappa$.

###  4.3 3D Field Example

Our last example is a simulation of a 3D field benchmark with a realistic
fracture network [3]. The domain is
$\Omega=(-500,350)\times(100,1500)\times(-100,500)$, and a cross-section of
the domain is shown in Figure 2 (Right) where we can see the complex fracture
network and computational grid. In this example, we again set the permeability
in the matrix to be $1$ and vary the values of the tangential and normal
fracture permeability and the mesh size in our numerical experiments, which
are the same as in previous examples. However, due to the complexity of this
example and computational cost consideration, we fix the far-field mesh size
of the matrix to be $100$ and vary the degree of mesh refinement towards the
fractures, mesh size in the fractures (denoted by $h_{f}$) only. Larger values
of $h_{f}$ give more irregular the elements, especially near the intersection
of the fractures and many tightly packed fractures, thus, the performance of
the linear solver is expected to improve as $h_{f}$ is decreased.

The results for the 3D field example are reported in Table 1(Right). As
expected, when $h_{f}$ is large, the mesh quality influences the overall
performance, and the number of GMRes iterations varies quite a bit, see the
columns of $h_{f}=80$ in Table 1(Right). When the fracture mesh size $h_{f}$
gets smaller, the number of GMRes iterations stabilizes, and the performance
is robust with respect to the discretization and physical parameters as we
expected. Due to the complex geometry, the overall number of iterations is
higher than the 3D regular fracture example. Therefore, for complex fracture
networks, we suggest investing in constructing a more regular mesh of the
fractured porous medium and then applying the proposed block preconditioners
in the iterative solvers.

##  5 Conclusions

Based on the block structure of the linear systems arising from discretizing
the mixed-dimensional scalar elliptic problems, we are able to develop block
preconditioners based on approximate factorization. We first properly
approximate the Schur complement to obtain a block preconditioner and then
apply the AMG methods to invert the diagonal blocks in our practical
implementation. Several benchmarks in 2D and 3D are considered. From the
numerical results, the GMRes methods accelerated by our proposed
preconditioner are robust with respect to the physical and discretization
parameters and complex fracture structures, making it attractive for real-
world applications.

## Acknowledgments

This work was financed in part by Norwegian Research Council grant number
308733.

