# Adjoints of sums of m-accretive operators and applications to non-autonomous
evolutionary equations

Rainer Picard Institut für Analysis, TU Dresden, Germany Sascha Trostorff
Mathematisches Seminar, CAU Kiel, Germany and Marcus Waurick Institut für
Angewandte Analysis, TU Bergakademie Freiberg, Germany

###### Abstract

We provide certain compatibility conditions for m-accretive operators such
that the adjoint of the sum is given by the closure of the sum of the
respective adjoint. We revisit the proof of well-posedness of the abstract
class of partial differential-algebraic equations known as evolutionary
equations. We show that the general mechanism provided here can be applied to
establish well-posedness for non-autonomous evolutionary equations with
$L_{\infty}$-coefficients thus not only generalising known results but opening
up new directions other methods such as evolution families have a hard time to
come by.

Keywords Evolutionary Equations, Non-autonomous Equations, m-accretive
Operators
MSC2020 35P05 (Primary), 47D99, 35Q61, 35Q59, 35K05, 35L05 (Secondary)

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Adjoints of Sums of m-accretive Operators
  4. 4 Applications to Evolutionary Equations
  5. 5 Conclusion

##  1 Introduction

Evolutionary Equations as introduced in the seminal paper [7] provide a
Hilbert space perspective towards numerous (both linear and non-linear) time-
dependent phenomena in mathematical physics. We refer to the monographs [8,
12] for a set of examples as well as further development of the theory. It is
instrumental for the success of the theory of evolutionary equations that many
(if not all) equations from mathematical physics can be written as a time-
dependent partial differential-algebraic equation. Then, establishing the
time-derivative as an m-accerive, normal operator in some weighted Hilbert
space and gathering all the other unbounded operators (i.e., spatial
derivative operators) in an abstract m-accretive operator $A$ defined on some
Hilbert space enconding the spatial variables, one can write evolutionary
equation as an operator equation in the following form

$\left(\partial_{0}\mathcal{M}_{0}+\mathcal{M}_{1}+A\right)U=F,$ (1)

where $\partial_{0}$ is the time-derivative, $U$ is the unknown, $F$ models
external forces and $\mathcal{M}_{0}$ and $\mathcal{M}_{1}$ are linear
operators in the considered space-time Hilbert space, which is a tensor
product Hilbert space putting together temporal and spatial variables. Any
standard solution theory for evolutionary equations of the form (1) provides
conditions on the so-called material law oparators, $\mathcal{M}_{0}$ and
$\mathcal{M}_{1}$, so that

$\overline{\left(\partial_{0}\mathcal{M}_{0}+\mathcal{M}_{1}+A\right)}-c$

becomes m-accretive in the space-time Hilbert space. Looking into the
different proofs under the various assumptions on the material law operators,
one realises that the principal mechanism of showing m-accretivity is based on
the following general well-known fact. Quickly recall that an operator $T$ in
some Hilbert space is accretive111We assume every Hilbert space to be real.,
if for all $u\in\operatorname{dom}(T),$

$\langle u,Tu\rangle\geq 0.$

$T$ is m-accretive, if $T$ is accretive and $T+\lambda$ is onto for all
$\lambda>0$ (or equivalently for some $\lambda>0$).

######  Theorem 1.1 (see also [4, Chapter 3, Theorem 1.43]).

Let $H$ be a Hilbert space, $T\colon\operatorname{dom}(T)\subseteq H\to H$
densely defined and closed linear operator. Then the following conditions are
equivalent:

  1. 1.

$T$ is m-accretive;

  2. 2.

$T$ and $T^{*}$ are accretive.

If $T-c$ is m-accretive for some $c>0$, then $0\in\rho(T)$.

In order to apply the last theorem showing m-accretivity of
$T\coloneqq\overline{\left(\partial_{0}\mathcal{M}_{0}+\mathcal{M}_{1}+A\right)}-c$,
it is necessary to compute the adjoint of $T$, which, applying standard
results, boils down to computing the adjoint of
$\left(\partial_{0}\mathcal{M}_{0}+\mathcal{M}_{1}+A\right)$ in the space-time
Hilbert space. Since both $\partial_{0}\mathcal{M}_{0}$ and $A$ are
generically speaking unbounded operators, this is a non-trivial task. We refer
to [5] and the references therein for a general account of computing adjoints
of sums of unbounded operators (focussing on situations complemented to the
present one here). In any case, once a formula of the type

$\left(\partial_{0}\mathcal{M}_{0}+\mathcal{M}_{1}+A\right)^{*}=\overline{\left(\left(\partial_{0}\mathcal{M}_{0}+\mathcal{M}_{1}\right)^{*}+A^{*}\right)}$

is established, the accretivity of $T$ (_and_ of $T^{*}$) follow from
m-accretivity of $\partial_{0}\mathcal{M}_{0}+\mathcal{M}_{1}$. Thus,
conditions for (1) being well-posed need to address the two facts: computing
the adjoint in the way sketched above needs to be possible and the problem
needs to be m-accretive if $A=0$.

The aim of this article is to understand the situation for the case
$\mathcal{M}_{0}=M_{0}(m_{0})$ and $\mathcal{M}_{1}=M_{1}(m_{0})$ are
multiplication operators of multiplying in the time-variable by $t\mapsto
M_{0}(t)$ and $t\mapsto M_{1}(t)$, respectively. It is known that Lipschitz
continuous $M_{0}$ allows for computing the adjoint as above and suitable
positive definiteness conditions for $M_{0}$ together with its (a.e. existing)
derivative $M_{0}^{\prime}$ and $M_{1}$ lead to m-accretivity for the case
$A=0$, see [10] or [12, Chapter 16].

A particular instance of the perspective of using operator sums to understand
partial differential equations has been provided (at least) as early as [2].
However, the methods fail to apply in a straightforward manner as the
coefficient $M_{0}$ is allowed to have a non-trivial kernel here (at least in
the case of Lipschitz continuous $M_{0}$). We illustrate our findings in the
non Lipschitz case by means of an example later on; note that this particular
instance was addressed in [1]. Even though we were not able to fully rectify
the arguments mentioned in this reference, their major application is
concerned with continuous-in-time coefficients anyway. It seems that this
condition is crucial for evolution families to be applicable. Thus, in case of
non-Lipschitz continuous $M_{0}$, we establish well-posedness for a system of
equations other methods (such as evolution families or evolution semigroups,
see [3, Chapter VI, Section 9] and [6]) are structurally deemed to fail.

The next section is concerned with some functional analytic preliminaries,
which we shall find useful in the subsequent parts. The subsequent section
contains our main result concerning operator sums of m-accretive operators.
The second to last section deals with applications to evolutionary equations.
We summarise our findings and open problems in the conclusion section.

##  2 Preliminaries

Throughout this section, let $H_{0},H_{1},H_{2}$ be Hilbert spaces.

######  Lemma 2.1.

Let $T\colon\operatorname{dom}(T)\subseteq H_{0}\to H_{1}$ be a closed linear
operator and $B\colon H_{1}\to H_{2}$ bounded and linear. Then, if $B$ is one-
to-one and has closed range, $BT$ is closed.

