# Generic power series on subsets of the unit disk

Balázs Maga  and  Péter Maga  Eötvös Loránd University, Department of
Analysis, Pázmány Péter sétány 1/C, Budapest, H–1117 Hungary<EMAIL_ADDRESS>MTA Alfréd Rényi Institute of Mathematics, POB 127, Budapest H-1364, Hungary
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we examine the boundary behaviour of the generic power series
$f$ with coefficients chosen from a fixed bounded set $\Lambda$ in the sense
of Baire category. Notably, we prove that for any open subset $U$ of the unit
disk $D$ with a non-real boundary point on the unit circle, $f(U)$ is a dense
set of $\mathbb{C}$. As it is demonstrated, this conclusion does not
necessarily hold for arbitrary open sets accumulating to the unit circle. To
complement these results, a characterization of coefficient sets having this
property is given.

###### Key words and phrases:

complex power series, boundary behaviour, Baire category

###### 2010 Mathematics Subject Classification:

Primary 30B30; Secondary 28A05, 54H05

The first author was supported by the ÚNKP-20-3 New National Excellence
Program of the Ministry for Innovation and Technology from the source of the
National Research, Development and Innovation Fund, and by the Hungarian
National Research, Development and Innovation Office–NKFIH, Grant 124003. The
second author was supported by the Premium Postdoctoral Fellowship of the
Hungarian Academy of Sciences, the MTA Rényi Intézet Automorphic Research
Group, and by the Hungarian National Research, Development and Innovation
Office–NKFIH, Grants FK 135218, K 119528.

##  1\. Introduction

Let $\Lambda\subseteq\mathbb{C}$ be a bounded subset with at least two
elements, endowed with its usual subspace topology. Moreover, assume that the
product space

$\Omega:=\Omega_{\Lambda}:=\bigtimes_{n=0}^{\infty}\Lambda$

is a Baire space. Due to Alexandrov’s theorem (e.g. [5, p. 408]), this holds,
for example, if $\Lambda$ is $G_{\delta}$. These general conditions on
$\Lambda$ will be assumed throughout the paper.

For any $\bm{\lambda}=(\lambda_{n})_{n=0}^{\infty}\in\Omega$ we can define the
power series

$f(z):=f_{\bm{\lambda}}(z):=\sum_{n=0}^{\infty}\lambda_{n}z^{n}.$

The resulting function is clearly holomorphic in the open unit disk
$D:=\\{|z|<1\\}$. Roughly speaking, we are interested in the generic behaviour
of $f$ in terms of Baire category near the boundary $\partial D=S$. The
genericity is understood as follows: if a property holds for a set of power
series corresponding to a residual set of configurations in $\Omega$, we say
it is generic.

This work is a direct continuation of our previous paper [6], in which we
investigated the typical boundary behaviour of real power series with
coefficients chosen from a finite set $\Lambda$. (Actually, the set of
coefficients was denoted by $D$ in that paper, we opted to introduce this
notational modification as the symbol $D$ is customarily preserved for the
unit disk in this setup.) While the probabilistic aspects of the problem was
our main focus (considering the uniform distribution over $\Lambda$), we
proved straightforward results in terms of Baire category as well ([6, Theorem
3]). Notably, if $\Lambda$ has both positive and negative elements, for the
generic power series $f$ we have $\limsup_{1-}f=+\infty$ and
$\liminf_{1-}f=-\infty$, while if $\Lambda\subseteq[0,+\infty)$ (resp.
$\Lambda\subseteq(-\infty,0]$), we have $\lim_{1-}f=+\infty$ (resp.
$\lim_{1-}f=-\infty$). It was natural to consider the same problem in the
complex setup as well, which is a more natural habitat of power series. (We
note that while those results were stated for finite $\Lambda$ exclusively,
the proofs can be generalized in a straightforward manner for any $\Lambda$
for which $\bigtimes_{n=0}^{\infty}\Lambda$ is a Baire space.)

A direct prelude to our results is given in [1]. Even though the main focus of
that paper is the theory of random power series, where it presents spectacular
results about natural boundaries without assuming independence, it contains
the following result relevant to our setting:

######  Proposition 1 ([1, Theorem 1.9]).

For generic $\bm{\lambda}=(\lambda_{n})_{n=0}^{\infty}\in\Omega$, the power
series $f_{\bm{\lambda}}$ has a natural boundary on $\partial D$. That is it
cannot be analytically continued through any of the boundary points of the
convergence domain.

Another predecessor of this line of research is [2], in which results are
stated and proved about generic complex power series. For a more detailed
historical summary of the topic we also refer to that paper. We recall that
while the problem of the “general behaviour” dates back to Borel, the more
thorougly examined probabilistic question was worked out by several authors,
as presented in [3]. In terms of Baire category, the first results were
provided in [4]. The setup of [4] slightly differs from ours, as the
topological vector space $H(D)$ in which Baire category is investigated is the
space of all functions which are holomorphic in $D$ with the topology of
locally uniform convergence. (The $\Omega$ we consider corresponds to a
subspace of it.) Their main results stated that generically, $S$ is a natural
boundary and $f(D)=\mathbb{C}$. These results were generalized by [2], in
particular:

######  Proposition 2 ([2, Proposition 3.2]).

Assume that $U\subseteq D$ is open and $\overline{U}\cap S\neq\emptyset$. Then
for generic $f\in H(D)$ we have $f(U)=\mathbb{C}$ generically. (For a set
$B\subseteq\mathbb{C}$, its closure is denoted by $\overline{B}$ throughout
the paper.)

Our main goal is strengthening Proposition 1 in a similar manner. The proof of
Proposition 2 relies on Runge’s theorem on polynomial approximation, which is
out of reach in our setup that poses restrictions on the permissible
holomorphic functions. Consequently, when one would like to verify similar
results for $\Omega$, different techniques are required. As we will see, this
leads to somewhat weaker theorems: roughly, we could only verify that $f(U)$
is an open, dense set instead of being equal to $\mathbb{C}$. Our first main
result is the following.

######  Theorem 1.

Assume that $U\subseteq D$ is open with an accumulation point $\zeta\in S$
with non-vanishing imaginary part. Then for generic
$\bm{\lambda}=(\lambda_{n})_{n=0}^{\infty}\in\Omega$, the image
$f(U)\subseteq\mathbb{C}$ is dense and open.

As we will see below, the conclusion of Theorem 1 does not hold for all open
sets accumulating to $S$. In order to allow 1 or $-1$ to be the accumulation
point on the boundary, certain conditions on $\Lambda$ should be introduced.
The necessary and sufficient conditions are summarised by our other theorems.
By a real line, we mean a one-dimensional real affine subspace of
$\mathbb{C}$, while by a real half-plane we mean one of the two components of
the complement of a real line. We refer to its closure as a closed real half-
plane.

######  Theorem 2.

If $\Lambda$ is not contained by a real line, and $U\subseteq D$ is open with
$-1\in\overline{U}$, then for generic
$\bm{\lambda}=(\lambda_{n})_{n=0}^{\infty}\in\Omega$, the image
$f_{\bm{\lambda}}(U)\subseteq\mathbb{C}$ is dense and open. If $\Lambda$ is
contained by a real line, there exists an open $U\subseteq D$ with
$-1\in\overline{U}$ for which $f_{\bm{\lambda}}(U)$ evades a closed real half-
plane, regardless of the choice of $\bm{\lambda}$.

######  Theorem 3.

If $\Lambda$ is not contained by a closed real half-plane of the form
$\\{z:\text{ }\alpha\leq\arg z\leq\alpha+\pi\\}$, and $U\subseteq D$ is open
with $1\in\overline{U}$, then for generic
$\bm{\lambda}=(\lambda_{n})_{n=0}^{\infty}\in\Omega$, the image
$f_{\bm{\lambda}}(U)\subseteq\mathbb{C}$ is dense and open. If $\Lambda$ is
contained by a closed real half-plane of the form $\\{z:\text{ }\alpha\leq\arg
z\leq\alpha+\pi\\}$, then there exists an open $U\subseteq D$ with
$1\in\overline{U}$ for which $f_{\bm{\lambda}}(U)$ evades a closed real half-
plane, regardless of the choice of $\bm{\lambda}$.

We note that the openness of $f(U)$ for generic $f$ is a trivial consequence
of the open mapping theorem for analytic functions in each of the cases, as
such $f$ are non-constant. This implies that the real task in these questions
is proving the density of the images.

We fix some notation for the rest of the paper. Throughout,
$D_{r}=\\{|z|<r\\}$ for general disks centered at the origin, while its
boundary is denoted by $S_{r}$ (that is, $D=D_{1}$, $S=S_{1}$.) We use the
notation $\text{proj}_{n}(G)$ for the projection of
$G\subseteq(\lambda_{n})_{n=0}^{\infty}\in\Omega$ to the $n$th coordinate of
the product.

For any $\zeta\in S$ and any $N\in\mathbb{N}$, define

$\\{0,1\\}_{N}[\zeta]:=\left\\{\sum_{j=N}^{\infty}a_{j}\zeta^{j}:a_{j}\in\\{0,1\\},\
a_{j}=0\text{ with finitely many exceptions}\right\\}.$

A terminology we are going to use frequently is the following. For sets
$A,B\subseteq\mathbb{C}$ and some $\varepsilon>0$, we say that $A$ is an
$\varepsilon$-net of $B$, if for any $w\in B$, there exists some $z\in A$ such
that $|z-w|<\varepsilon$.

Below we will also use the notation $e(x)=e^{2\pi ix}$ for $x\in\mathbb{C}$.

Acknowledgement. We are grateful to the anonymous referee for their careful
reading of the manuscript and their suggestions to improve the exposition.

##  2\. Preliminary statements

We fix the following notation: for any $c\in\mathbb{C}$, set

$\Lambda+c:=\\{\lambda+c:\lambda\in\Lambda\\},\qquad
c\Lambda:=\\{c\lambda:\lambda\in\Lambda\\},$

and

$\bm{\lambda}+c=(\lambda_{n})_{n=0}^{\infty}+c:=(\lambda_{n}+c)_{n=0}^{\infty},\qquad
c\bm{\lambda}=c(\lambda_{n})_{n=0}^{\infty}c:=(c\lambda_{n})_{n=0}^{\infty}.$

Set also

$f_{\Lambda}(U):=\bigcup_{\bm{\lambda}\in\Omega}f_{\bm{\lambda}}(U).$

Our first lemma concerns the effect on $f_{\bm{\lambda}}(U)$ of certain
modifications on $\Lambda$, enabling us to circumvent some of the technical
burden in the proof of our theorems through replacing arbitrary $\Lambda$’s by
simpler ones.