## References

  * [1] Ivar Aavatsmark  “An introduction to multi-point flux approximations for quadrilateral grids”  In _Computational Geosciences_ 6 Springer, 2002, pp. 405–432  DOI: 10.1023/A:1021291114475
  * [2] Inga Berre, Florian Doster and Eirik Keilegavlen  “Flow in fractured porous media: A review of conceptual models and discretization approaches”  In _Transport in Porous Media_ 130.1 Springer, 2019, pp. 215–236  DOI: 10.1007/s11242-018-1171-6
  * [3] Inga Berre et al.  “Verification benchmarks for single-phase flow in three-dimensional fractured porous media”  In _Advances in Water Resources_ 147 Elsevier, 2021, pp. 103759  DOI: 10.1016/j.advwatres.2020.103759
  * [4] W.. Boon, J. Nordbotten and J.. Vatne  “Functional analysis and exterior calculus on mixed-dimensional geometries”  In _Annali di Matematica Pura ed Applicata (1923-)_ Springer, 2021, pp. 757–789  DOI: 10.1007/s10231-020-01013-1
  * [5] Wietse M Boon, Jan M Nordbotten and Ivan Yotov  “Robust discretization of flow in fractured porous media”  In _SIAM Journal on Numerical Analysis_ 56.4 SIAM, 2018, pp. 2203–2233  DOI: 10.1137/17M1139102
  * [6] Marian Brezina and Jan Mandel  “Convergence of algebraic multigrid based on smoothed aggregation”  In _Numerische Mathematik_ 88.3 Springer, 2001, pp. 559–579  DOI: 10.1007/s002110000226
  * [7] Marian Brezina, Petr Vaněk and Panayot S Vassilevski  “An improved convergence analysis of smoothed aggregation algebraic multigrid”  In _Numerical Linear Algebra with Applications_ 19.3 Wiley Online Library, 2012, pp. 441–469  DOI: 10.1002/nla.775
  * [8] Ana Budiša, Wietse M Boon and Xiaozhe Hu  “Mixed-dimensional auxiliary space preconditioners”  In _SIAM Journal on Scientific Computing_ 42.5 SIAM, 2020, pp. A3367–A3396  DOI: 10.1137/19M1292618
  * [9] Ana Budiša and Xiaozhe Hu  “Block preconditioners for mixed-dimensional discretization of flow in fractured porous media”  In _Computational Geosciences_ 25.2 Springer, 2021, pp. 671–686  DOI: 10.1007/s10596-020-09984-z
  * [10] Tony F Chan and Wing-Lok Wan  “Robust multigrid methods for nonsmooth coefficient elliptic linear systems”  In _Journal of Computational and Applied Mathematics_ 123.1-2 Elsevier, 2000, pp. 323–352  DOI: 10.1016/S0377-0427(00)00411-8
  * [11] Jocelyne Erhel, Jean-Raynald Dreuzy and Baptiste Poirriez  “Flow Simulation in Three-Dimensional Discrete Fracture Networks”  In _SIAM Journal on Scientific Computing_ 31.4, 2009, pp. 2688–2705  DOI: 10.1137/080729244
  * [12] Bernd Flemisch et al.  “Benchmarks for single-phase flow in fractured porous media”  In _Advances in Water Resources_ 111, 2018, pp. 239–258  DOI: 10.1016/j.advwatres.2017.10.036
  * [13] Najla Frih, Vincent Martin, Jean Elizabeth Roberts and Ali Saâda  “Modeling fractures as interfaces with nonmatching grids”  In _Computational Geosciences_ 16.4 Springer, 2012, pp. 1043–1060  DOI: 10.1007/s10596-012-9302-6
  * [14] Christophe Geuzaine and Jean-François Remacle  “Gmsh: A 3-D finite element mesh generator with built-in pre-and post-processing facilities”  In _International Journal for Numerical Methods in Engineering_ 79.11 Wiley Online Library, 2009, pp. 1309–1331  DOI: 10.1002/nme.2579
  * [15] Dennis Gläser, Martin Schneider, Bernd Flemisch and Rainer Helmig  “Comparison of cell-and vertex-centered finite-volume schemes for flow in fractured porous media”  In _Journal of Computational Physics_ 448 Elsevier, 2022, pp. 110715  DOI: 10.1016/j.jcp.2021.110715
  * [16] Ian G Graham and MJ Hagger  “Unstructured additive Schwarz–conjugate gradient method for elliptic problems with highly discontinuous coefficients”  In _SIAM Journal on Scientific Computing_ 20.6 SIAM, 1999, pp. 2041–2066  DOI: 10.1137/S1064827596305593
  * [17] SY Greer, JD Hyman and Daniel O’Malley  “A Comparison of Linear Solvers for Resolving Flow in Three-Dimensional Discrete Fracture Networks”  In _Water Resources Research_ 58.4 Wiley Online Library, 2022, pp. WR031188
  * [18] Xiaozhe Hu, James H. Adler and Ludmil T. Zikatanov  “HAZmath: A Simple Finite Element, Graph, and Solver Library”, 2022
  * [19] Xiaozhe Hu and Panayot S Vassilevski  “Modifying AMG coarse spaces with weak approximation property to exhibit approximation in energy norm”  In _SIAM Journal on Matrix Analysis and Applications_ 40.3 SIAM, 2019, pp. 1131–1152  DOI: 10.1137/18M1165190
  * [20] Jeffrey D Hyman et al.  “dfnWorks: A discrete fracture network framework for modeling subsurface flow and transport”  In _Computers & Geosciences_ 84 Elsevier, 2015, pp. 10–19  DOI: 10.1016/j.cageo.2015.08.001
  * [21] Mohammad Karimi-Fard, Luis J Durlofsky and Khalid Aziz  “An efficient discrete-fracture model applicable for general-purpose reservoir simulators”  In _SPE journal_ 9.02 OnePetro, 2004, pp. 227–236  DOI: 10.2118/88812-PA
  * [22] Eirik Keilegavlen and Xiaozhe Hu  “Runscripts for linear solvers for mixed-dimensional flow problems”, 2022  URL: https://doi.org/10.5281/zenodo.6593919
  * [23] Eirik Keilegavlen et al.  “Porepy: An open-source software for simulation of multiphysics processes in fractured porous media”  In _Computational Geosciences_ 25.1 Springer, 2021, pp. 243–265  DOI: 10.1007/s10596-020-10002-5
  * [24] Jan Mandel and Marian Brezina  “Balancing domain decomposition for problems with large jumps in coefficients”  In _Mathematics of Computation_ 65.216, 1996, pp. 1387–1401  DOI: 10.1090/s0025-5718-96-00757-0
  * [25] SV Nepomnyaschikh  “Preconditioning operators for elliptic problems with bad parameters”  In _Eleventh International Conference on Domain Decomposition Methods_ , 1999, pp. 82–88  Citeseer
  * [26] Jan Martin Nordbotten, Wietse M Boon, Alessio Fumagalli and Eirik Keilegavlen  “Unified approach to discretization of flow in fractured porous media”  In _Computational Geosciences_ 23.2 Springer, 2019, pp. 225–237  DOI: 10.1007/s10596-018-9778-9
  * [27] Peter Oswald  “On the robustness of the BPX-preconditioner with respect to jumps in the coefficients”  In _Mathematics of Computation_ 68.226, 1999, pp. 633–650
  * [28] T.. Sandve, I. Berre and J.. Nordbotten  “An efficient multi-point flux approximation method for Discrete Fracture–Matrix simulations”  In _Journal of Computational Physics_ 231.9, 2012, pp. 3784–3800  DOI: 10.1016/j.jcp.2012.01.023
  * [29] Ulrich Trottenberg, Cornelius W Oosterlee and Anton Schuller  “Multigrid”  Elsevier, 2000
  * [30] Petr Vaněk, Jan Mandel and Marian Brezina  “Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems”  In _Computing_ 56.3 Springer, 1996, pp. 179–196  DOI: 10.1007/BF02238511