###### Proof.

The closed graph theorem yields that the adstriction $\widetilde{B}\colon
H_{1}\to\operatorname{ran}(B)$ of $B$ is a continuously invertible operator.
Next, let $(x_{n})_{n}$ in $\operatorname{dom}(BT)$ such that $x_{n}\to x$ and
$BTx_{n}\to y$ in $H_{0}$ and $H_{2}$ as $n\to\infty$ for some $x\in H_{0}$
and $y\in H_{2}$. Then, by the continuity of $(\widetilde{B})^{-1}$, we infer
$Tx_{n}=\left(\widetilde{B}\right)^{-1}BTx_{n}\to\left(\widetilde{B}\right)^{-1}y$
in $H_{1}$ as $n\to\infty$. By the closedness of $T$, we obtain
$x\in\operatorname{dom}(T)\subseteq\operatorname{dom}(BT)$ and
$Tx=\left(\widetilde{B}\right)^{-1}y.$ Applying $\widetilde{B}$ to both sides
of the latter equality, we infer $y=BTx$ as desired. ∎

######  Theorem 2.2.

Let $T\colon\operatorname{dom}(T)\subseteq H_{0}\to H_{1}$ be closed, $B\colon
H_{2}\to H_{0}$ be a bounded linear operator. Then, as an identity of
relations, we have

$(TB)^{*}=\overline{B^{*}T^{*}}.$

Moreover, $TB$ is densely defined if and only if $B^{*}T^{*}$ is a closable
operator. If, $B^{*}$ is one-to-one and has closed range and $T$ is densely
defined, then $B^{*}T^{*}$is closed. In particular, in this case, we have $TB$
is densely defined and

$(TB)^{*}=B^{*}T^{*}.$

###### Proof.

The first statement is a consequence of [12, Theorem 2.3.4]. The stated
equivalence follows from $(TB)^{*}=\overline{B^{*}T^{*}}$ in conjunction with
[12, Lemma 2.2.7]. Finally, the statement containing $B^{*}T^{*}$ closed is a
direct consequence of Lemma 2.1. ∎

######  Remark 2.3.

The assumptions on $B^{*}$ are equivalent to $B$ being onto. Indeed, by the
closed range theorem, the closed range of $B$ is then inherited by $B^{*}$ and
the equation $H_{0}=\ker(B^{*})\oplus\overline{\operatorname{ran}}(B)$ yields
that (almost) surjectivity of $B$ is equivalent to injectivity of $B^{*}$.

######  Theorem 2.4.

Let $T\colon\operatorname{dom}(T)\subseteq H_{0}\to H_{0}$ be densely defined
and closed, $B\colon H_{0}\to H_{0}$ be bounded, linear operator mapping onto
$H_{0}$. Then

$\left(B^{*}TB\right)^{*}=B^{*}T^{*}B.$

###### Proof.

By Lemma 2.1 and Remark 2.3, $B^{*}T$ is closed. It is – trivially – densely
defined as so is $T$. Hence, by Theorem 2.2, we deduce

$\left((B^{*}T)B\right)^{*}=\overline{B^{*}(B^{*}T)^{*}}.$

Next, since $\left(B^{*}T\right)^{*}$ is a closed linear operator as the
adjoint of a densely defined operator and as $B^{*}$ is one-to-one with closed
range (see Remark 2.3), $B^{*}(B^{*}T)^{*}$ is closed by Lemma 2.1; i.e.,
$\overline{B^{*}(B^{*}T)^{*}}=B^{*}(B^{*}T)^{*}$. Next, we compute
$(B^{*}T)^{*}$. For this, using Theorem 2.2 again, we deduce

$\left(T^{*}B\right)^{*}=\overline{B^{*}T^{**}}=\overline{B^{*}\overline{T}}=\overline{B^{*}T}=B^{*}T,$

as $B^{*}T$ is closed, again by Lemma 2.1. Computing adjoints on both sides,
we infer

$T^{*}B=\left(T^{*}B\right)^{**}=\left(B^{*}T\right)^{*},$

where we used that $T^{*}B$ is closed. As a consequence of the above, we get

$\left((B^{*}T)B\right)^{*}=B^{*}(B^{*}T)^{*}=B^{*}T^{*}B.$

######  Proposition 2.5.

Let $T\colon\operatorname{dom}(T)\subseteq H_{0}\to H_{0}$ be densley defined
and closed and $B\colon H_{0}\to H_{0}$ be a topological isomorphism. If
$D\subseteq\operatorname{dom}(T)$ is a core for $T$, then $B^{-1}[D]$ is a
core for $B^{*}TB.$

###### Proof.

$B$ being a topological isomorphism, $B^{-1}$ maps dense sets onto dense sets;
thus $B^{-1}[D]$ is dense in $H_{0}$. Also it is elementary to see that
$B^{-1}[D]\subseteq\operatorname{dom}(B^{*}TB)$. Finally, let
$x\in\operatorname{dom}(B^{*}TB)=\operatorname{dom}(TB).$ Then
$Bx\in\operatorname{dom}(T).$ By assumption, we find $(y_{n})_{n}$ in $D$ such
that $y_{n}\to Bx$ and $Ty_{n}\to TBx$ in $H_{0}$ as $n\to\infty$. Defining
$x_{n}\coloneqq B^{-1}y_{n}\in B^{-1}[D]$, we get $x_{n}\to B^{-1}Bx=x$ and
$TBx_{n}=Ty_{n}\to TBx$ as $n\to\infty.$ The continuity of $B^{*}$ yields the
assertion. ∎

##  3 Adjoints of Sums of m-accretive Operators

This section is devoted to computing the adjoint of a sum of two (unbounded)
operators $T$ and $S$ both densely defined and closed on a Hilbert space $H$.
The aim is to provide conditions so that

$(S+T)^{*}=\overline{{S^{*}+T^{*}}}.$

We refer to [9], where several conditions for this equality were given. The
main theorem, which in turn is relevant to the applications we have in mind,
reads as follows:

######  Theorem 3.1.

Let $T\colon\operatorname{dom}(T)\subseteq H_{0}\to H_{1}$ and
$S\colon\operatorname{dom}(S)\subseteq H_{0}\to H_{1}$ be two densely defined
closed operators such that $\operatorname{dom}(S)\cap\operatorname{dom}(T)$ is
dense. Moreover, we assume that there exist families
$(L_{\varepsilon})_{\varepsilon>0}$
$\left(K_{\varepsilon}\right)_{\varepsilon>0}$ and
$(\widetilde{K}_{\varepsilon})_{\varepsilon>0}$ in $L(H_{1})$ and
$\left(R_{\varepsilon}\right)_{\varepsilon>0}$, in $L(H_{0})$ such that
$L_{\varepsilon}\to 1_{H_{1}},\,R_{\varepsilon}\to 1_{H_{0}}$ and
$K_{\varepsilon},\widetilde{K}_{\varepsilon}\to 0$ in the weak operator
topology as $\varepsilon\to 0.$ Moreover, assume that

$\displaystyle L_{\varepsilon}S$ $\displaystyle\subseteq
SR_{\varepsilon}+K_{\varepsilon},$ $\displaystyle L_{\varepsilon}T$
$\displaystyle\subseteq TR_{\varepsilon}+\widetilde{K}_{\varepsilon}$ (1)

and

$L_{\varepsilon}^{\ast}[\operatorname{dom}\left((S+T)^{\ast}\right)]\subseteq\operatorname{dom}(S^{\ast})\cap\operatorname{dom}(T^{\ast}).$
(2)

Then

$(S+T)^{\ast}=\overline{S^{\ast}+T^{\ast}}.$

###### Proof.

Since in general $\overline{S^{\ast}+T^{\ast}}\subseteq(S+T)^{\ast}$ (see [12,
Theoerem 2.3.2]) it suffices to prove the remaining inclusion. So, let
$u\in\operatorname{dom}(S+T)^{\ast}$ and set $u_{\varepsilon}\coloneqq
L_{\varepsilon}^{\ast}u\in\operatorname{dom}(S^{\ast})\cap\operatorname{dom}(T^{\ast}).$
Let $v\in\operatorname{dom}(S)\cap\operatorname{dom}(T).$ We compute