######  Lemma 1.

  1. (a)

Let $U\subseteq D$ be open. Assume that $f_{\bm{\lambda}}(U)$ is dense in
$\mathbb{C}$. Then for any $0\neq c\in\mathbb{C}$, $f_{c\bm{\lambda}}(U)$ is
also dense in $\mathbb{C}$.

  2. (b)

Let $(U_{k})_{k=1}^{\infty}$ be a shrinking sequence of open subsets of $D$
such that $\text{diam}(U_{k})\to 0$ and none of them accumulates to $1$.
Moreover, assume that all of the sets $f_{\bm{\lambda}}(U_{k})$ are dense in
$\mathbb{C}$. Then for any $c\in\mathbb{C}$, all of the sets
$f_{\bm{\lambda}+c}(U_{k})$ are dense in $\mathbb{C}$.

  3. (c)

Assume that $f_{\Lambda}(U)$ evades a closed real half-plane for some open
$U\subseteq D$ with $1\notin\overline{U}$. Then for any $c\in\mathbb{C}$, the
same holds for $f_{\Lambda+c}(U)$.

  4. (d)

If the assumption of a (resp. b) is a generic property in $\Omega_{\Lambda}$,
then its implication is a generic property in $\Omega_{c\Lambda}$ (resp.
$\Omega_{\Lambda+c}$).

######  Remark 1.

The assumption of Lemma 1 b looks a bit complicated and one may wonder if it
can be formulated in a much simpler way, akin to Lemma 1 a. Notably, one can
intuitively believe that assuming $\overline{U}\cap
S\setminus\\{1\\}\neq\emptyset$, the density of $f_{\bm{\lambda}}(U)$ in
$\mathbb{C}$ implies that $f_{\bm{\lambda}+c}(U)$ is also dense in
$\mathbb{C}$ for any $c\in\mathbb{C}$. (This formulation would be directly
usable in the proof of Theorem 1 and 2 to translate $\Lambda$.) However, this
claim is false: a simple counterexample is given by

$\Lambda=\\{0,1\\},\qquad U=D$

and

$f_{\bm{\lambda}}(z)=\sum_{k=0}^{\infty}z^{2k+1}=\frac{z}{1-z^{2}}.$

Indeed, $f_{\bm{\lambda}}(z)=\alpha$ leads to a quadratic equation such that
its roots has product -1. Consequently, it has a root in $\overline{D}$, which
quickly yields the density of $f_{\bm{\lambda}}(D)$. However,

$f_{\bm{\lambda}-\frac{1}{2}}(z)=f_{\bm{\lambda}}(z)-\frac{1}{2}\cdot\frac{1}{1-z}=-\frac{1}{2(1+z)},$

for which $f_{\bm{\lambda}-\frac{1}{2}}(U)$ is clearly not dense in
$\mathbb{C}$.

###### Proof of Lemma 1.

Statement a follows trivially from the relation
$f_{c\lambda}(U)=cf_{\lambda}(U)$.

To prove b and c, consider the mapping

(1) $f_{\bm{\lambda}}\mapsto f_{\bm{\lambda}+c},\qquad
f_{\bm{\lambda}+c}(z)=f_{\bm{\lambda}}(z)+\frac{c}{1-z}=f_{\bm{\lambda}}(z)+g(z).$

Now if $(U_{k})_{k=1}^{\infty}$ is a sequence satisfying the conditions of b,
we clearly have $\text{diam}(g(U_{k}))\to 0$, which easily implies the
statement due to (1).

As for c, observe that in (1), $g(U)$ is bounded under the assumptions.
Therefore if $f_{\Lambda}(U)$ is a subset of a closed real half-plane, then so
is $f_{\Lambda+c}(U)$.

Finally, let us observe that d is obvious. ∎

The following three lemmata serve as a preparation to the proof of Theorem 1.

######  Lemma 2.

Assume that $(H_{j})_{j=1}^{\infty}$ is a sequence of dense subsets of $S$.
Then $\bigcup_{k=1}^{\infty}\sum\limits_{j=1}^{k}H_{j}$ is dense in
$\mathbb{C}$, where $\sum\limits_{j=1}^{k}H_{j}$ denotes the Minkowski sum of
$H_{1},...,H_{k}$.

###### Proof.

The proof follows from three simple observations:

  * •

for any $k\geq{1}$,

$\overline{\left(\sum_{j=1}^{k}H_{j}\right)}=\sum_{j=1}^{k}\overline{H_{j}};$

  * •

$S_{1}+S_{1}=\overline{D_{2}}$;

  * •

$D_{r}+S_{r^{\prime}}=D_{r+r^{\prime}}$ for $0<r^{\prime}<r$.

Putting together these claims yields that

$\overline{\left(\sum_{j=1}^{k}H_{j}\right)}=\overline{D_{k}}$

for $k\geq 2$. Consequently,

$\overline{\left(\bigcup_{k=1}^{\infty}\sum\limits_{j=1}^{k}H_{j}\right)}=\mathbb{C}$

clearly holds. ∎

######  Lemma 3.

Let $\zeta\in S$ be different from $\pm 1,\pm i,\pm\omega,\pm\omega^{2}$,
where $i=e(1/4)$, $\omega=e(1/3)$. Then for any $N\in\mathbb{N}$,
$\\{0,1\\}_{N}[\zeta]$ is dense in $\mathbb{C}$, i.e. $\\{0,1\\}_{N}[\zeta]$
is an $\varepsilon$-net of $\mathbb{C}$ for any $\varepsilon>0$.

###### Proof.

If $\zeta=e(x)$ with $x\in\mathbb{R}\setminus\mathbb{Q}$, then the statement
follows simply from Lemma 2. Indeed, if $w\in\mathbb{C}$ and $\varepsilon>0$
be given, our goal is to approximate $w$ with error smaller than $\varepsilon$
with a finite sum of the form given in the statement. Setting

$H_{1}:=H_{2}:=\ldots:=\\{\zeta^{N},\zeta^{N+1},\zeta^{N+2},\ldots\\}$

in Lemma 2, we obtain a certain $z:=\zeta^{n_{1}}+\ldots+\zeta^{n_{k}}$ such
that $|z-w|<\varepsilon$. Possibly there are repetitions among the $n_{j}$’s,
but any $\zeta^{n_{j}}$ can be replaced with another $\zeta^{n_{j^{\prime}}}$
on the cost of an arbitrarily small error, which altogether verifies the
claim.

If $\zeta=e(x)$ with $x\in\mathbb{Q}$, a root of unity different from $\pm
1,\pm i,\pm\omega,\pm\omega^{2}$, then its degree over $\mathbb{Q}$ is greater
than $2$. In particular, $\zeta+\zeta^{-1}\in\mathbb{R}\setminus\mathbb{Q}$,
and since $\zeta,\zeta^{-1}$ can be obtained as arbitrarily large powers of
$\zeta$, we see that $\\{0,1\\}_{N}[\zeta]$ is dense in $\mathbb{R}$ for any
$N\in\mathbb{N}$. Then obviously $\\{0,1\\}_{N}[\zeta]$ is dense also in
$\zeta\mathbb{R}$, hence in $\mathbb{C}$, too. ∎

######  Lemma 4.

Let $\zeta\in S$ be different from $\pm 1$. Then $\\{0,1\\}_{N}[\zeta]$ is a
$1$-net of $\mathbb{C}$ for any $N\in\mathbb{N}$.

###### Proof.

If $\zeta\neq\pm i,\pm\omega,\pm\omega^{2}$, then the statement is obvious
from Lemma 3. Otherwise, we may assume $N=0$, and if $\zeta=\pm i$ (resp.
$\zeta=\pm\omega$ or $\zeta=\pm\omega^{2}$), then
$\mathbb{Z}[\zeta]\subset\mathbb{C}$ is the lattice of Gaussian (resp.
Eulerian) integers, which are known from elementary geometry to satisfy that
for any $w\in\mathbb{C}$, there exist $a_{0},a_{1}\in\mathbb{Z}$ such that

$\left|w-(a_{0}+a_{1}\zeta)\right|<1.$

Again, the potientially negative coefficients can be switched to a sum of
positive ones by recording

$-1=\sum_{j=1}^{11}\zeta^{j},\qquad-\zeta=\sum_{j=2}^{12}\zeta^{j},$

and repetitions can be treated via $\zeta^{k}=\zeta^{k+12}$. ∎

The following three lemmata serve as a preparation to the proof of Theorem 3.

######  Lemma 5.

If $\Lambda\subseteq\mathbb{C}$ is not contained by a closed real half-plane
of the form $\\{z:\text{ }\alpha\leq\arg z\leq\alpha+\pi\\}$, then we can find
$\Delta_{0}=\Delta_{4},\Delta_{1},\Delta_{2},\Delta_{3}$ elements of $\Lambda$
such that

(2) $0\leq\arg{\frac{\Delta_{j+1}}{\Delta_{j}}}<\pi$

for any $0\leq j\leq 3$.