# Recurrent Neural Networks for Modelling Gross Primary Production

David Montero1,2,* Miguel D. Mahecha1,2,3,4 Francesco Martinuzzi1,4
César Aybar5 Anne Klosterhalfen6 Alexander Knohl6
Franziska Koebsch6 Jesús Anaya7 Sebastian Wieneke1
1RSC4Earth, IEF, Leipzig University 2iDiv 3UFZ 4ScaDS.AI 5IPL, Universitat de
València
6Bioclimatology, University of Göttingen 7GEMA, Universidad de Medellín
*Corresponding author<EMAIL_ADDRESS>
###### Abstract

Accurate quantification of Gross Primary Production (GPP) is crucial for
understanding terrestrial carbon dynamics. It represents the largest
atmosphere-to-land CO2 flux, especially significant for forests. Eddy
Covariance (EC) measurements are widely used for ecosystem-scale GPP
quantification but are globally sparse. In areas lacking local EC
measurements, remote sensing (RS) data are typically utilised to estimate GPP
after statistically relating them to in-situ data. Deep learning offers novel
perspectives, and the potential of recurrent neural network architectures for
estimating daily GPP remains underexplored. This study presents a comparative
analysis of three architectures: Recurrent Neural Networks (RNNs), Gated
Recurrent Units (GRUs), and Long-Short Term Memory (LSTMs). Our findings
reveal comparable performance across all models for full-year and growing
season predictions. Notably, LSTMs outperform in predicting climate-induced
GPP extremes. Furthermore, our analysis highlights the importance of
incorporating radiation and RS inputs (optical, temperature, and radar) for
accurate GPP predictions, particularly during climate extremes.