$\displaystyle\langle(S^{\ast}+T^{\ast})u_{\varepsilon},v\rangle$
$\displaystyle=\langle L_{\varepsilon}^{\ast}u,(S+T)v\rangle$
$\displaystyle=\langle u,L_{\varepsilon}(S+T)v\rangle$ $\displaystyle=\langle
u,(S+T)R_{\varepsilon}v+(K_{\varepsilon}+\widetilde{K}_{\varepsilon})v\rangle$
$\displaystyle=\langle\left(R_{\varepsilon}^{\ast}(S+T)^{\ast}+(K_{\varepsilon}+\widetilde{K}_{\varepsilon})^{\ast}\right)u,v\rangle$

and since $\operatorname{dom}(S)\cap\operatorname{dom}(T)$ is dense, we infer

$(S^{\ast}+T^{\ast})u_{\varepsilon}=\left(R_{\varepsilon}^{\ast}(S+T)^{\ast}+(K_{\varepsilon}+\widetilde{K}_{\varepsilon})^{\ast}\right)u\rightharpoonup(S+T)^{\ast}u,$

where we have used $R_{\varepsilon}^{\ast}\to 1_{H_{0}}$ and
$(K_{\varepsilon}+\widetilde{K}_{\varepsilon})^{\ast}\to 0$ in the weak
operator topology. Since also $u_{\varepsilon}\rightharpoonup u$ (use again
$L_{\varepsilon}^{\ast}\to 1_{H_{1}}$ in the weak operator topology), we infer
that $u\in\operatorname{dom}\left(\overline{S^{\ast}+T^{\ast}}\right)$ with

$\left(\overline{S^{\ast}+T^{\ast}}\right)u=(S+T)^{\ast}u.$

######  Remark 3.2.

If $SR_{\varepsilon}$ is bounded and
$\operatorname{dom}(S)\cap\operatorname{dom}(T)$ is a core for $T$ in the
above theorem, then (2) holds true. Indeed, from (1) we infer

$\left(SR_{\varepsilon}+K_{\varepsilon}\right)^{\ast}\subseteq\left(L_{\varepsilon}S\right)^{\ast}=S^{\ast}L_{\varepsilon}^{\ast}.$

If now $SR_{\varepsilon}$ is bounded, the operator on the left-hand side in
the above inclusion is bounded, and hence, the operator on the right-hand side
is defined on $H_{1}$, meaning that
$\operatorname{ran}(L_{\varepsilon}^{\ast})\subseteq\operatorname{dom}(S^{\ast})$.
Hence, for $u\in\operatorname{dom}(T)\cap\operatorname{dom}(S)$ and
$v\in\operatorname{dom}((S+T)^{\ast})$ we compute

$\displaystyle\langle Tu,L_{\varepsilon}^{\ast}v\rangle$
$\displaystyle=\langle(S+T)u-Su,L_{\varepsilon}^{\ast}v\rangle$
$\displaystyle=\langle L_{\varepsilon}(S+T)u,v\rangle-\langle
u,S^{\ast}L_{\varepsilon}^{\ast}v\rangle$
$\displaystyle=\langle(S+T)R_{\varepsilon}u+(K_{\varepsilon}+\widetilde{K}_{\varepsilon})u,v\rangle-\langle
u,S^{\ast}L_{\varepsilon}^{\ast}v\rangle$ $\displaystyle=\langle
u,R_{\varepsilon}^{\ast}(S+T)^{\ast}v+(K_{\varepsilon}+\widetilde{K}_{\varepsilon})^{\ast}v-S^{\ast}L_{\varepsilon}^{\ast}v\rangle$

and since $\operatorname{dom}(T)\cap\operatorname{dom}(S)$ is a core for $T$,
we infer that also $L_{\varepsilon}^{\ast}v\in\operatorname{dom}(T^{\ast}).$

######  Theorem 3.3.

Let $S\colon\operatorname{dom}(S)\subseteq H\to H$ and
$T\colon\operatorname{dom}(T)\subseteq H\to H$ both m-accretive. If
$(1+T)^{-1}(1+S)^{-1}=(1+S)^{-1}(1+T)^{-1}$, then
$\operatorname{dom}(S)\cap\operatorname{dom}(T)$ is dense in $H$ and

$(S+T)^{*}=\overline{{S^{*}+T^{*}}}.$

Before we prove this result, we draw some elementary consequences of the
commutator condition. A first consequence of this will be that $S+T$ is
densely defined.

######  Proposition 3.4.

Under the conditions of Theorem 3.3, the following holds:

  1. 1.

For all $\varepsilon>0$, we have

$(1+\varepsilon S)^{-1}T\subseteq T(1+\varepsilon S)^{-1}.$

  2. 2.

$\operatorname{dom}(S)\cap\operatorname{dom}(T)$ is a core for $T$. In
particular, $\operatorname{dom}(S)\cap\operatorname{dom}(T)$ is dense in $H$.

###### Proof.

For the first statement, we observe that
$(1+T)^{-1}(1+S)^{-1}=(1+S)^{-1}(1+T)^{-1}$ yields

$(1+S)^{-1}(1+T)\subseteq(T+1)(1+S)^{-1}.$

As a consequence,

$(1+S)^{-1}T\subseteq T(1+S)^{-1}.$

By [12, Lemma 9.3.3 (a)], we deduce for all $\varepsilon>0$ that

$(1+\varepsilon
S)^{-1}T=\frac{1}{\varepsilon}(\frac{1}{\varepsilon}+S)^{-1}T\subseteq
T\frac{1}{\varepsilon}(\frac{1}{\varepsilon}+S)^{-1}=T(1+\varepsilon S)^{-1}.$

The second statement is based on the observation that $(1+\varepsilon
S)^{-1}\to 1$ as $\varepsilon\to 0+$ in the strong operator topology (this
follows from the strong convergence on $\operatorname{dom}(S)$ and the uniform
boundedness of the resolvents). Let now $x\in\operatorname{dom}(T)$ and define
$x_{\varepsilon}\coloneqq(1+\varepsilon S)^{-1}x.$ Then, $x_{\varepsilon}\to
x$ as $\varepsilon\to 0+$ and by part 1 of the present proposition, we deduce
$x_{\varepsilon}\in\operatorname{dom}(T)$ and

$Tx_{\varepsilon}=(1+\varepsilon S)^{-1}Tx\to Tx,$

yielding that $\operatorname{dom}(T)\cap\operatorname{dom}(S)$ is a core for
$T$. ∎

###### Proof of Theorem 3.3.

We apply Theorem 3.1. By Proposition 3.4 we see that for
$L_{\varepsilon}\coloneqq R_{\varepsilon}\coloneqq(1+\varepsilon S)^{-1}$ and
$K_{\varepsilon}=\widetilde{K}_{\varepsilon}=0$ the relations (1) are
satisfied. Furthermore, by Proposition 3.4 we have that
$\operatorname{dom}(S)\cap\operatorname{dom}(T)$ is dense and a core for $T.$
Since clearly $SR_{\varepsilon}$ is bounded, Remark 3.2 gives that also (2) is
satisfied. Thus, the assertion follows from Theorem 3.1. ∎

##  4 Applications to Evolutionary Equations

This section is devoted to apply the previous findings to operator equations
in weighted, vector-valued $L_{2}$-type spaces. The general setting can be
found in [8, 12]. Throughout, let $H$ be a Hilbert space and for
$\rho\in\mathbb{R}$ we let

$L_{2,\rho}(\mathbb{R};H)\coloneqq\\{f\in
L_{2,\textnormal{loc}}(\mathbb{R};H);\int_{\mathbb{R}}\|f(t)\|_{H}^{2}\exp(-2\rho
t)dt<\infty\\},$

endowed with the obvious norm and corresponding scalar product. We define

$\partial_{0}\colon H_{\rho}^{1}(\mathbb{R};H)\subseteq
L_{2,\rho}(\mathbb{R};H)\to
L_{2,\rho}(\mathbb{R};H),\phi\mapsto\phi^{\prime}.$

For $\rho>0$, it can be shown that $\partial_{0}$ is m-accretive with
$\operatorname{Re}\partial_{0}=\rho$.