###### Proof.

Multiplying $\Lambda$ by a nonzero scalar does not change the assumption, nor
the implication. Consequently, we can assume $1\in\Lambda$. Let
$\Delta_{0}=1$. By the condition on $\Lambda$, $\\{\lambda:\Im\lambda>0\\}$ is
nonempty. Consequently, we can define

$\beta:=\sup_{\lambda\in\Lambda,\text{ }\Im\lambda>0}\arg{\lambda}.$

Now by the same argument, $\\{\lambda\in\Lambda:\text{
}\beta<\arg\lambda<\beta+\pi\\}$ is nonempty. However, due to the definition
of $\beta$, we can deduce that $\\{\lambda\in\Lambda:\text{
}\pi\leq\arg\lambda<\beta+\pi\\}$ is nonempty. Let $\Delta_{2}$ be an element
of it. Due to the definition of $\beta$, we can find $\Delta_{1}$ such that
(2) is satisfied for $j=0,1$. Now if $\arg{\Delta_{2}}>\pi$, the choice
$\Delta_{3}=\Delta_{2}$ guarantees that it is also satisfied for $j=2,3$ and
we are done. Otherwise we can choose $\Delta_{3}$ to be any element of the
necessarily nonempty $\\{\lambda\in\Lambda,\text{ }\Im\lambda<0\\}$, which
yields (2) for $j=2,3$. ∎

######  Lemma 6.

Assume that $\Lambda\subseteq\mathbb{C}$ is not contained by a closed real
half-plane of the form $\\{z:\text{ }\alpha\leq\arg z\leq\alpha+\pi\\}$. Then
there exists an appropriate $R>0$ with the property that for any
$z\in\mathbb{C}$ satisfying $|z|>R$, we can find $\lambda\in\Lambda$ such that
$|z+\lambda|<|z|$.

###### Proof.

Fix $\Delta_{0},\Delta_{1},\Delta_{2},\Delta_{3}$ as guaranteed by Lemma 5.
Let

$\alpha_{0}:=\max_{0\leq j\leq 3}\arg{\frac{\Delta_{j+1}}{\Delta_{j}}}<\pi.$

Now, if $z\neq 0$ is arbitrary, we can find $0\leq j\leq 3$ such that for
$\Delta_{j}$ we have
$\frac{\pi+\alpha_{0}}{2}\leq\arg\frac{\Delta_{j}}{z}<\frac{3\pi-\alpha_{0}}{2}$.
Consequently, if we consider the triangle determined by $0,z,z+\Delta_{j}$, we
have that the angle at $z$ is smaller than the right angle, and its size is
bounded away from $\frac{\pi}{2}$ by some positive quantity. As the set of all
$\Delta_{j}$’s is bounded, this implies that if $|z|\geq R$ for large enough
$R$, then the side $[0,z]$ of the triangle is larger than the side
$[0,z+\Delta_{j}]$. Defining $R$ accordingly proves the lemma. ∎

Note that if $R$ is sufficient for some $\Lambda$ in the setup of Lemma 6,
then $cR$ is sufficient for $c\Lambda$. In particular, if $c<1$, the same $R$
can be used.

######  Lemma 7.

Assume that $\Lambda\subseteq\mathbb{C}$ is not contained by a closed real
half-plane of the form $\\{z:\text{ }\alpha\leq\arg z\leq\alpha+\pi\\}$. Then
there exists $R^{*}>0$ such that for any $z$, $|z|<1$, and $0\leq
n_{0}<n_{1}<\ldots$, there exists $(\lambda_{n_{j}})_{j=0}^{\infty}$,
$\lambda_{n_{j}}\in\Lambda$ such that
$\left|\sum_{j=0}^{\infty}\lambda_{n_{j}}z^{n_{j}}\right|\leq R^{*}$.

###### Proof.

We prove that $R^{*}=R+\sup_{\lambda\in\Lambda}|\lambda|$ is sufficient, where
$R$ is the one guaranteed by Lemma 6. Due to the note following its proof, the
same $R$ can be used for any coefficient set of the form $z^{n}\Lambda$.

For $z=0$, the claim is trivial, regardless of the choice of
$(\lambda_{n_{j}})_{j=0}^{\infty}$. Hence fix $z\neq 0$, $|z|<1$. The proof
depends on a recursive construction of the sequence $(\lambda_{n_{j}})$.
Notably, let $\lambda_{n_{0}}\in\Lambda$ be arbitrary, and assume
$\lambda_{n_{0}},\ldots,\lambda_{n_{k}}$ are already defined. If

$\left|\sum_{j=0}^{k}\lambda_{n_{j}}z^{n_{j}}\right|\leq R,$

then $\lambda_{n_{k+1}}\in\Lambda$ can be chosen arbitrarily as well.
Otherwise, we apply Lemma 6 to $z^{n_{k+1}}\Lambda$ to define
$\lambda_{n_{k+1}}$ such that

$\left|\sum_{j=0}^{k+1}\lambda_{n_{j}}z^{n_{j}}\right|<\left|\sum_{j=0}^{k}\lambda_{n_{j}}z^{n_{j}}\right|.$

These choices obviously guarantee that

$\left|\sum_{j=0}^{k}\lambda_{n_{j}}z^{n_{j}}\right|\leq
R+\max_{\lambda\in\Lambda}|\lambda|=R^{*},$

regardless of the value of $k$. Consequently, the same bound holds for the sum
of the series as well. ∎

##  3\. Proof of Theorem 1

Assume first that $0,1\in\Lambda$. For any fixed $w\in\mathbb{C}$ and
$\varepsilon>0$, we introduce

$A_{w,\varepsilon}:=\\{\bm{\lambda}=(\lambda_{n})_{n\in\mathbb{N}}:\text{there
exists some $\tau\in U$ such that
$|f_{\bm{\lambda}}(\tau)-w|<\varepsilon$}\\}.$

Fixing $w\in\mathbb{C}$ and $\varepsilon>0$, we introduce the abbreviation
$A:=A_{w,\varepsilon}$, and prove below that it is open and dense in $\Omega$.

To see that $A$ is open in $\Omega$, let
$\bm{\lambda}=(\lambda_{n})_{n\in\mathbb{N}}\in A$, i.e. for some $\tau\in U$,
$\varepsilon_{0}:=|f_{\bm{\lambda}}(\tau)-w|<\varepsilon$. Let $N$ be large
enough to satisfy that

$\sum_{n>N}\sup\\{|\lambda|:\lambda\in\Lambda\\}\tau^{n}<\frac{\varepsilon-\varepsilon_{0}}{2}.$

Also, choose $\delta>0$ in such a way that if
$|\lambda_{n}^{\prime}-\lambda_{n}|<\delta$ for all $n\leq N$, then

$\left|\sum_{n\leq N}\lambda_{n}\tau^{n}-\sum_{n\leq
N}\lambda_{n}^{\prime}\tau^{n}\right|<\frac{\varepsilon-\varepsilon_{0}}{2}.$

Clearly, if

$\bm{\lambda}^{\prime}=(\lambda_{n}^{\prime})_{n\in\mathbb{N}}\in\bigtimes_{n\leq
N}\\{\lambda_{n}^{\prime}:|\lambda_{n}^{\prime}-\lambda_{n}|<\delta\\}\times\bigtimes_{n>N}\Lambda\subseteq\Omega,$

then

$|f_{\bm{\lambda}^{\prime}}(\tau)-w|<\varepsilon,$

hence $\bm{\lambda}^{\prime}\in A$, which shows that $A$ is open.

Now we prove that $A$ is dense in $\Omega$. It suffices to show that $A$
intersects any set of the form

$G:=\\{\lambda_{0}\\}\times\ldots\times\\{\lambda_{N}\\}\times\bigtimes_{n>N}\Lambda\subseteq\Omega.$

Let us fix some $\pm 1\neq\zeta\in\overline{U}\cap S$ throughout the proof.
Our goal is to find an element $\bm{\lambda}=(\lambda_{n})_{n\in\mathbb{N}}\in
G$ and some $\tau\in U$ such that $|f_{\bm{\lambda}}(\tau)-w|<\varepsilon$. We
immediately prescribe $|\tau-\zeta|<\delta$ with $0<\delta<2/5$ chosen in such
a way that

$\left|\sum_{n\leq N}\lambda_{n}\tau^{n}-\sum_{n\leq
N}\lambda_{n}\zeta^{n}\right|<\frac{\varepsilon}{2}$

is guaranteed by $|\tau-\zeta|<\delta$.

Relabeling our original $w$ (shifting it by $-\sum_{n\leq
N}\lambda_{n}\zeta^{n}$), and rescaling $\varepsilon$, we have to find a
sequence $(\lambda_{n})_{n>N}\in\\{0,1\\}^{n>N}$ and some $\tau\in U$ in such
a way that

(3)
$\left|\sum_{n>N}\lambda_{n}\tau^{n}-w\right|<\varepsilon,\qquad|\tau-\zeta|<\delta.$

Now we go by cases according to Lemmata 3–4 about the nature of $\zeta$. To
avoid notational difficulties, assume that $w\neq 0$ (which is not a real
restriction, since if anything but zero can be arbitrarily approximated, then
so can zero).

If $\zeta\neq\pm i,\pm\omega,\pm\omega^{2}$, then by Lemma 3, we may find and
fix a _finite_ sequence $(\lambda_{n})_{N<n<K}$ satisfying

$\left|\sum_{N<n<K}\lambda_{n}\zeta^{n}-w\right|<\frac{\varepsilon}{2},\qquad\lambda_{n}\in\\{0,1\\},$

and then if $|\tau-\zeta|$ is small enough (consistently with the earlier
prescribed $|\tau-\zeta|<\delta$),

$\left|\sum_{N<n<K}\lambda_{n}\tau^{n}-\sum_{N<n<K}\lambda_{n}\zeta^{n}\right|<\frac{\varepsilon}{2},\qquad|\tau-\zeta|<\delta,$

which, setting $\lambda_{K}:=\lambda_{K+1}:=\ldots:=0$, together clearly imply
(3).

Now assume that $\zeta\in\\{\pm i,\pm\omega,\pm\omega^{2}\\}$. Fix a finite
set $W$ which is a $1$-net of $D_{10|w|/\varepsilon}$. By Lemma 4, for any
$w^{\prime}\in W$, we may find and fix a _finite_ sequence
$(\lambda_{n})_{N<n<K}(w^{\prime})$ satisfying

$\left|\sum_{N<n<K}\lambda_{n}(w^{\prime})\zeta^{n}-w^{\prime}\right|<1,\qquad\lambda_{n}(w^{\prime})\in\\{0,1\\},$

where the upper bound $K$ on the coefficient indices is uniform over
$w^{\prime}\in W$ (this can be achieved, since $W$ is finite). Now if
$|\tau-\zeta|$ is small enough (consistently with the earlier prescribed
$|\tau-\zeta|<\delta$), then

$\left|\sum_{N<n<K}\lambda_{n}(w^{\prime})\tau^{n}-\sum_{N<n<K}\lambda_{n}(w^{\prime})\zeta^{n}\right|<1\text{
for all $w^{\prime}\in W$},\qquad|\tau-\zeta|<\delta.$

Fix now $M$ in such a way that $\varepsilon/5<|\tau^{M}|<\varepsilon/3$ (this
is possible, since $\delta<2/5$ implies $|\tau|>3/5$, and that $\tau$ has a
power in the indicated annulus). Then

$\left\\{\sum_{N<n<K}\lambda_{n}(w^{\prime})\tau^{n+M}:w^{\prime}\in
W\right\\}$

gives rise to an $\varepsilon$-net of $D_{2|w|}$. In particular, choosing the
appropriate $w^{\prime}$ (for which the sum in the last display is closest to
$w$), and setting

$\lambda_{n}:=\begin{cases}\lambda_{n},\qquad&\text{if $n\leq N$,}\\\
\lambda_{n-M}(w^{\prime}),&\text{if $N+M<n<K+M$,}\\\
0&\text{otherwise,}\end{cases}$

(3) is achieved.