##  1 Introduction

Terrestrial ecosystems are pivotal in carbon sequestration (Friedlingstein et
al., 2023), with forests contributing significantly to this process (Bonan,
2008). The single largest flux in this context is the gross uptake of CO2 by
vegetation “Gross Primary Production” (GPP). Hence, accurately quantifying
forest GPP is essential for monitoring terrestrial carbon dynamics.
Traditionally, GPP at the ecosystem scale is estimated using in-situ
measurements of net CO2 exchange (NEE) obtained through Eddy Covariance (EC)
systems. Despite the growing deployment of EC measurements within global
networks (e.g. Fluxnet, Baldocchi, 2019) , which are built on regional or
continental initiatives (e.g. ICOS, Asiaflux, Ameriflux), these networks still
exhibit a sparse global distribution (Schimel et al., 2015). Remote sensing
data, particularly Vegetation Indices (VIs), are commonly employed as weak
proxies for GPP estimation where no local EC estimations are performed (Zeng
et al., 2022). Additionally, integrating Artificial Intelligence (AI) methods
with remote sensing data, sometimes in conjunction with climate data, has
facilitated GPP product generation (Jung et al., 2019). However, while
recurrent neural network architectures, which exploit temporal dependencies,
have been used for modelling VIs (Martinuzzi et al., 2023) and land-atmosphere
interactions (Besnard et al., 2019), their potential for modelling daily
forest GPP remains underexplored.

This study conducts a comparative analysis of three recurrent architectures:
1) Recurrent Neural Networks, RNNs, 2) Gated Recurrent Units, GRUs, and 3)
Long-Short Term Memory networks, LSTMs. Additionally, we explore the models’
predictive performance under climate-induced GPP extreme values and provide
insights into the importance of the utilised features. The paper is organised
as follows: Sec. 2 details the data preprocessing and AI approaches, Sec. 3
presents and discusses the results, and Sec. 4 outlines the conclusions.

##  2 Methods

###  2.1 Data preparation

The study period spanned from 2016 to 2020, aligning with the availability of
Sentinel-2 (S2) data and the ICOS 2020 Warm Winter dataset (Warm Winter 2020
Team and ICOS Ecosystem Thematic Centre, 2022). The data preprocessing
procedures were conducted as follows:

Gross Primary Production. Daily GPP data were sourced from the ICOS 2020 Warm
Winter dataset. Forest cover information was derived from CORINE 2018 Land
Cover (CLC2018), and sites with less than 70% forest cover within a 1 km
radius of the EC tower were excluded. Sites with urban development within the
buffer and locations outside Europe were excluded. Daily GPP values (g C m-2
d-1) for each site were obtained from the nighttime partitioning method
(GPP_NT_VUT_REF, Reichstein et al., 2005). Timesteps with less than 70% high-
quality measurements (using NEE_VUT_REF_QC) and GPP values below zero were
discarded. Additionally, sites with less than 60% valid data during the study
period were excluded. A total of 19 forest sites, categorised as 12 Evergreen
Needleleaf Forests (ENF), 4 Deciduous Broadleaf Forests (DBF), and 3 Mixed
Forests (MF), were retained for analysis. Values below the 10% threshold of
the negative tail of the anomalies distribution were flagged. Anomalies were
computed as the deviation from the raw time series to the mean seasonal cycle,
calculated from all available years in the dataset. Segments of 5 or more
consecutive days with flagged anomalies were identified as climate-induced GPP
extremes.

Sentinel-2. S2 L2A data cubes centred on each EC tower location were created.
Non-10 m bands were resampled to this resolution via nearest neighbours. Cloud
and cloud shadow pixels were masked using a CloudSEN12-based model (Aybar et
al., 2022), while snow pixels were masked with Fmask v3.2. Surface
reflectances underwent conversion to Nadir BRDF Adjusted Reflectance (NBAR)
using the c-factor method (Roy et al., 2017). The CLC2018 was employed to mask
out non-forest pixels. Timesteps with less than 1% clear surface pixels within
the 1 km buffer were excluded, and NBAR values were spatially averaged within
it. A total of 122 VIs from Awesome Spectral Indices v0.5.0 (Montero et al.,
2023), designed for S2, were computed using the NBAR values. To manage the
substantial correlation among VIs and ensure interpretability, Principal
Component Analysis (PCA) was performed. The first 18 Principal Components
(PCs), jointly explaining $>$99% of the variance, were selected as predictors
in the GPP modelling. S2 features are denoted as S2i, where $i$ is one of the
PCs.