For a bounded, strongly measurable, operator-valued function
$M\colon\mathbb{R}\to L(H)$, we denote by

$M(m_{0})\in L(L_{2,\rho}(\mathbb{R};H))$

the associated multiplication operator of multiplying by $M$. In applications,
$M$ will be induced by scalar-valued measurable functions; that is, $M\in
L_{\infty}(\mathbb{R}).$ We will work under the following standing
assumptions.

  1. 1.

Let $A\colon\operatorname{dom}(A)\subseteq H\to H$ be $m$-accretive.

  2. 2.

Let $M_{0},M_{1}\colon\mathbb{R}\to L(H)$ be strongly measurable and uniformly
bounded.

  3. 3.

$M_{0}(m_{0})^{*}=M_{0}(m_{0})$.

Here we have employed the custom to re-use the notation $A$ for the
(canonically) extended operator defined on $L_{2,\rho}(\mathbb{R};H)$ with
domain $L_{2,\rho}(\mathbb{R};\operatorname{dom}(A))$. The aim is to study the
well-posedness of non-autonomous problems of the form

$\left(\partial_{0}M_{0}(m_{0})+M_{1}(m_{0})+A\right)U=F$ (3)

under suitable commutator conditions of $M_{0}(m_{0})$ with $\partial_{0}$ or
with $A$.

### Bounded Commutator with $\partial_{0}$

We begin to study the case when $M_{0}(m_{0})$ and $\partial_{0}$ have a
bounded commuator; that is, we assume there exists a strongly measurable and
uniformly bounded mapping $M^{\prime}_{0}\colon\mathbb{R}\to L(H)$ such that

$M_{0}(m_{0})\partial_{0}\subseteq\partial
M_{0}(m_{0})-M_{0}^{\prime}(m_{0}).$

######  Remark 4.1.

In [10] it was shown that this assumption is equivalent to the Lipschitz-
continuity of $M_{0}$. In this case, $M_{0}$ is differentiable almost
everywhere and $M_{0}^{\prime}$ is just the so-defined derivative of $M_{0}.$

Moreover, we impose the following accretivity condition on $M_{0}(m_{0})$ and
$M_{1}(m_{0})$:

$\exists c>0,\,\rho_{0}>0\,\forall\rho\geq\rho_{0}:\,\rho
M_{0}(t)+\frac{1}{2}M_{0}^{\prime}(t)+M_{1}(t)\geq c\quad(t\in\mathbb{R}\text{
a.e.}).$ (4)

######  Lemma 4.2.

Assume (4). Then for each $\rho\geq\rho_{0}$ the operator

$\partial_{0}M_{0}(m_{0})+M_{1}(m_{0})-c$

is accretive. Moreover, $\operatorname{dom}(\partial_{0})$ is a core for this
operator.

###### Proof.

If $u\in\operatorname{dom}(\partial_{0})$ we infer

$\displaystyle 2\langle(\partial_{0}M_{0}(m_{0})+M_{1}(m_{0}))u,u\rangle$
$\displaystyle=\langle\left(\partial_{0}M_{0}(m_{0})+M_{0}(m_{0})\partial_{0}\right)u,u\rangle+\langle
M_{0}^{\prime}(m_{0})u,u\rangle+2\langle M_{1}(m_{0})u,u\rangle.$

Moreover, with $\partial_{0}^{*}=-\partial_{0}+2\rho$ (see [12, Corollary
3.2.6])

$\displaystyle\langle\partial_{0}M_{0}(m_{0})u,u\rangle$
$\displaystyle=\langle u,M_{0}(m_{0})\partial_{0}^{\ast}u\rangle$
$\displaystyle=-\langle u,M_{0}(m_{0})\partial_{0}u\rangle+2\rho\langle
u,M_{0}(m_{0})u\rangle,$

which gives

$\langle\left(\partial_{0}M_{0}(m_{0})+M_{0}(m_{0})\partial_{0}\right)u,u\rangle=2\rho\langle
u,M_{0}(m_{0})u\rangle.$

Summarising, we obtain

$\langle(\partial_{0}M_{0}(m_{0})+M_{1}(m_{0}))u,u\rangle=\langle\left(\rho
M_{0}(m_{0})+\frac{1}{2}M_{0}^{\prime}(m_{0})+M_{1}(m_{0})\right)u,u\rangle\geq
c\|u\|^{2}.$

It remains to prove that $\operatorname{dom}(\partial_{0})$ is a core for
$\partial_{0}M_{0}(m_{0}).$ This however follows from

$(1+\varepsilon\partial_{0})^{-1}M_{0}(m_{0})=M_{0}(m_{0})(1+\varepsilon\partial_{0})^{-1}-\varepsilon(1+\varepsilon\partial_{0})^{-1}M_{0}^{\prime}(m_{0})(1+\varepsilon\partial_{0})^{-1},$

which gives

$(1+\varepsilon\partial_{0})^{-1}\partial_{0}M_{0}(m_{0})=\partial_{0}M_{0}(m_{0})(1+\varepsilon\partial_{0})^{-1}-\varepsilon\partial_{0}(1+\varepsilon\partial_{0})^{-1}M_{0}^{\prime}(m_{0})(1+\varepsilon\partial_{0})^{-1}.$

If now $u\in\operatorname{dom}(\partial_{0}M_{0}(m_{0}))$ we set
$u_{\varepsilon}\coloneqq(1+\varepsilon\partial_{0})^{-1}u\in\operatorname{dom}(\partial_{0})$
and since $(1+\varepsilon\partial_{0})^{-1}\to 1$ strongly, the latter
equality proves that $u_{\varepsilon}\to u$ with respect to the graph norm of
$\partial_{0}M_{0}(m_{0}).$ ∎

We obtain [12, Theoerem 16.3.1] or the main result of [10] as a special case:

######  Theorem 4.3.

Assume (4). Then for each $\rho\geq\rho_{0}$ the operator

$\overline{\partial_{0}M_{0}(m_{0})+M_{1}(m_{0})+A}-c$

is m-accretive and hence, $\overline{\partial_{0}M_{0}(m_{0})+M_{1}(m_{0})+A}$
is boundedly invertible in $L_{2,\rho}(\mathbb{R};H)$ yielding the well-
posedness of (3).