To sum up, any $A_{w,\varepsilon}$ is open and dense, which in turn implies
that the set

$\bigcap_{w\in\mathbb{Q}+\mathbb{Q}i}\bigcap_{k=1}^{\infty}A_{w,1/k}$

is residual, hence the proof of Theorem 1 is complete, at least, when
$0,1\in\Lambda$. This, however, immediately gives rise to the general case by
applying Lemma 1 a–b. Indeed, the proof of the density of $A$ presented above
guarantees $\tau$’s arbitrarily close to $\zeta$, which means that the
condition of Lemma 1 b is satisfied.

A fairly straightforward consequence of Theorem 1 is the following:

######  Corollary 1.

For the generic $(\lambda_{n})_{n=0}^{\infty}\in\Omega$, for any $\zeta\in S$
and $w\in\mathbb{C}$ there exists $(\zeta_{k})_{k=1}^{\infty}\subseteq D$ with
$\zeta_{k}\to\zeta$ such that $f(\zeta_{k})\to w$.

The proof is left as a simple exercise to the reader, it suffices to rely on
the case of irrational arguments. With a slightly different formulation,
proving this statement was Problem 9 at the prestigious Miklós Schweitzer
Memorial Competition for Hungarian university students in 2020, proposed by
the authors. Complete solutions were given by Márton Borbényi and Attila
Gáspár, who were awarded the two first prizes of the contest, and to whom we
congratulate hereby. A direct solution is available at [7] in Hungarian.

##  4\. Proof of Theorem 2

Due to the open mapping theorem, it suffices to prove that the density of
$f(U)$ holds generically.

Due to Lemma 1 a–b, we can assume $0,1\in\Lambda$. First we will consider the
case when $\Lambda$ is contained by a real line. Following our assumption,
this means $\Lambda\subseteq\mathbb{R}$.

We will define $U=\bigcup_{k=1}^{\infty}U_{k}$ where

$U_{k}=\left\\{z:-\frac{k-1}{k}<\Re z<0,\text{
}\pi-\alpha_{k}<\arg{z}<\pi+\alpha_{k}\right\\}$

for $\alpha_{k}>0$ to be fixed later. (The lower bound on $\Re z$ is somewhat
arbitrary, separation from $-1$ is relevant only.) Right now we specify only
that $\alpha_{k}$ is small enough to guarantee that $\overline{U_{k}}\subseteq
D$.

As each $U_{k}$ is open, the same holds for $U$. Now consider any $z\in
U_{k}$. As $\overline{U_{k}}\subseteq S$, we can choose $N$ large enough to
have

(4) $\left|\sum_{n=2N}^{\infty}z^{n}\right|<1.$

We choose $\alpha_{k}$ based on the choice of $N$ such that

$2N\alpha_{k}<\arcsin\left(\frac{1}{N}\right).$

This clearly implies that

$-\arcsin\left(\frac{1}{N}\right)<\arg{\left(\sum_{n=0}^{N-1}z^{2n}\right)}<\arcsin\left(\frac{1}{N}\right)$

and

$\pi-\arcsin\left(\frac{1}{N}\right)<\arg{\left(\sum_{n=0}^{N-1}z^{2n+1}\right)}<\pi+\arcsin\left(\frac{1}{N}\right).$

However, the absolute value of each of these partial sums is at most $N$,
which yields that their imaginary part is at most $1$. Consequently,

$\Im\left(\sum_{n=0}^{2N-1}z^{n}\right)\leq 2.$

By the same argument for any $(\lambda_{n})_{n=0}^{\infty}$ we have

$\Im\left(\sum_{n=0}^{2N-1}\lambda_{n}z^{n}\right)\leq
2\sup_{\lambda\in\Lambda}|\lambda|.$

Taking (4) into consideration implies

$\Im\left(\sum_{n=0}^{\infty}\lambda_{n}z^{n}\right)\leq
3\sup_{\lambda\in\Lambda}|\lambda|.$

As it holds for any $k$ and $z\in U_{k}$, we have it for any $z\in U$, which
concludes the proof of the first part.

In the other direction, our argument will be similar to the one we have given
in the proof of Theorem 1. Indeed, defining the set $A$ as above, and
following the argument verbatim, it suffices to find a sequence
$(\lambda_{n})_{n>N}\in\\{0,1\\}^{n>N}$ and some $\tau\in U$ in such a way
that

$\left|\sum_{n>N}\lambda_{n}\tau^{n}-w\right|<\varepsilon,\qquad|\tau-(-1)|<\delta.$

Fix an element $\lambda\in\Lambda$ with non-vanishing imaginary part; its
existence is guaranteed by the assumption that $0,1\in\Lambda$ and
$\Lambda\nsubseteq\mathbb{R}$. Consider the lattice

$\\{a+b\lambda:\text{ }a,b\in\mathbb{Z}\\}.$

It obviously gives a $\delta$-net of $\mathbb{C}$ for some $\delta>0$.
Consequently,

$\\{\xi(a+b\lambda):\text{ }a,b\in\mathbb{Z}\\}$

gives an $\varepsilon/2$-net of $\mathbb{C}$ for $|\xi|=\varepsilon_{0}$ if
$\varepsilon_{0}>0$ is small enough. Fix $\varepsilon_{0}$ accordingly, and
fix $R$ such that $|w|<\varepsilon_{0}R$. Now it is clear that one can find
$m_{R}$ such that for

$C^{m_{R}}=\\{a+b\lambda:\text{ }a,b\in\mathbb{Z},\text{ }|a|,|b|<m_{R}\\},$

$\xi C^{m_{R}}$ gives an $\varepsilon/2$-net of $D_{R}$ for
$|\xi|=\varepsilon_{0}$.

Now let us notice that for large enough $M_{R}$, any element of $C^{m_{R}}$
can be written in the form

$\lambda\sum_{j=1}^{k}(-1)^{n_{j}}+\sum_{j^{\prime}=1}^{l}(-1)^{n^{\prime}_{j^{\prime}}},$

where the exponents used are pairwise distinct, and
$N<n_{j},n^{\prime}_{j^{\prime}}<M_{R}$ for $j=1,\ldots,k$ and
$j^{\prime}=1,\ldots,l$. Denote the set of such sums by $C(-1)$, and motivated
by this, let

$C(z)=\left\\{\lambda\sum_{j=1}^{k}z^{n_{j}}+\sum_{j^{\prime}=1}^{l}z^{n^{\prime}_{j^{\prime}}}:\text{
}N<n_{j},n^{\prime}_{j^{\prime}}<M_{R}\text{ are distinct}\right\\}.$

As $C(z)$ is determined by finitely many continuous functions of $z$, if
$|(-1)-\tau|$ is small enough, $\xi C(\tau)$ gives a $\varepsilon$-net of
$D_{\varepsilon_{0}R}$ for any $\xi,\text{ }|\xi|=\varepsilon_{0}$. On the
other hand, we can choose $\tau$ in any neighborhood of $-1$ so that
$|\tau|^{M}=\varepsilon_{0}$ for some $M>0$. Consequently, we have that
$\tau^{M}C(\tau)$ forms a $\varepsilon$-net of $D_{\varepsilon_{0}R}$.

By definition, this implies that there exist pairwise distinct numbers
$N<n_{j},n^{\prime}_{j^{\prime}}<M_{R}$ for $j=1,...,k$, $j^{\prime}=1,...,l$
such that

$\left|\left(\lambda\sum_{j=1}^{k}\tau^{n_{j}+M}+\sum_{j^{\prime}=1}^{l}\tau^{n^{\prime}_{j^{\prime}}+M}\right)-w\right|<\varepsilon.$

Now if we define $(\lambda_{n})_{n=N+1}^{\infty}$ such that
$\lambda_{n}=\lambda$ if and only if $n=n_{j}+M$ for some $1\leq j\leq k$,
moreover, $\lambda_{n}=1$ if and only if $n=n^{\prime}_{j^{\prime}}+M$ for
some $1\leq j^{\prime}\leq l$, and otherwise $\lambda_{n}=0$, then we
immediately obtain $|f_{\bm{\lambda}}(\tau)-w|<\varepsilon$, which concludes
the proof.

##  5\. Proof of Theorem 3

Due to the open mapping theorem, it suffices to prove that the density of
$f(U)$ holds generically.

First we will consider the case when $\Lambda$ is contained by a closed real
half-plane of the form $\\{z:\text{ }\alpha\leq\arg z\leq\alpha+\pi\\}$. Due
to Lemma 1 a, we can assume that this half-plane is $\\{\Re z\geq 0\\}$.

We will define $U=\bigcup_{k=1}^{\infty}U_{k}$ where

$U_{k}=\left\\{z:0<\Re z<\frac{k-1}{k},\text{
}-\alpha_{k}<\arg{z}<\alpha_{k}\right\\}$

for $\alpha_{k}>0$ to be fixed later. (The upper bound on $\Re z$ is somewhat
arbitrary, separation from $1$ is relevant only.) Right now we specify only
that $\alpha_{k}$ is small enough to guarantee that $\overline{U_{k}}\subseteq
S$.