Sentinel-1. Sentinel-1 (S1) data cubes with a pixel size of 10 m and centred
on each EC tower were created. We used the Radiometrically Terrain Corrected
(RTC) product, which consists of terrain-flattened $\gamma^{0}$ backscattering
values. Both $\gamma^{0}$ values ($\gamma^{0}_{\textrm{VV}}$ and
$\gamma^{0}_{\textrm{VH}}$) were used. The Dual Polarised Radar Vegetation
Index for S1 (DpRVI${}_{\textrm{VV}}$) was computed from the $\gamma^{0}$
values in linear units. $\gamma^{0}$ values were also scaled to dB units. The
CLC2018 was employed to mask out non-forest pixels. All features were
spatially averaged within 50 and 1,000 m from each EC tower, aimed at
minimising potential signal contributions from the tower itself. S1 features
are denoted as S1i, where $i$ is one of the $\gamma^{0}$ values or the
DpRVI${}_{\textrm{VV}}$.

Land Surface Temperature (LST). MODIS daily LST v6.1 data were obtained from
Terra (MOD11A1) and Aqua (MYD11A1). LST values were spatially smoothed using a
mean $3\times 3$ kernel. The pixel values intersecting the coordinates of each
EC tower were retrieved. LST features are denoted as MOD11A1i or MYD11A1i,
where $i$ is one of the daytime (dt) or nightime (nt) LST values.

Simulated clear-sky radiation. The daily mean clear-sky radiation (W m-2) was
simulated individually for each site (Reda et al., 2008).

###  2.2 Modelling and evaluation approach

We interpolated missing data in the predictor features linearly, while we did
not interpolate GPP values, even when missing. A temporal split was
implemented, utilising 2016-2018 as the training set and 2019-2020 as the test
set. The three models (RNNs, GRUs, and LSTMs) were hyper-tuned using
HyperBand, employing a sequence-to-one approach with a sequence length of 90
days. The hyperparameter configuration included a maximum of 5 layers and a
maximum of 512 units per layer. Dropout layers with a 20% dropout rate were
incorporated. Weights were initialised by sampling from a random uniform
distribution within $[-\sqrt{n},\sqrt{n}]$, where $n$ is the number of
features. The Adam optimiser was used, and the learning rate was hyper-tuned
from 0.0001 to 0.01 on a logarithmic scale. The Mean Absolute Error (MAE)
served as the loss function. Following hyperparameter tuning, the best
configurations were employed, and the models were trained for 300 epochs,
saving the checkpoint for the best performance on the test set.

All models were evaluated using the Normalised Root Mean Squared Error
(NRMSE), calculated as
$\textrm{NRMSE}=\textrm{RMSE}/(y_{\textrm{max}}-y_{\textrm{min}})$, where RMSE
is the Root Mean Squared Error, and $y_{\textrm{max}}$ and $y_{\textrm{min}}$
indicate the maximum and minimum observed values. This evaluation was
performed on the test set using three different settings: 1) full period, 2)
growing season (May-Sep), and 3) climate-induced GPP extremes.

Figure 1: Comparison of models performance. Median NRMSE per model for a) full
period, b) growing season, and c) climate-induced GPP extremes. Black boxes
represent the range from Q1 to Q3 while error bars denote the range between
the 5th and 95th percentiles.

###  2.3 XAI: Feature Importance

Feature importance (FI) for a specific feature $f$ was calculated as
$\textrm{FI}_{f}=E_{f}-E_{b}$, where $E_{f}$ is the NRMSE of predictions
derived from a permuted version of the test set, where the feature array was
randomly shuffled; and $E_{b}$ is the baseline NRMSE of predictions derived
from the non-permuted test set. This process was iterated ten times, and the
mean was calculated per site.

##  3 Results and Discussions

###  3.1 Comparison of models

The three models exhibit comparable predictive performance throughout the
entire test period (Fig. 1a). The NRMSE increases for all models when
considering predictions solely during the growing season, with comparable
outcomes. LSTMs consistently maintain the lowest error variance (Fig. 1b).
Upon examining predictions related to climate-induced GPP extremes, all models
exhibit increased NRMSE and a higher error variance. However, this increase is
comparatively lower for LSTMs, followed by RNNs, while GRUs show the most
significant escalation among the three models (Fig. 1c).