###### Proof.

It is clear that $\partial_{0}M_{0}(m_{0})+M_{1}(m_{0})+A-c$ is accretive as
it is the sum of two accretive operators. In order to show that its closure is
m-accretive, it suffices to show that its adjoint is also accretive. For this,
we compute its adjoint with the help of Theorem 3.1. We set
$S\coloneqq\partial_{0}M_{0}(m_{0})+M_{1}(m_{0})$ and $T\coloneqq A$. Then
$C_{c}^{\infty}(\mathbb{R};\operatorname{dom}(A))\subseteq\operatorname{dom}(S)\cap\operatorname{dom}(T)$
is dense in $L_{2,\rho}(\mathbb{R};H)$ and it is even a core for $T.$ Setting
$L_{\varepsilon}\coloneqq(1+\varepsilon\partial_{0})^{-1}$, we obtain (1) with
$R_{\varepsilon}=L_{\varepsilon},$ $\widetilde{K}_{\varepsilon}=0$ and

$K_{\varepsilon}=\varepsilon\partial_{0}(1+\varepsilon\partial_{0})^{-1}M_{0}^{\prime}(m_{0})(1+\varepsilon\partial_{0})^{-1}.$

Finally,

$SR_{\varepsilon}=\partial_{0}M_{0}(m_{0})\left(1+\varepsilon\partial_{0}\right)^{-1}=(1+\varepsilon\partial_{0})^{-1}\partial_{0}M_{0}(m_{0})+K_{\varepsilon}$

is bounded, and hence, (2) holds by Remark 3.2. Thus, we can apply Theorem 3.1
and obtain

$\left(\partial_{0}M_{0}(m_{0})+M_{1}(m_{0})+A\right)^{\ast}=\overline{\left(\partial_{0}M_{0}(m_{0})+M_{1}(m_{0})\right)^{\ast}+A^{\ast}}.$

Since clearly $A^{\ast}$ is accretive, it remains to prove the strict
accretivity of
$\left(\partial_{0}M_{0}(m_{0})+M_{1}(m_{0})\right)^{\ast}=\left(\partial_{0}M_{0}(m_{0})\right)^{\ast}+M_{1}(m_{0})^{\ast}$.
In order to compute the first adjoint we recall that
$\operatorname{dom}(\partial_{0})$ is a core for $\partial_{0}M_{0}(m_{0})$
and hence

$\left(\partial_{0}M_{0}(m_{0})\right)^{\ast}=\left(M_{0}(m_{0})\partial_{0}+M_{0}^{\prime}(m_{0})\right)^{\ast}=\partial_{0}^{\ast}M_{0}(m_{0})+M_{0}^{\prime}(m_{0}).$

Now, as in Lemma 4.2 one proves that (4) yields the accretivity of
$\partial_{0}^{\ast}M_{0}(m_{0})+M_{0}^{\prime}(m_{0})+M_{1}(m_{0})-c.$ ∎

### Commutator with $A$

Here, we assume a commutator condition with $A$. To keep things simple, we
assume that there exists $d>0$ such that $M_{0}(t)\geq d$ for almost every
$t\in\mathbb{R}$ and that

$M_{0}(m_{0})A\subseteq AM_{0}(m_{0}).$ (5)

######  Lemma 4.4.

Under the standing assumptions together with (5), we have

$M_{0}(m_{0})A=AM_{0}(m_{0}).$

###### Proof.

The inclusion $M_{0}(m_{0})A\subseteq AM_{0}(m_{0})$ leads to

$M_{0}(m_{0})(A+1)\subseteq(A+1)M_{0}(m_{0}).$

Now, the right-hand side operator is one-to-one and the left-hand side is
onto. Hence,

$M_{0}(m_{0})(A+1)=(A+1)M_{0}(m_{0}),$

which yields the assertion∎

######  Remark 4.5.

It is a consequence of the definition of the square root (see also [8,
Theoerem B.8.2 and its proof]) that

$M_{0}(m_{0})A\subseteq AM_{0}(m_{0})$

leads to

$M_{0}(m_{0})^{1/2}A\subseteq AM_{0}(m_{0})^{1/2};$

thus, by Lemma 4.4, it follows

$M_{0}(m_{0})^{1/2}A=AM_{0}(m_{0})^{1/2}.$

In particular, we obtain

$M_{0}(m_{0})^{-1/2}A=AM_{0}(m_{0})^{-1/2}.$

For motivating the main result of this section, we consider the following
evolutionary equation

$\left(\partial_{0}M_{0}(m_{0})+M_{1}(m_{0})+A\right)U=F.$

After multiplication by $M_{0}(m_{0})^{1/2}$ the latter can be rewritten

$\displaystyle\left(M_{0}(m_{0})^{1/2}\partial_{0}M_{0}(m_{0})^{1/2}+M_{0}(m_{0})^{1/2}M_{1}(m_{0})M_{0}(m_{0})^{-1/2}+M_{0}(m_{0})^{1/2}AM_{0}(m_{0})^{-1/2}\right)M_{0}(m_{0})^{1/2}U$
$\displaystyle=M_{0}(m_{0})^{1/2}F.$

Using the latter remark and (5), we get

$\left(M_{0}(m_{0})^{1/2}\partial_{0}M_{0}(m_{0})^{1/2}+M_{0}(m_{0})^{1/2}M_{1}(m_{0})M_{0}(m_{0})^{-1/2}+A\right)M_{0}(m_{0})^{1/2}U=M_{0}(m_{0})^{1/2}F.$

######  Remark 4.6.

If instead of the above equation, we consider

$\left(M_{0}(m_{0})\partial_{0}+M_{1}(m_{0})+A\right)U=F,$

we may multiply by $M_{0}(m_{0})^{-1/2}$ instead and obtain

$\displaystyle\left(M_{0}(m_{0})^{1/2}\partial_{0}M_{0}(m_{0})^{1/2}+M_{0}(m_{0})^{-1/2}M_{1}(m_{0})M_{0}(m_{0})^{1/2}+M_{0}(m_{0})^{-1/2}AM_{0}(m_{0})^{1/2}\right)M_{0}(m_{0})^{-1/2}U$
$\displaystyle=M_{0}(m_{0})^{-1/2}F,$

which in turn leads to

$\left(M_{0}(m_{0})^{1/2}\partial_{0}M_{0}(m_{0})^{1/2}+M_{0}(m_{0})^{-1/2}M_{1}(m_{0})M_{0}(m_{0})^{1/2}+A\right)M_{0}(m_{0})^{-1/2}U=M_{0}(m_{0})^{-1/2}F,$

being basically of the same shape of equation as the one above with
$\partial_{0}M_{0}(m_{0})$.

The main result of this section is the following.

######  Theorem 4.7.

Under the standing assumptions,

$\displaystyle\left(M_{0}(m_{0})^{1/2}\partial_{0}M_{0}(m_{0})^{1/2}+M_{0}(m_{0})^{1/2}M_{1}(m_{0})M_{0}(m_{0})^{-1/2}+A\right)^{*}$
$\displaystyle=\overline{\left(M_{0}(m_{0})^{1/2}\partial_{0}^{*}M_{0}(m_{0})^{1/2}+M_{0}(m_{0})^{-1/2}M_{1}(m_{0})^{*}M_{0}(m_{0})^{-1/2}+A^{*}\right)}.$

###### Proof.

First of all note that $M_{0}(m_{0})^{1/2}M_{1}(m_{0})M_{0}(m_{0})^{-1/2}$ is
a bounded linear operator and can, thus, be assumed to be $0$ when computing
the adjoint. Next,

$\left(M_{0}(m_{0})^{1/2}\partial_{0}M_{0}(m_{0})^{1/2}\right)^{*}=M_{0}(m_{0})^{1/2}\partial_{0}^{*}M_{0}(m_{0})^{1/2}$

by Theorem 2.4 applied to $T=\partial_{0}$ and $B=M_{0}(m_{0})^{1/2}=B^{*}$.
In particular, note that it particularly follows that

$\left(M_{0}(m_{0})^{1/2}\partial_{0}M_{0}(m_{0})^{1/2}\right)$

is m-accretive. Thus, for proving the present theorem, it suffices to apply
Theorem 3.3 to $T=M_{0}(m_{0})^{1/2}\partial_{0}M_{0}(m_{0})^{1/2}$ and $S=A$.
What remains is to show the commutativity of the resolvents:

$\displaystyle\left(1+T\right)^{-1}\left(1+S\right)^{-1}$
$\displaystyle=(1+M_{0}(m_{0})^{1/2}\partial_{0}M_{0}(m_{0})^{1/2})^{-1}(1+A)^{-1}$
$\displaystyle=(M_{0}(m_{0})^{1/2}(M_{0}(m_{0})^{-1}+\partial_{0})M_{0}(m_{0})^{1/2})^{-1}(1+A)^{-1}$
$\displaystyle=M_{0}(m_{0})^{-1/2}(M_{0}(m_{0})^{-1}+\partial_{0})^{-1}M_{0}(m_{0})^{-1/2}(1+A)^{-1}$
$\displaystyle=M_{0}(m_{0})^{-1/2}(M_{0}(m_{0})^{-1}+\partial_{0})^{-1}(1+A)^{-1}M_{0}(m_{0})^{-1/2}$
$\displaystyle=M_{0}(m_{0})^{-1/2}\left((1+A)(M_{0}(m_{0})^{-1}+\partial_{0})\right)^{-1}M_{0}(m_{0})^{-1/2}$
$\displaystyle=M_{0}(m_{0})^{-1/2}\left((M_{0}(m_{0})^{-1}+\partial_{0})(1+A)\right)^{-1}M_{0}(m_{0})^{-1/2}$
$\displaystyle=\left(1+S\right)^{-1}\left(1+T\right)^{-1},$

where we used $\left(1+A\right)\partial_{0}=\partial_{0}(1+A)$ and Lemma 4.4
for $M_{0}(m_{0})^{-1}(1+A)=(1+A)M_{0}(m_{0})^{-1}$. Hence, Theorem 3.3 is
applicable and the assertion follows. ∎

######  Lemma 4.8.

For each $c>0$ there exists $\rho_{0}>0$ such that for each $\rho\geq\rho_{0}$
the operator

$M_{0}(m_{0})^{1/2}\partial_{0}M_{0}(m_{0})^{1/2}+M_{0}(m_{0})^{1/2}M_{1}(m_{0})M_{0}(m_{0})^{-1/2}-c$

is m-accretive.

###### Proof.

For $u\in\operatorname{dom}(\partial_{0}M_{0}(m_{0})^{1/2})$ we compute

$\displaystyle\langle\left(M_{0}(m_{0})^{1/2}\partial_{0}M_{0}(m_{0})^{1/2}+M_{0}(m_{0})^{1/2}M_{1}(m_{0})M_{0}(m_{0})^{-1/2}\right)u,u\rangle$
$\displaystyle=\rho\|M_{0}(m_{0})^{1/2}u\|^{2}-\|M_{0}(m_{0})^{1/2}M_{1}(m_{0})M_{0}(m_{0})^{-1/2}\|\|u\|^{2}$
$\displaystyle\geq\left(\rho
d-\|M_{0}(m_{0})^{1/2}M_{1}(m_{0})M_{0}(m_{0})^{-1/2}\|\right)\|u\|^{2}.$

Choosing now $\rho$ large enough, we infer the strict accretivity of the
operator. Since its adjoint is of the form

$M_{0}(m_{0})^{1/2}\partial_{0}^{\ast}M_{0}(m_{0})^{1/2}+M_{0}(m_{0})^{-1/2}M_{1}(m_{0})^{\ast}M_{0}(m_{0})^{1/2}$

the same argument shows that for $\rho$ large enough, this operator is also
accretive, and hence, the assertion follows. ∎

######  Corollary 4.9.

Under the standing assumptions, the operator

$\overline{\partial_{0}M_{0}(m_{0})+M_{1}(m_{0})+A}$

is boundedly invertible in $L_{2,\rho}(\mathbb{R};H)$ for large enough $\rho.$

###### Proof.

In the present situation, consider

$\widetilde{T}\coloneqq\overline{T+S},$

where $T\coloneqq
M_{0}(m_{0})^{1/2}\partial_{0}M_{0}(m_{0})^{1/2}+M_{0}(m_{0})^{1/2}M_{1}(m_{0})M_{0}(m_{0})^{-1/2}$
and $S\coloneqq A$. We will show that $\widetilde{T}-c$ is m-accretive for
some $c>0$. By assumption and Lemma 4.8, it is not difficult to see that
$\widetilde{T}-c$ is accretive for some $c>0$ and all large enough $\rho>0$.
Using the formula for the adjoint in Theorem 4.7 and taking into account the
accretivtiy of $T^{*}-c$, we have that $\left(\widetilde{T}\right)^{*}-c$ is,
too, accretive. Hence, $0\in\rho(\widetilde{T})$. The reformulation just
before Remark 4.6 yields the assertion by multiplying $\widetilde{T}$ by the
topological isomorphism $M_{0}(m_{0})^{-1/2}$ from the left and
$M_{0}(m_{0})^{1/2}$ from the right. ∎

######  Remark 4.10.

A similar result holds under the assumption that

$M_{0}(m_{0})^{1/2}\partial_{0}M_{0}(m_{0})^{1/2}+M_{0}(m_{0})^{-1/2}M_{1}(m_{0})M_{0}(m_{0})^{1/2}-c$

is m-accretive for some $c>0$. Then

$0\in\rho\left(\overline{M_{0}(m_{0})\partial_{0}+M_{1}(m_{0})+A}\right).$

######  Example 4.11.

Let $d\in\mathbb{N}$ and consider $H\coloneqq L_{2}(0,1)^{d}.$ Moreover, let
$B\in\mathbb{R}^{d\times d}$ with $\|B\|\leq 1$ and consider the operator $A$
given by

$\displaystyle\operatorname{dom}(A)$ $\displaystyle\coloneqq\\{u\in
H^{1}(0,1)^{d}\,;\,u(0)=Bu(1)\\},$ $\displaystyle Au$ $\displaystyle\coloneqq
u^{\prime}.$

Then $A$ is m-accretive by [11, Theorem 4.1]. Moreover, let
$c_{1},\ldots,c_{d}\in L_{\infty}(\mathbb{R})$ such that
$c_{1},\ldots,c_{d}\geq k>0$ almost everywhere. We set
$M_{0}(t)\coloneqq\operatorname{diag}(c_{j}(t))$ and assume that
$BM_{0}(t)=M_{0}(t)B$ for almost every $t\in\mathbb{R}.$ Then we clearly have
for $u\in L_{2,\rho}(\mathbb{R};\operatorname{dom}(A))$ that $M_{0}(m_{0})u\in
L_{2,\rho}(\mathbb{R};H^{1}(0,1)^{d})$ and that

$BM_{0}(t)u(1)=M_{0}(t)Bu(1)=M_{0}(t)u(0);$

that is, $M_{0}(m_{0})u\in L_{2,\rho}(\mathbb{R};\operatorname{dom}(A)).$
Moreover,

$AM_{0}(m_{0})u=M_{0}(m_{0})u^{\prime}=M_{0}(m_{0})Au,$

which shows $M_{0}(m_{0})A\subseteq AM_{0}(m_{0}).$ Hence, by our findings
above, the non-autonomous problem

$\left(\partial_{0}M_{0}(m_{0})+A\right)u=f$

is well-posed in $L_{2,\rho}(\mathbb{R};L_{2}(0,1)^{d})$ if we choose $\rho$
large enough.

##  5 Conclusion

We provided applicable conditions for m-accretive operators so that the
adjoint of the sum can be represented as the closure of the sum of adjoints of
the individual operators. We applied this observation to evolutionary
equations and developed well-posedness criteria for the same. The case of the
operator $M_{0}$ being $L_{\infty}$-in time only and having non-trivial
(possibly time-independent) kernel remains open for easily applicable
conditions establishing well-posedness. Thus, to properly address the general
situation of time-dependent partial differential algebraic equations appears
to be a challenge for future research.