From this point, the proof of this part is basically a simplified version of
the proof of the same part of the proof of Theorem 2. Notably, for $z\in
U_{k}$ we can find a threshold index such that the tail sum is very small due
to $|z|$ being bounded away from 1, and then by choosing $\alpha_{k}$ to be
small enough, we can control the argument of the preceding terms. (The
relative simplicity in this case is due to the fact these arguments are all
near 0, instead of being near to 0 and $\pi$ alternatingly.) Based on these
estimates, the real part of $f_{\bm{\lambda}}(z)$ can be bounded from below,
regardless of $\bm{\lambda}=(\lambda_{n})_{n=0}^{\infty}$ and $z\in U_{k}$,
which concludes the proof of the first part.

We now prove the second part of the statement of the theorem. Defining the set
$A$ as in the proof of Theorem 1 and following the argument verbatim, it
suffices to find a sequence $(\lambda_{n})_{n>N}\in\Lambda^{n>N}$ and some
$\tau\in U$ in such a way that

$\left|\sum_{n>N}\lambda_{n}\tau^{n}-w\right|<\varepsilon,\qquad|\tau-1|<\delta.$

Define $\Delta_{0},\Delta_{1},\Delta_{2},\Delta_{3}\in\Lambda$ as guaranteed
by Lemma 5. Denote their set by $V$, and define the convex polygon
$P=\text{conv}(V)$. Due to the choice of $V$, $0\in\operatorname{int}(P)$,
that is $D_{r}\subseteq V$ for small enough $r$.

Notice that if $P$ has diameter $\delta$, then $V$ is clearly a $\delta$-net
of $P$. Moreover, for any $m$ we have that the Minkowski sum $\sum_{i=1}^{m}V$
is a $\delta$-net of $\sum_{i=1}^{m}P$: the proof proceeds by induction,
capitalizing on the simple observation that
$\sum_{i=1}^{m}P=V+\sum_{i=1}^{m-1}P$. It clearly yields that
$\sum_{i=1}^{m}V$ is a $\delta$-net of $D_{mr}$ as well for any $m$.
Consequently, $\xi\cdot\sum_{i=1}^{m}V$ gives an $\varepsilon/2$-net of
$D_{\varepsilon_{0}mr}$ for any $m$ and for $|\xi|=\varepsilon_{0}$, if
$\varepsilon_{0}>0$ is small enough. Fix $\varepsilon_{0}$ accordingly, noting
that it does not depend on $m$. Now fix $R^{*}$ as guaranteed by Lemma 7, and
based on the choice of $R^{*}$ and $\varepsilon_{0}$, fix $m$ such that
$|w|+R^{*}<\varepsilon_{0}mr$. Let us remark that any element of
$\sum_{i=1}^{m}V$ is expressible in the seemingly complicated form

$\sum_{j=0}^{3}\Delta_{j}k_{j}=\sum_{j=0}^{3}\Delta_{j}\sum_{l=1}^{k_{j}}1^{n^{(j)}_{l}},$

where each $0\leq k_{j}\leq m$, and the exponents $n^{(j)}_{l}$ are pairwise
distinct and their union equals $\\{N+1,N+2,...,N+m\\}$. Let us denote the set
of such combinations by $C(1)$, and motivated by this, let

$C(z)=\left\\{\sum_{j=0}^{3}\Delta_{j}\sum_{l=1}^{k_{j}}z^{n^{(j)}_{l}}:\text{
}0\leq k_{j}\leq m,\text{ }N<n^{(j)}_{l}\leq N+m\text{ are all distinct and
their union is }\\{N+1,N+2,...,N+m\\}\right\\}.$

As $C(z)$ is determined by finitely many continuous functions of $z$, if
$|1-\tau|$ is small enough, $\xi C(\tau)$ gives a $\varepsilon$-net of
$D_{\varepsilon_{0}mr}$ for any $\xi,\text{ }|\xi|=\varepsilon_{0}$. On the
other hand, we can choose $\tau$ in any neighborhood of $1$ so that
$|\tau|^{M}=\varepsilon_{0}$ for some $M>0$. Consequently, we have that
$\tau^{M}C(\tau)$ forms an $\varepsilon$-net of $D_{\varepsilon_{0}mr}$.

So far the coefficients of the power series we would like to define are fixed
for the indices $(i)_{i=0}^{N}$. Motivated by the previous paragraph, we would
like to set aside the indices $(N+i+M)_{i=1}^{m}$. Notably, these are the
indices which are intimately connected to the lastly defined
$\tau^{M}C(\tau)$. Consequently, we apply Lemma 7 at this point for $\tau$ and
the complementary sequence $(N+1,N+2,...,N+M,N+m+M+1,N+m+M+2,...)$: we can
find elements of $\Lambda$ corresponding to these indices,
$(\lambda_{n})_{n=N+1}^{N+M}$, $(\lambda_{n})_{n=N+m+M+1}^{\infty}$ such that

$|w_{1}|\leq R^{*},\qquad\text{where}\qquad
w_{1}:=\sum_{n=N+1}^{N+M}\lambda_{n}z^{n}+\sum_{n=N+m+M+1}^{\infty}\lambda_{n}z^{n}.$

Consequently, $|w-w_{1}|\leq|w|+R^{*}<\varepsilon_{0}R$. This implies that
there exist numbers

$N<n^{(j)}_{l}\leq N+m$

for $j=0,1,2,3$ and $l=1,...,k_{j}$, such that their union fills
$\\{N+1,N+2,...,N+m\\}$ without any repetitions (that is the numbers
$n^{(j)}_{l}$ are pairwise distinct for all the possible choices of $j,l$),
and

(5)
$\left|\sum_{j=0}^{3}\Delta_{j}\sum_{l=1}^{k_{j}}\tau^{n^{(j)}_{l}+M}-(w-w_{1})\right|<\frac{\varepsilon}{2}.$

What remains from the definition of $(\lambda_{n})$ is fixing
$(\lambda_{n})_{n=N+1+M}^{N+m+M}$, which we carry out now based on (5).
Notably let $\lambda_{n}=\Delta_{j}$ if and only if $n=n^{(j)}_{l}+M$ for some
$1\leq l\leq k_{j}$. Then by the definition of $w_{1}$ we obtain
$|\sum_{n=N+1}^{\infty}\lambda_{n}\tau^{\prime n}-w|<\varepsilon$. This
concludes the proof.

##  6\. Concluding remarks

Even though with a careful separation of cases we managed to generalize the
most natural result given by Theorem 1, our results are far from being
complete. In our view, the most interesting open problem related to them is
whether $f(U)=\mathbb{C}$ holds generically in the setup of our theorems,
similarly to what is proved in [2]. As $f$ is uniformly locally bounded in
$D$, we clearly cannot rely on techniques similar to the ones seen there,
hence answering this question requires additional ideas.

Another interesting aspect partially inspired by this paper is whether we can
rearrange the quantifiers in our statements to some extent. More explicitly,
each of our theorems addresses the question of what the generic image is of a
fixed open set. It would be desirable to find extensions of this result, for
example a nontrivial family of open sets such that generically, the image of
each of them is dense.

## References

  * [1] J. Breuer , B. Simon, Natural boundaries and spectral theory, Adv. Math. 226, (2011), 4902–4920.
  * [2] J.-P. Kahane, Baire’s category theorem and trigonometric series, J. Anal. Math. 80, (2000), 143–182.
  * [3] J.-P. Kahane, Some Random Series of Functions, Heath, Mass., 1968; 2nd edn. Cambridge Univ. Press, 1985, (1993).
  * [4] S. Kierst, E. Szpilrajn, Sur certaines singularités desfonctions analytiques uniformes, Fund. Math. 21, (1933), 267-294.
  * [5] K. Kuratowski, Topologie, Vol. 1, 4th ed., PWN, Warsaw, 1958; English transl., Academic Press, New York; PWN, Warsaw, (1966).
  * [6] B. Maga, P. Maga, Random power series near the endpoint of the convergence interval, Publ. Math. Debrecen, 93 (3-4). (2018), 413-424. ISSN 0033-3883
  * [7] https://www.bolyai.hu/Schweitzer2020_elozetes_megoldasok.pdf


# Learning and Autonomy for Extraterrestrial Terrain Sampling: An Experience
Report from OWLAT Deployment

Pranay Thangeda111Graduate Student, Department of Aerospace Engineering,
<EMAIL_ADDRESS>University of Illinois Urbana-Champaign, Urbana,
Illinois 61801  Yifan Zhu222Graduate Student, Department of Computer Science,
<EMAIL_ADDRESS>University of Illinois Urbana-Champaign, Urbana, Illinois
61801  Kris Hauser333Professor, Department of Computer Science,
<EMAIL_ADDRESS>University of Illinois Urbana-Champaign, Urbana,
Illinois 61801  and Melkior Ornik444Assistant Professor, Department of
Aerospace Engineering<EMAIL_ADDRESS>University of Illinois Urbana-
Champaign, Urbana, Illinois 61801  Ashish Goel555Robotics Technologist,
Robotic Surface Mobility Group<EMAIL_ADDRESS>Jet Propulsion
Laboratory, California Institute of Technology, Pasadena, California 91109
Erica L Tevere666Robotics Technologist, Robotic Surface Mobility Group,
<EMAIL_ADDRESS>Jet Propulsion Laboratory, California Institute of
Technology, Pasadena, California 91109  Adriana Daca777Robotics Technologist,
Robotic Surface Mobility Group<EMAIL_ADDRESS>Jet Propulsion
Laboratory, California Institute of Technology, Pasadena, California 91109
and Hari D Nayar888Member of Technical Staff, Robotic Surface Mobility Group,
<EMAIL_ADDRESS>Jet Propulsion Laboratory, California Institute of
Technology, Pasadena, California 91109  Erik Kramer999Graduate Student,
Department of Mechanical and Aerospace Engineering<EMAIL_ADDRESS>University of California, Los Angeles, Los Angeles, California 90095

###### Abstract

Extraterrestrial autonomous lander missions increasingly demand adaptive
capabilities to handle the unpredictable and diverse nature of the terrain.
This paper discusses the deployment of a Deep Meta-Learning with Controlled
Deployment Gaps (CoDeGa) trained model for terrain scooping tasks in Ocean
Worlds Lander Autonomy Testbed (OWLAT) at NASA Jet Propulsion Laboratory. The
CoDeGa-powered scooping strategy is designed to adapt to novel terrains,
selecting scooping actions based on the available RGB-D image data and limited
experience. The paper presents our experiences with transferring the scooping
framework with CoDeGa-trained model from a low-fidelity testbed to the high-
fidelity OWLAT testbed. Additionally, it validates the method’s performance in
novel, realistic environments, and shares the lessons learned from deploying
learning-based autonomy algorithms for space exploration. Experimental results
from OWLAT substantiate the efficacy of CoDeGa in rapidly adapting to
unfamiliar terrains and effectively making autonomous decisions under
considerable domain shifts, thereby endorsing its potential utility in future
extraterrestrial missions.