Figure 2: Feature Importances (FIs) per model. Median FIs (expressed in
increased units of NRMSE) for a) GRUs, b) LSTMs, and c) RNNs. Features are
sorted in decreasing order by model.

###  3.2 Feature Importance

Notably, the simulated clear-sky radiation is a pivotal factor for all models,
evident from its highest FI ranking (Fig. 2). This is attributed to its role
in providing site-specific physical radiation values crucial for modulating
the GPP prediction range, as illustrated by predictions across example sites
characterised by variations in latitude and forest type (Fig. 3). Following
radiation, S2${}_{\textrm{PC16}}$, S2${}_{\textrm{PC1}}$, and
S2${}_{\textrm{PC2}}$ exhibit notably higher FIs in GRUs and RNNs compared to
other features. S2${}_{\textrm{PC16}}$ is primarily linked to chlorophyll-
related VIs (e.g. Modified Chlorophyll Absorption in Reflectance Index, MCARI)
with a slight association with some water-related VIs (e.g. Disease-Water
Stress Index, DSWI). Meanwhile, S2${}_{\textrm{PC1}}$ is highly correlated
with productivity-related VIs (e.g. Kernel Normalised Difference Vegetation
Index, kNDVI), and S2${}_{\textrm{PC2}}$ is closely associated with water-
related VIs (e.g. Normalised Difference Moisture Index, NDMI). In the case of
LSTMs, these S2 features gain importance shortly after most LST-based
features. We speculate that this FI explains the higher predictive performance
of LSTMs during extreme events, where LST plays a more pivotal role than
optical-based features for forests, especially during climate extremes like
droughts (Hoek van Dijke et al., 2023). Following these features, in LSTMs,
the significance of most S1 features is notable, with FIs surpassing those of
the remaining S2 features (a distinction less clear in RNNs and GRUs).

Figure 3: GPP predictions for three example sites. The first column shows
predictions for a) DE-Hai (DBF), c) BE-Vie (MF), and e) FI-Var (ENF). The
second column displays zoom-in panels depicting extreme events at each site.

##  4 Conclusions

We assessed recurrent neural networks for daily GPP modelling using remote
sensing data. Our key observation is that all models perform similarly in
predicting daily GPP over the growing season or the entire year. However,
errors escalate when predicting climate-induced GPP extremes. Here, LSTMs
demonstrate a slightly lower error rate in predicting climate-induced GPP
extremes. Moreover, our analysis underscores the importance of simulated
clear-sky radiation in modulating the GPP response. Additionally, we find that
LST data proves to be effective for predicting climate-induced GPP extremes,
complementing the information derived from optical and radar data obtained
from S2 and S1.