## References

  * [1] C. Budde and M. K. Fijavž.  Well-posedness of non-autonomous transport equation on metric graphs, 2023\.  arXiv:2206.02540.
  * [2] G. da Prato and P. Grisvard.  Sommes d’opérateurs linéaires et équations différentielles opérationnelles.  J. Math. Pures Appl. (9), 54:305–387, 1975.
  * [3] K.-J. Engel and R. Nagel.  One-parameter semigroups for linear evolution equations, volume 194 of Grad. Texts Math. Berlin: Springer, 2000.
  * [4] S. Hu and N. S. Papageorgiou.  Handbook of multivalued analysis. Volume I: Theory, volume 419 of Math. Appl., Dordr. Dordrecht: Kluwer Academic Publishers, 1997.
  * [5] M. H. Mortad.  On the adjoint and the closure of the sum of two unbounded operators.  Can. Math. Bull., 54(3):498–505, 2011.  doi:10.4153/CMB-2011-041-7.
  * [6] G. Nickel.  Evolution semigroups for nonautonomous Cauchy problems.  Abstr. Appl. Anal., 2(1-2):73–95, 1997.
  * [7] R. Picard.  A structural observation for linear material laws in classical mathematical physics.  Math. Methods Appl. Sci., 32(14):1768–1803, 2009.
  * [8] R. Picard, D. McGhee, S. Trostorff, and M. Waurick.  A primer for a secret shortcut to PDEs of mathematical physics.  Front. Math. Cham: Springer, 2020.
  * [9] R. Picard and S. Trostorff.  Dynamic first order wave systems with drift term on Riemannian manifolds.  J. Math. Anal. Appl., 505(1):30, 2022.
  * [10] R. Picard, S. Trostorff, M. Waurick, and M. Wehowski.  On non-autonomous evolutionary problems.  J. Evol. Equ., 13(4):751–776, 2013.
  * [11] R. H. Picard, S. Trostorff, B. Watson, and M. Waurick.  A structural observation on port-Hamiltonian systems.  SIAM J. Control Optim., 61(2):511–535, 2023.
  * [12] C. Seifert, S. Trostorff, and M. Waurick.  Evolutionary equations. Picard’s theorem for partial differential equations, and applications, volume 287 of Oper. Theory: Adv. Appl. Cham: Birkhäuser, 2022.


# DataGarden: Exploring Our Community in a VR Data Visualization

Joy Kondo  Boston CollegeChestnut HillUSA<EMAIL_ADDRESS>,  Justin Park
Boston CollegeChestnut HillUSA<EMAIL_ADDRESS>,  Josiah Kondo  Boston
CollegeChestnut HillUSA<EMAIL_ADDRESS>and  Nam Wook Kim  Boston
CollegeChestnut HillUSA<EMAIL_ADDRESS>
(2023)

###### Abstract.

As our society is becoming increasingly data-dependent, more and more people
rely on charts and graphs to understand and communicate complex data. While
such visualizations effectively reveal meaningful trends, they unavoidably
aggregate data into points and bars that are overly simplified depictions of
ourselves and our communities. We present DataGarden, a system that supports
embodied interactions with humane data representations in an immersive VR
environment. Through the system, we explore ways to rethink the traditional
visualization approach and allow people to empathize more deeply with the
people behind the data.

data visualization, immersion, virtual environment, community data, humane
representation

††copyright: acmcopyright††journalyear: 2023††doi:
XXXXXXX.XXXXXXX††conference: Make sure to enter the correct conference title
from your rights confirmation email; April 23–28, 2023; Hamburg,
Germany††booktitle: Hamburg ’23: ACM XXX, April 23–28, 2023, Hamburg,
Germany††price: 0.00††isbn: 978-1-4503-XXXX-X/18/06††ccs: Human-centered
computing Interactive systems and tools††ccs: Human-centered computing
Visualization systems and tools

Figure 1. An overview of the DataGarden virtual environment. Each flower or
tree represents an individual person, and the various characteristics
including the flower color and number of clouds are determined by users’
survey responses.

Wide view showing the DataGarden virtual environment. A focused view of a
tree. A person using a Meta Quest device to look up the tree.

##  1\. Introduction

Traditional approaches to data visualization have commonly relied on abstract
visualizations, such as bars, lines, and pies, to objectively represent data
in a minimalist manner. While these methods have demonstrated perceptual
effectiveness in conveying statistical information, such abstract
representations tend to overlook the human aspect underlying the data (Boy et
al., 2017). Moreover, conventional design guidelines such as avoidance of
visual clutter have reduced the stories and emotions represented by data
visualizations to mere graphical marks (Hullman et al., 2011). However, given
that the data we gather from individuals’ lived experiences offers a glimpse
into our world and the people it comprises, it is crucial that data
visualizations elicit an emotional connection with the individuals represented
by the data (Giorgia, 2016).

In this paper, we present DataGarden, a novel data visualization system
leveraging immersiveness and humanistic representations of data to help people
reflect on the people behind the data. Our system is based on the empirical
foundations demonstrating the benefits of immersion and pictorial
representations (Borkin et al., 2016; Romat et al., 2020). Designers can
collect qualitative questionnaire data and construct mappings from the data to
virtual objects and their visual properties in DataGarden. Users can then use
VR devices like Meta Quest to interact with the data in an immersive
environment. The current system supports garden objects such as trees and
flowers, but the visual vocabulary can be expanded and customized to
incorporate additional visual metaphors. The closest work to ours is
DataSelfie (Kim et al., 2019), but we extend the idea to a VR environment and
community data rather than 2D and individual responses. Below, we describe our
design goals and system components in detail.

Figure 2. DataGarden’s data visualization pipeline. Left: personal and
community data is collected via a questionnaire. Middle: responses are mapped
to unique visual objects. Right: legends and interactive groupings facilitate
data interpretation.

##  2\. DataGarden Design and System

###  2.1. Design Goals

####  2.1.1. Capturing human elements in data

: we want to collect data that contains rich human elements (e.g., bullet
journaling (Ayobi et al., 2018)). We are not interested in purely quantitative
data, such as economic indicators or numerical measurements, as they might
work better with standard charts and graphs (Kim et al., 2019). Instead, we
aim to use a questionnaire to collect individual responses. Each data point is
personal, and the collection of data points can represent the community.

####  2.1.2. Creating humanistic representations of data

To elicit an emotional response from users, we intend to create humanistic
data visualizations. Instead of using traditional visual marks, we resort to
using metaphors and rhetoric in our data visualizations. Such visual
representations were used in personal data visualizations (Lupi and Posavec,
2016; Kim et al., 2019) and were also shown to promote better engagement and
memorability of the data (Borkin et al., 2016; Morais et al., 2022) without
necessarily hindering comprehension (Bateman et al., 2010).

####  2.1.3. Utilizing immersive embodied interactions

We take one step further to harness the immersive potential of VR
environments. They can provide a more visceral data experience by enabling
people to use familiar senses, such as sight, sound, and touch, in a real-
world-like setting (Donalek et al., 2014; Romat et al., 2020; Ivanov et al.,
2018). Such a heightened sense of immersion can not only enhance the overall
enjoyment and engagement but also foster a holistic understanding of the
underlying data (Dwyer et al., 2018).

###  2.2. System Components: Mapping & Interaction

The DataGarden system consists of three main components.

####  2.2.1. Data Collection

Qualitative community data is collected using a questionnaire. For the
application in this paper, we surveyed students and faculty at a college in
the U.S. by asking questions about their backgrounds, personalities (e.g.,
MBTIs), views of the world (e.g., gloomy and bright), and other opinions and
experiences. The responses are collected in a spreadsheet to be fed to the
visual mapping module.

####  2.2.2. Visual Mapping

Answers to multiple questions in each response are mapped to visual objects
and their properties in the virtual environment. For instance, using the
college survey, students were visualized as flowers and faculty as trees.
Their plastic usage levels are encoded using the number of clouds. Moreover,
their MBTI personality type determines the color of one’s primary object; the
variety of colors in the environment would represent the diversity of
personalities in the community. Configuring a new visual mapping can be easily
done by supplying new models and specifying mapping parameters in the system.