##  1 Introduction

The exploration of ocean worlds stands as a pivotal element in humanity’s
exploration of our solar system, encompassing critical research objectives
including the quest for potential signs of life and the comprehensive
understanding of conditions fostering habitability [1], [2], [3]. Robotic
exploration missions are essential for the exploration of potentially
habitable ocean worlds. Past lander and rover missions including the Mars
exploration program [4] and the Perseverance rover mission [5] are human-in-
the-loop systems with expert teams on Earth supervising the terrain sampling
process and controlling them based on the collected data. However, unlike Mars
missions, many of the ocean world missions, including the Europa Lander
mission concept [6], are anticipated to have short durations, on the order of
tens of days, due to the intensity of the radiation environment, adverse
thermal conditions, low availability of solar energy, and using battery as the
sole power source. The limited mission duration combined with the long
communication delays between Earth and the ocean worlds necessitates a high
degree of autonomy for the lander’s success [7].

The Europa lander’s primary objectives include collecting terrain samples for
in situ analysis of surface and sub-surface materials. Autonomy in terrain
sampling missions is challenging due to the high degree of uncertainty in the
surface topology at the landing site, terrain material properties,
composition, and appearance. Constraints on the number of samples that can be
analyzed in-situ, coupled with the risk of system failures, further limits the
extent of exploration [8]. Any realistic sampling strategy needs to be able to
make decisions under uncertainty and rapidly adapt to the environment in order
to maximize the scientific return from the mission. To address these
challenges, recent efforts have focused on the development of capabilities for
autonomous excavation site selection [9], [10], autonomy software prototype to
execute complex and highly constrained missions with limited human
intervention [11], and field testing the functional autonomy stack in
terrestrial analog environments [12]. Specifically, our prior work [10]
proposed Deep Meta-Learning with Controlled Deployment Gaps (CoDeGa), an
adaptive scooping strategy that uses deep Gaussian processes trained with a
novel meta-learning approach. CoDeGa learns online from very limited
experience on target terrains despite large domain shifts from the training
set. These prior experiments were conducted on a low-fidelity testbed at the
University of Illinois Urbana-Champaign (UIUC) designed to rapidly collect
large-scale data for training and testing the models on a wide range of
terrains.

Building upon this foundational work, the next crucial step is to validate and
refine our strategy within a more sophisticated and representative
environment. The Ocean Worlds Lander Autonomy Testbed (OWLAT) [13] at NASA Jet
Propulsion Laboratory (JPL) provides such an environment. OWLAT is a high-
fidelity testbed developed to validate autonomy algorithms for future ocean
world missions. It serves as a state-of-the-art platform for simulating
various potential future planetary missions over a wide range of dynamic
environments, including surface operations on small bodies where recreating
the dynamics in low gravity is critical. By integrating CoDeGa with OWLAT, we
aim to assess the robustness of the proposed autonomy algorithms in conditions
that closely mimic those of actual extraterrestrial landscapes, thereby
bridging the gap between preliminary tests and real-world deployment.

In this paper, we report our experiences deploying the CoDeGa-trained adaptive
scooping model on the OWLAT testbed. Our contribution is threefold: (1) we
assess the feasibility of transferring the model across systems with similar
sensor suites, end-effectors, and primitive actions, (2) we validate the
model’s adaptability in novel, realistic environments and (3) we share the
lessons learned from designing and deploying learning-based autonomy
algorithms in the context of space exploration. Experimental results provide
strong evidence that CoDeGa-trained model adapts to the significant domain
shifts presented by the OWLAT testbed, reinforcing its applicability and
promising role in autonomous terrain sampling for future off-world missions.

##  2 Preliminaries and Background

This section describes the scooping problem setting in detail and provides a
brief overview of the solution approach using the CoDeGa-trained model as
proposed in [10]. We also highlight the differences between the UIUC testbed
and OWLAT and the importance of deploying and validating the solution approach
on the OWLAT testbed.

###  2.1 Scooping Problem

We study the problem of scooping in which the goal is to collect high-volume
samples from the lander’s workspace with a limited budget of attempts. The
problem is formulated as a sequential decision-making problem where the robot
observes the terrain RGB-D image $o\in\mathcal{O}$, uses a scooping policy to
apply action $a\in\mathcal{A}(o)$ where $\mathcal{A}(o)$ is a discrete set of
parameterized scooping motions dependent on the RGB-D image, and receives
reward $r\in\mathcal{R}$, which is the scooped volume.

Given a target terrain $T_{*}$, the robot’s goal is to find a series of
scooping motions that maximize the total reward from scoops across the first
$k$ attempts. During the $n$-th attempt, for $n\leq k$, the robot has access
to the history of scoops on this terrain
$H=\\{(o^{j},a^{j},r^{j})\,|\,j=1,\ldots,n-1\\}$. The robot also has access to
prior scooping experience, which consists of a set of $M$ terrains
$\\{T_{1},\dots,T_{M}\\}$, and a training dataset
$D_{i}=\\{(o^{j},a^{j},r^{j})\,|\,j=1,\ldots,N_{i}\\}$ of past scoops and
their rewards for each terrain $i=1,...,M$.

We suppose that a latent variable $\alpha$ characterizes a given terrain’s
indirectly observed properties including composition, material properties, and
topography. Let $\alpha_{*}$ characterize terrain $T_{*}$ and $\alpha_{i}$
characterize terrain $T_{i}$ for $i=1,\ldots,M$. The observation depends on
$\alpha$ and action rewards are unknown functions of both action and $\alpha$.
Standard supervised learning for modeling $r\approx f(o,a)$ is effective when
$\alpha$ aligns with training terrains and is deducible from observation $o$,
or if rewards are not heavily tied to unobserved latent characteristics.
Performance drops when $T_{*}$ is outside the training distribution or when
observation $o$ inadequately indicates the terrain’s latent aspects affecting
rewards.

We propose an adaptive online learning strategy for better sampling
performance on a terrain $T_{*}$ despite significant differences from training
terrains. This method utilizes real-time data from $H$, adapting to terrain
$T_{*}$ and countering domain shifts from the training set.

###  2.2 Adaptive Scooping using CoDeGa-trained Model

Figure 1: Overview of the CoDeGa-trained model with deep mean, deep kernel,
and feature encoder modules.

The solution approach proposed in [10], Deep Meta-Learning with Controlled
Deployment Gaps (CoDeGa), leverages a deep Gaussian process (GP) model to
capture the relationship between the observation-action pair $(o,a)$ and the
reward $r$. The deep Gaussian process model employs deep mean functions and
deep kernels where the input to the GP kernel is transformed by a neural
network. We note that, in addition to the observation-action pair, the model
is also conditioned on the online support set that contains the history of
previous scoops and their outcomes on the same terrain. In the context of a
scooping task, this model predicts the scooped volume given an input
consisting of a local patch of an RGB-D image and action parameters, as
illustrated in Fig. 1.

In the CoDeGa training procedure, the training terrains are split into mean
and kernel training sets containing different materials. Doing so encourages
the kernel to encounter residuals representative of those in out-of-
distribution tasks. The deep mean is first trained on the mean training set to
minimize error, and the GP kernel is then trained on the residuals of the deep
mean model applied to the kernel training set. This process is repeated,
similar to $k$-fold cross-validation, with a common kernel trained over
aggregated losses across folds. The strength of CoDeGa lies in its ability to
generate a model that performs well under deployment gaps, which are common in
real-world applications.

Given a model to robustly predict the scooped volume, [10] employs a Bayesian
optimization approach for selecting the scooping action. Rather than simply
using the mean prediction of the model, it utilizes an acquisition function
that also takes uncertainty into account. This function serves as a scoring
system that guides the selection of actions and encourages the exploration of
actions with uncertain outcomes, allowing for a more robust performance under
varying conditions.

The CoDeGa-based scooping strategy, while effective, requires a large amount
of data across different terrains for training the deep mean and kernel. In
the next section, we describe the UIUC testbed designed to collect such data
with minimal human supervision and the different terrains and compositions on
which the data is collected.

###  2.3 UIUC Testbed

The data collection testbed at the University of Illinois Urbana-Champaign
(UIUC) is designed for large-scale data collection and testing of learning-
based approaches for scooping tasks. The setup includes a UR5e arm with a
scoop mounted on the end-effector, an overhead Intel RealSense L515 RGB-D
camera, and a wheeled simulant bin that is approximately 0.9 m x 0.7 m x 0.2
m. A scoop action in this setup is a parameterized trajectory for the scoop
end-effector tracked by an impedance controller. Fig. 2 shows an illustration
of the testbed.

Figure 2: Comparison of the UIUC testbed (left) and the Ocean Worlds Lander
Autonomy Testbed (OWLAT) (right).

A terrain is defined as a unique composition of one or more materials. The
setup uses multiple wheeled simulant bins to parallelly collect data in one
bin and set up a new terrain in another that can be quickly swapped. For each
new terrain, data is collected by executing different scooping actions and
measuring the volume of scooped material. A detailed description of the
different materials considered and the scoop action parameters is provided in
[10].

The UIUC testbed enables rapid data collection at the cost of limited
representation of the ocean world lander system complexity. In the next
section, we describe the OWLAT testbed designed for validating autonomy
algorithms for ocean worlds missions.

###  2.4 Ocean Worlds Lander Autonomy Testbed

The Ocean Worlds Lander Autonomy Testbed (OWLAT) [13] is a testbed developed
at NASA Jet Propulsion Laboratory (JPL) to test and validate the performance
of various autonomy algorithms and architectures for future missions to ocean
worlds. As shown in Fig. 2, the testbed hardware consists of a 7-DOF Barrett
WAM7 robotic arm with a host of interchangeable end-effector tools
representing the manipulator, an intel Realsense D415 mounted on a pan-tilt
mount for 3D perception, and force-torque sensors located at the interface
between the arm and the platform and also at the end of the arm’s wrist. The
testbed also has a 6-DOF Stewart platform that is used to simulate the lander
and a simulant area that hosts the testbed’s terrains with different simulants
and surface features.