## References

  * Aybar et al. (2022) C. Aybar et al.  CloudSEN12, a global dataset for semantic understanding of cloud and cloud shadow in sentinel-2.  _Scientific Data_ , 9(1), Dec. 2022.  doi: 10.1038/s41597-022-01878-2.  URL https://doi.org/10.1038/s41597-022-01878-2.
  * Baldocchi (2019) D. D. Baldocchi.  How eddy covariance flux measurements have contributed to our understanding of global change biology.  _Global Change Biology_ , 26(1):242–260, Sept. 2019.  ISSN 1365-2486.  doi: 10.1111/gcb.14807.  URL http://dx.doi.org/10.1111/gcb.14807.
  * Besnard et al. (2019) S. Besnard et al.  Memory effects of climate and vegetation affecting net ecosystem co2 fluxes in global forests.  _PLOS ONE_ , 14(2):e0211510, Feb. 2019.  ISSN 1932-6203.  doi: 10.1371/journal.pone.0211510.  URL http://dx.doi.org/10.1371/journal.pone.0211510.
  * Bonan (2008) G. B. Bonan.  Forests and climate change: Forcings, feedbacks, and the climate benefits of forests.  _Science_ , 320(5882):1444–1449, June 2008.  ISSN 1095-9203.  doi: 10.1126/science.1155121.  URL http://dx.doi.org/10.1126/science.1155121.
  * Friedlingstein et al. (2023) P. Friedlingstein et al.  Global carbon budget 2023.  _Earth System Science Data_ , 15(12):5301–5369, Dec. 2023.  ISSN 1866-3516.  doi: 10.5194/essd-15-5301-2023.  URL http://dx.doi.org/10.5194/essd-15-5301-2023.
  * Hoek van Dijke et al. (2023) A. J. Hoek van Dijke et al.  Comparing forest and grassland drought responses inferred from eddy covariance and earth observation.  _Agricultural and Forest Meteorology_ , 341:109635, Oct. 2023.  ISSN 0168-1923.  doi: 10.1016/j.agrformet.2023.109635.  URL http://dx.doi.org/10.1016/j.agrformet.2023.109635.
  * Jung et al. (2019) M. Jung et al.  The fluxcom ensemble of global land-atmosphere energy fluxes.  _Scientific Data_ , 6(1), May 2019.  ISSN 2052-4463.  doi: 10.1038/s41597-019-0076-8.  URL http://dx.doi.org/10.1038/s41597-019-0076-8.
  * Martinuzzi et al. (2023) F. Martinuzzi et al.  Learning extreme vegetation response to climate forcing: A comparison of recurrent neural network architectures.  _EGUsphere_ , pages 1–32, Oct. 2023.  doi: 10.5194/egusphere-2023-2368.  URL http://dx.doi.org/10.5194/egusphere-2023-2368.
  * Montero et al. (2023) D. Montero et al.  A standardized catalogue of spectral indices to advance the use of remote sensing in earth system research.  _Scientific Data_ , 10(1), Apr. 2023.  doi: 10.1038/s41597-023-02096-0.  URL https://doi.org/10.1038/s41597-023-02096-0.
  * Reda et al. (2008) I. Reda et al.  _Solar Position Algorithm for Solar Radiation Applications (Revised)_.  Jan. 2008.  doi: 10.2172/15003974.  URL http://dx.doi.org/10.2172/15003974.
  * Reichstein et al. (2005) M. Reichstein et al.  On the separation of net ecosystem exchange into assimilation and ecosystem respiration: review and improved algorithm.  _Global Change Biology_ , 11(9):1424–1439, Sept. 2005.  doi: 10.1111/j.1365-2486.2005.001002.x.  URL https://doi.org/10.1111/j.1365-2486.2005.001002.x.
  * Roy et al. (2017) D. P. Roy et al.  Examination of Sentinel-2A multi-spectral instrument (MSI) reflectance anisotropy and the suitability of a general method to normalize MSI reflectance to nadir BRDF adjusted reflectance.  _Remote Sensing of Environment_ , 199:25–38, 2017.  ISSN 00344257.  doi: 10.1016/j.rse.2017.06.019.  URL http://dx.doi.org/10.1016/j.rse.2017.06.019.
  * Schimel et al. (2015) D. Schimel et al.  Observing terrestrial ecosystems and the carbon cycle from space.  _Global Change Biology_ , 21(5):1762–1776, Feb. 2015.  ISSN 1365-2486.  doi: 10.1111/gcb.12822.  URL http://dx.doi.org/10.1111/gcb.12822.
  * Warm Winter 2020 Team and ICOS Ecosystem Thematic Centre (2022) Warm Winter 2020 Team and ICOS Ecosystem Thematic Centre.  Warm winter 2020 ecosystem eddy covariance flux product for 73 stations in fluxnet-archive format—release 2022-1, 2022.  URL https://www.icos-cp.eu/data-products/2G60-ZHAK.
  * Zeng et al. (2022) Y. Zeng et al.  Optical vegetation indices for monitoring terrestrial ecosystems globally.  _Nature Reviews Earth & Environment_, 3(7):477–493, May 2022.  ISSN 2662-138X.  doi: 10.1038/s43017-022-00298-5.  URL http://dx.doi.org/10.1038/s43017-022-00298-5.