####  2.2.3. Interaction & Exploration

Users can freely walk or fly around the environment using a VR device, and
switch viewing modes from ground-level to aerial bird’s-eye to top-down views.
They can open up a visual legend to better understand the visual mapping. In
addition, they can perform interactive groupings to facilitate interactive
exploration, e.g., organic organization of objects into an intuitive axis-
aligned layout of the objects. Finally, a tooltip appears upon touching an
object to explain the underlying data. Through this interactive support,
DataGarden not only gains the benefit of immersive visuals but also retains
comprehension of the data.

##  3\. Preliminary User Study

We conducted a user study with five participants to evaluate DataGarden. They
were asked to perform five tasks using the system, including counting and
comparing elements, analyzing relationships, and summarizing trends. After the
tasks, they filled out a System Usability Scale survey and engaged in a short
interview.

All participants successfully completed the tasks. The overall usability score
was 66.5 on a scale from 0 to 100. Overall, participants enjoyed the
experience and were engaged throughout. One participant (P2) was particularly
engaged by immersion and said, “It’s the feeling of being in a game”.
Participants also mentioned that visual aesthetics and physical movements made
the experience more positive and meaningful. Some participants also noted that
the system could be used in different contexts, such as a social impact
application, by humanizing data to encourage empathy and understanding.

However, they often felt overwhelmed by the amount of information presented
and had difficulty keeping track of it, which indicates a design trade-off of
DataGarden compared to traditional visualizations. Some participants expressed
difficulty in navigation, as well as user controls such as the legend and
menu. We observed that people with prior experience in gaming and VR figured
out the navigation and interface controls quicker than the others. Based on
this feedback, we improved DataGarden, whose latest version is described in
the previous system description.

##  4\. Conclusion and Future Work

In this work, we demonstrated a humanistic and immersive approach to data
visualization via DataGarden. For future work, we plan to further improve
DataGarden’s interpretability and scalability through enhanced user
interactions (e.g., multi-dimensional filtering) and visual mappings (e.g.,
adding sound as a variable). We also plan to improve extensibility by making
the visual vocabulary fully configurable for different application contexts.

## References

  * (1)
  * Ayobi et al. (2018) Amid Ayobi, Tobias Sonne, Paul Marshall, and Anna L. Cox. 2018\.  Flexible and Mindful Self-Tracking: Design Implications from Paper Bullet Journals. In _Proceedings of the 2018 CHI Conference on Human Factors in Computing Systems_ (Montreal QC, Canada) _(CHI ’18)_. Association for Computing Machinery, New York, NY, USA, 1–14.  https://doi.org/10.1145/3173574.3173602
  * Bateman et al. (2010) Scott Bateman, Regan L. Mandryk, Carl Gutwin, Aaron Genest, David McDine, and Christopher Brooks. 2010\.  Useful Junk? The Effects of Visual Embellishment on Comprehension and Memorability of Charts. In _Proceedings of the SIGCHI Conference on Human Factors in Computing Systems_ (Atlanta, Georgia, USA) _(CHI ’10)_. Association for Computing Machinery, New York, NY, USA, 2573–2582.  https://doi.org/10.1145/1753326.1753716
  * Borkin et al. (2016) M. A. Borkin, Z. Bylinskii, N. Kim, C. Bainbridge, C. S. Yeh, D. Borkin, H. Pfister, and A. Oliva. 2016.  Beyond Memorability: Visualization Recognition and Recall.  _IEEE Transactions on Visualization and Computer Graphics_ 22, 01 (jan 2016), 519–528.  https://doi.org/10.1109/TVCG.2015.2467732
  * Boy et al. (2017) Jeremy Boy, Anshul Vikram Pandey, John Emerson, Margaret Satterthwaite, Oded Nov, and Enrico Bertini. 2017.  Showing People Behind Data: Does Anthropomorphizing Visualizations Elicit More Empathy for Human Rights Data?. In _Proceedings of the 2017 CHI Conference on Human Factors in Computing Systems_ (Denver, Colorado, USA) _(CHI ’17)_. Association for Computing Machinery, New York, NY, USA, 5462–5474.  https://doi.org/10.1145/3025453.3025512
  * Donalek et al. (2014) Ciro Donalek, S. G. Djorgovski, Alex Cioc, Anwell Wang, Jerry Zhang, Elizabeth Lawler, Stacy Yeh, Ashish Mahabal, Matthew Graham, Andrew Drake, Scott Davidoff, Jeffrey S. Norris, and Giuseppe Longo. 2014.  Immersive and collaborative data visualization using virtual reality platforms. In _2014 IEEE International Conference on Big Data (Big Data)_. 609–614.  https://doi.org/10.1109/BigData.2014.7004282
  * Dwyer et al. (2018) Tim Dwyer, Kim Marriott, Tobias Isenberg, Karsten Klein, Nathalie Riche, Falk Schreiber, Wolfgang Stuerzlinger, and Bruce H Thomas. 2018.  Immersive analytics: An introduction.  _Immersive analytics_ (2018), 1–23.
  * Giorgia (2016) Lupi Giorgia. 2016\.  Data Humanism, the Revolution will be Visualized.  _Print Magazine_ 3 (2016), 76–83.
  * Hullman et al. (2011) Jessica Hullman, Eytan Adar, and Priti Shah. 2011.  Benefitting InfoVis with Visual Difficulties.  _IEEE Transactions on Visualization and Computer Graphics_ 17, 12 (2011), 2213–2222.  https://doi.org/10.1109/TVCG.2011.175
  * Ivanov et al. (2018) Alexander Ivanov, Kurtis Thorvald Danyluk, and Wesley Willett. 2018.  Exploration & Anthropomorphism in Immersive Unit Visualizations. In _Extended Abstracts of the 2018 CHI Conference on Human Factors in Computing Systems_ (Montreal QC, Canada) _(CHI EA ’18)_. Association for Computing Machinery, New York, NY, USA, 1–6.  https://doi.org/10.1145/3170427.3188544
  * Kim et al. (2019) Nam Wook Kim, Hyejin Im, Nathalie Henry Riche, Alicia Wang, Krzysztof Gajos, and Hanspeter Pfister. 2019.  DataSelfie: Empowering People to Design Personalized Visuals to Represent Their Data. In _Proceedings of the 2019 CHI Conference on Human Factors in Computing Systems_ (Glasgow, Scotland Uk) _(CHI ’19)_. Association for Computing Machinery, New York, NY, USA, 1–12.  https://doi.org/10.1145/3290605.3300309
  * Lupi and Posavec (2016) Giorgia Lupi and Stefanie Posavec. 2016.  _Dear data_.  Chronicle books.
  * Morais et al. (2022) Luiz Morais, Yvonne Jansen, Nazareno Andrade, and Pierre Dragicevic. 2022.  Showing Data About People: A Design Space of Anthropographics.  _IEEE Transactions on Visualization and Computer Graphics_ 28, 3 (2022), 1661–1679.  https://doi.org/10.1109/TVCG.2020.3023013
  * Romat et al. (2020) Hugo Romat, Nathalie Henry Riche, Christophe Hurter, Steven Drucker, Fereshteh Amini, and Ken Hinckley. 2020.  Dear Pictograph: Investigating the Role of Personalization and Immersion for Consuming and Enjoying Visualizations. In _Proceedings of the 2020 CHI Conference on Human Factors in Computing Systems_ (Honolulu, HI, USA) _(CHI ’20)_. Association for Computing Machinery, New York, NY, USA, 1–13.  https://doi.org/10.1145/3313831.3376348