OWLAT hardware is complemented by a software interface to command the
manipulator and the camera along with its pan-tilt mount for carrying out
surface operations and collecting data. The force-torque sensors located at
the end of the wrist and at the interface between the arm and the Stewart
platform play a critical role in replicating the dynamical environment such
landers are likely to experience on the low-gravity icy moons of Jupiter and
Saturn. As the tool interacts with the simulant in the testbed, the reaction
forces measured are fed into a dynamics model of the system. The computed
motion is imposed on the Stewart platform in real-time. The use of high
bandwidth Ethercat force-torque sensors allows OWLAT to close the control-
sensing loop at 500 Hz and study test cases demonstrating how interaction with
the surface on objects with gravity as low as Enceladus
($g=0.13\,\text{m/s}^{2}$) can cause the legs of the lander to lift off the
ground, thereby achieving Earth gravity compensation without the use of
suspension cables and gantry mechanisms.

In the next section, we describe the process of deploying the CoDeGa-trained
model on the OWLAT testbed, highlighting the challenges of interfacing a
learned model with new hardware configurations vastly different from the
training setting.

##  3 Deployment

This section details the process of deploying the CoDeGa-trained model on
OWLAT testbed for selecting scooping actions based on RGB-D information of the
robot’s workspace and past experience in the deployed environment.

Figure 3: Remote deployment of the adaptive scooping strategy using UIUC-
trained model on OWLAT.

Deploying the adaptive scooping strategy with the CoDeGa-trained model
involves taking a sequence of scooping actions where the scoop volume
estimation model adapts based on the reward from the previously executed
actions. A single scooping attempt involves the following steps:

  1. 1.

collect the RGB-D data capturing the current state of the simulant bin,

  2. 2.

pass the data and the set of all candidate scoop actions to the client,

  3. 3.

evaluate the candidate actions using the model, taking into account the
history of previous actions and their outcomes,

  4. 4.

measure the scooped volume, and

  5. 5.

record the outcome along with the executed action and corresponding
observation in the online support set.

Fig. 3 shows an overview of the deployment process with the OWLAT testbed.
OWLAT provides a ROS interface with access to the RGB-D camera data and ROS
actions over the ROS middleware. We implemented ROS actions that allow us to
execute scoop actions parameterized similar to the UIUC testbed. The RGB-D
data captured by the camera is passed to the client over the ROS middleware.
The deployment client preprocesses the RGB-D data and passes it to the CoDeGa-
trained model along with the candidate actions. The decision maker then
decides the next action to execute based on the model’s evaluations and sends
the action to the testbed over the ROS middleware. The testbed executes the
action on the robot and measures the scooped volume. Given that the testing on
OWLAT involves far fewer overall scooping attempts when compared to the data
collection process in the UIUC testbed, we manually measure the scooped
volume. We note that a deployment-ready ocean world mission is expected to
include the instrumentation to analyze the scientific value of the scooped
material, thereby addressing the issue of scooped volume measurement.

We now discuss in detail the steps involved in processing RGB-D data to be
compatible with the CoDeGa-trained model and generating the candidate action
set that can be easily implemented on any robotic system with a scoop end-
effector.

###  3.1 Processing RGB-D Data

The CoDeGa-trained model utilizes a convolutional neural network-based feature
encoder to extract relevant information from the RGB-D data. While effective,
these encoders are susceptible to variations in input data, requiring a degree
of consistency in feature scale and camera orientation relative to the
training set for optimal performance. The UIUC testbed standardizes data
capture across materials with varying feature scales by using a static
overhead RGB-D camera, thereby streamlining data collection and preprocessing
by minimizing environmental inconsistencies.

In contrast, real-world systems rarely offer such controlled conditions. For
example, in the OWLAT testbed, the placement of the RGB-D camera is akin to
the camera placement in real-world landers. It is mounted on a pan-tilt unit
at the base of the robot, capturing data from this unique perspective, as
shown in Fig. 2. Additionally, real-world data is often further complicated by
variations in camera resolution and quality across systems, and it commonly
includes noise, occlusions, and other artifacts.

Figure 4: Overview of the steps involved in processing the RGB-D data captured
by the OWLAT testbed.

To align the OWLAT data with the training conditions of the CoDeGa-trained
model, we developed a preprocessing pipeline as shown in Fig. 4. This pipeline
includes (i) reprojecting the point cloud to a top-down view to emulate the
UIUC testbed perspective, (ii) reconstructing missing values to fill in gaps
in the RGB and depth data while accounting for occluded and out-of-range
regions, and (iii) using system state and constraints to filter out anomalies.
The resulting images are then used to produce localized image patches based on
the action parameters.

In the next section, we discuss how our choice of parameterized motion
primitives as actions enables us to easily deploy the CoDeGa-trained model on
any robotic system with a scoop end-effector. Additionally, we detail our
approach of generating valid candidate actions while accounting for system
constraints.

###  3.2 Generating Action Candidates

In our implementation, the scoop action is a motion primitive characterized by
an initial $x,y$ position, the yaw angle $\theta$ of the scoop, the scooping
depth $d$, and the impedance controller stiffness parameter $b$ that takes a
value out of the set {low, high}. The z-coordinate is deduced from the depth
map at the outset of the action. This action primitive follows the
parameterized trajectory described in [10], which is executed via an impedance
controller. By defining the action in the end effector space, we maintain a
consistent action interpretation across varied systems. We adopt the same
trajectory for the scoop action primitive as established in [10] for
deployment. Moreover, empirical evidence from [10] indicates that higher
stiffness values for the impedance controller yield superior performance on
diverse terrains; consequently, we fix the stiffness parameter $b$ to high
value within the OWLAT action primitive to leverage these observed benefits.

Figure 5: Illustration of the candidate action set generation process using
OWLAT testbed data.

To synthesize the set of candidate actions, we use the processed depth data
and system constraints to delineate regions in the simulant bin amenable to
scooping. Within these identified regions, we establish a uniform grid to
represent potential action locations. At each grid point, we evaluate eight
possible yaw angles—excluding any that are deemed infeasible—coupled with four
distinct scooping depths ($d$ values of 0.2 cm, 0.4 cm, 0.6 cm, and 0.8 cm)
and a single ’high’ stiffness value ($b$). We note that the size of the
resultant action set is dynamic, as it adjusts with the changes in the
environment. For each candidate action, we generate corresponding RGB and
depth image patches centered at the action $x,y$ and aligned with the scoop
orientation $\theta$, as illustrated in Fig. 5. The image patches along with
the stiffness parameter and the selected depth value are then used to evaluate
the action candidate, as illustrated in Fig. 1.

The next section describes the experiments conducted on the OWLAT testbed to
evaluate the performance of the CoDeGa-trained model.

##  4 Experimental Evaluation and Discussion

We performed the experiments using an active remote connection with the OWLAT
testbed operating at the Jet Propulsion Laboratory (JPL), Pasadena,
California, and the CoDeGa-trained model deployed for inference at the
University of Illinois Urbana-Champaign (UIUC), Champaign, Illinois. Table 1
lists all the materials used for training the model along with their
corresponding physical properties. For evaluation, we used a terrain designed
by subject matter experts using out-of-distribution materials, Comet and
Regolith. Comet is an unscoopable composition of grey comet simulant material
[14] surrounded by 3D printed PLA features with rugged terrain features from a
3D scan of Devil’s Golf Course in Death Valley National Park, painted to match
the Regolith’s color. Regolith is a fine sand-like material that is visually
distinct from the sand used in training. The two materials were composed
together to create a hypothetical representation of the ocean world terrain,
as depicted in Fig. 6.

Figure 6: (left) The terrain composition used for testing in OWLAT testbed. (right) The terrain composition with labels for the Regolith (green) and Comet (red) materials. Table 1: Materials used in the experiments along with their corresponding grain sizes, categorized by training and testing sets, with U.S. quarter coin for scale. Training | Testing
---|---
Sand fine play sand, $<<$ 1 mm |  Pebbles rocks, 0.8 – 1.0 cm |  Slate flat rocks, 2.0–4.0 cm |  Gravel rocks, 1.5–3.0 cm |  Comet unscoopable
|  |  |  |
Paper Balls crumpled paper, 4.0 – 6.0 cm |  Corn dry corn kernels, 0.3–0.7 cm |  Shredded Cardboard cardboard, 1.0 – 8.0 cm |  Mulch red wood landscape mulch |  Regolith fine sand, 0.1 mm – 0.5 mm
|  |  |  |

Following the experimental procedure in [10], we compared the proposed method
to the non-adaptive baseline (Non-Adaptive), i.e., only the deep mean
component of the CoDeGa-trained model, which is a neural network that predicts
the mean value of the estimated scoop volume, and a volume maximizing (Vol-
Max) policy, where the action is chosen to maximize the intersection between
the scoop’s swept volume and the terrain following a strategy proposed in the
excavation literature [15].

We conducted three iterations of the experiment using each method across three
different scenarios, each featuring varying terrain topologies. Scenarios 1
and 2 consist of flat Regolith regions and higher unscoopable Comet areas,
with minor terrain features in the Regolith region that occur naturally during
resets. In Scenario 3, the scoopable Regolith region is designed with terrain
features that have heights comparable to those of the unscoopable Comet
regions. The depth map in Fig. 7 illustrates the terrain topology for Scenario
3, where the three circular green areas represent mounds created in the
Regolith material.

The objective is to maximize the total volume of material scooped in $k=5$
attempts. Given that only one of the two materials in the testing terrain is
scoopable, we measured mass in each attempt rather than volume to enhance
measurement accuracy and ease. We used the average scooped mass over a run as
the metric for comparing the efficacy of different methods. For all three
scooping approaches, if the robot trajectory planning for a selected action
during an attempt fails, the subsequent highest scoring action is selected
until planning succeeds. We used the candidate action set outlined in Section
3.2.

Table 2: Results comparing CoDeGa with two baselines in OWLAT testbed. Higher scooped mass is better. | Vol-Max (gram) | Non-Adaptive (gram) | CoDeGa (gram)
---|---|---|---
Scenario 1 | 0.0 | 3.5 | 52.2
Scenario 2 | 0.0 | 18.8 | 64.2
Scenario 3 | 5.6 | 43.6 | 75.4
Average | 1.9 | 22.0 | 63.9

(a) Vol-Max

(b) Non-Adaptive

(c) CoDeGa

Figure 7: Example scooping attempts on the OWLAT testbed using three different
methods. The cyan-to-yellow colors represent regions with higher terrain
elevations.

The average scooped mass for all methods across the three different scenarios
is reported in Table 2. The proposed approach that utilized CoDeGa-trained
model significantly outperformed the baseline methods in all scenarios. Fig. 7
depicts the scooping actions executed during a representative trial for each
method. The Vol-Max method consistently selected scoop locations near the
Comet regions due to its inherent preference for areas with steep terrain
gradients. While the Non-Adaptive baseline initially targeted the Regolith
mounds, it failed to modify its policy in response to the data observed
online, eventually resorting to ineffective scooping attempts in the Comet
region, akin to Vol-Max. On the other hand, our approach initially engaged the
Comet region but rapidly adapted its strategy in response to the low mass of
scooped material, shifting focus to the Regolith mounds and thereby maximizing
the total scooped mass.

##  5 Conclusion

This paper presents the deployment of a CoDeGa-trained adaptive scooping model
on the high-fidelity OWLAT testbed. We evaluated the model, originally
developed on a low-fidelity UIUC testbed, on novel terrains in OWLAT testbed
with out-of-distribution materials. Experimental results demonstrate the
model’s capability to rapidly adapt to unfamiliar environments and make
effective decisions despite significant domain shifts. Specifically, the model
achieved several times higher average scooped mass compared to non-adaptive
baselines across varying terrain topologies by adapting its strategy online
based on limited experience.

The successful deployment substantiates the potential utility of learning-
based autonomy for maximizing scientific return under uncertainty in ocean
world missions. Moreover, it provides insights into real-world integration
challenges, including data preprocessing and motion planning. Overall, this
work endorses the feasibility of transferring learning-based systems from
idealized training settings to realistic deployment environments.

## Acknowledgments

This work was supported by NASA Grant 80NSSC21K1030. Part of this work was
carried out at the Jet Propulsion Laboratory, California Institute of
Technology, under a contract with the National Aeronautics and Space
Administration (80NM0018D0004).

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# DUNE as an Example of Sustainable Open Source Scientific Software
Development

Markus Blatt Dr. Markus Blatt - HPC-Simulation-Software & Service, Hans-Bunte-
Str. 8-10, 69123 Heidelberg, Email<EMAIL_ADDRESS>
In this paper we describe how DUNE, an open source scientific software
framework, is developed. Having a sustainable software framework for the
solution of partial differential equations is the main driver of DUNE’s
development. We take a look how DUNE strives to stay sustainable software.

##  1 The Cause for DUNE’s Development

DUNE, the “Distributed and Unified Numerics Environment” [5, 2, 1, 3] is a
modular software toolbox for solving partial differential equations. When
initiating DUNE the rule in scientific software development was that PhD
students were given a legacy software and further extended it during their PhD
until it did fit their needs. The purpose of this custom software was to
produce publishable scientific results. The software itself was only a
(unwanted) by-product. As such it was neither designed for maintainability nor
well documented.

Even institutions that have an inhouse software or framework developed by a
group of individual researchers suffer from such development practices. This
has produced many rather unmaintainable software dinosaurs. They have a lot of
hidden undocumented features unknown to current developers. This leads to new
researchers reimplementing functionality that they need.

Often the developing institutions have a rather narrow application area for
their software. Therefore it is almost impossible to use the software in a
slightly different area as originally intended. This again leads to
researchers having to reimplement the wheel if they want to switch the
application area or even the underlying numerics.

This situation was a thorn in the eyes of the DUNE developers. At the time of
DUNE’s inaugurating meeting 2002 they decided to attack these problems. The
developers come from different institutions and have different (numerical)
backgrounds. Quite a few of them had their own legacy inhouse software when
DUNE started, at that time mostly grid managers. The purpose was to develop a
unified interface to use them all underneath for different purposes. This
interface should be slim. That is provide only absolutely necessary
functionality. Nevertheless it should be universally applicable to different
numerical methods. In contrast to many older interfaces it should be a fine
grained interface in the sense that it is possible to access individual
entries in a container (e.g. vertices of a computational grid or matrix
entries) rather then just providing functions that work on the whole
containers. Given the rise of modern C++ with templates this was possible
without sacrificing computational efficiency.

##  2 The Development Cycle: History, Current and Future

It took approximately three years for fixing the initial interface and
developing various prototypes. Some were full grids, others were based on
legacy code but only exposed the interface of a DUNE grid. Another two years
went by until we released the first stable version and published the papers
[2, 1, 3] describing the interfaces.

Already at that time the DUNE community was divided into three groups: the so-
called core developers that are allowed to vote on interface decisions, normal
DUNE developers that have write access to the source code repositories, and
other DUNE users that contribute with bug report and patches. This division
exists until now. Active patch providers can become developers, if they find a
core developer willing to mentor and take responsibility for them. A DUNE
developer can become core developers with the support of the majority of the
core developers.

Currently DUNE users play a more and more important role collaborating with
the developers and we search for ways to raise their participation. This will
be done with more regular user meetings. The first one was in 2010 and the
next one will be in September 2013. This time there will also be a developer
meeting directly afterwards. This will facilitate an even closer interaction
between users and developers and make sure that future developments fit the
need of an even more varied user base.

The main current aim is to establish a more forseeable release cycle. We are
planning to release roughly twice per year. This allows users to focus their
own development on the current stable release and still be able to use most of
the most recent features.

##  3 Key Sustainability Drivers of DUNE

###  3.1 Scientific Developers with Varied Backgrounds

The core developers of DUNE all come from various institutions, have different
scientific backgrounds (e.g. Navier Stokes, linear elasticity, flow in porous
media, multigrid methods), and use different numerical methods. This makes
sure that the software and its interface is usable for various numerical
methods.

###  3.2 Open Source

From the beginning DUNE is available as open source. Its license is GPL-2 with
the so-called “runtime exception”. This exception allows the template code of
the DUNE modules to be used even in closed source projects. This was a
prerequisite of some of the initial DUNE developers because there are German
funding agencies that sponsor projects where both industry and academia
collaborate. Currently there are big companies that adopted DUNE as their
development platform due to its openness and even open sourced some of their
own products.

###  3.3 Modularity

DUNE consists of various modules, each for its own purpose. Currently there
are seven core modules supported by the DUNE community and two discretization
modules developed by some DUNE developers. This lets users choose just the
functionality that they need or want.

When moving from one monolithic source tree to various modules, each with its
own source tree, the developers also implemented a rather elaborate build
system on top of autoconf and automake. This makes it simple for users to
create their own DUNE modules by simply calling a script. The new module now
has the same build system support as the core DUNE modules and allows for code
and build system reuse.

Today there are various other large frameworks based on DUNE: DUMUX, a
framework for multi-physics, multi-phase, and multi-domain simulation, [4],
and OPM, the open porous media initiative, [6], that strives to build an open
source simulator suite for flow and transport in porous media for the oil
industry.

###  3.4 Semi Open Development Model

The further development of DUNE is open. There are regular developer and user
meetings. Even the developer meetings are open to others upon request. At the
developer meetings important decisions, such as interface changes, are made.
The core developer, currently nine from eight different institutions in
Germany, the UK, and the USA, are able to vote during decisions and a majority
of them is needed for a decision. Most decisions are discussed a priori on the
mailing list or the bug tracker. Smaller once are even decided there. This
procedure makes sure that only changes or additions are made that make sense
to and are really needed by the majority. Flaws in an initial proposal are
easily detected because of the diverse application areas of the developers and
users.

In recent years the cooperation of the DUNE users with the DUNE development
has increased. We receive various important bugfixes from them. Due to the
stability of the development branch of DUNE experienced users often choose to
use it in their everyday work and development. This leads to many human
testers and helps to further improve the stability of DUNE.

###  3.5 Code and Interface Reviews

Since the beginning DUNE was developed publically using modern version control
systems (VCS): first subversion and since 2013 git. All committed patches are
also send to a special mailing list. Most of the core DUNE developers read
this list and thus review the patches. This often helps to detect bugs,
possible problems with downstream user modules, or not aproofed interface
changes very early in the development process. If developers want to change
the interface, then they have to provide a proposal that has to be aproofed by
the core developers at one of their meetings.

###  3.6 Joint Development by Industry and Academia

Since 2011 one the core developers has become an entrepreneur and is now
providing DUNE contract work, and support. Commercial entities participating
in the DUNE development naturally have a long term interest in the
development. Often they have resources available for infrastructure work (e.g.
build system, testing, and usability features) and naturally do quality
management. This reduces the work for scientific developers and lets them
focus more on the science side. In the end this will be advantageous for all.

## References

  * [1] Peter Bastian, Markus Blatt, Andreas Dedner, Christia n Engwer, Robert Klöfkorn, Ralf Kornhuber, Mario Ohlberger, and Oliver Sander.  A generic grid interface for parallel and adaptive scientific computing. part II: implementation and test in DUNE.  Computing, 82(2–3):121–138, 2008.
  * [2] Peter Bastian, Markus Blatt, Andreas Dedner, Christia n Engwer, Robert Klöfkorn, Mario Ohlberger, and Oliver Sander.  A generic grid interface for parallel and adaptive scientific computing. part I: abstract framework.  Computing, 82(2–3):103–119, 2008.
  * [3] Markus Blatt and Peter Bastian.  The iterative solver template library.  In Bo Kågström, Erik Elmroth, Jack Dongarra, and Jerzy Waśniewski, editors, Applied Parallel Computing. State of the Art in Scientific Computing, volume 4699 of Lecture Notes in Computer Science, pages 666–675. Springer, 2007.
  * [4] DUMUX.  http://www.dumux.org/.
  * [5] DUNE.  http://www.dune-project.org/.
  * [6] OPM.  http://www.opm-project.org/.