# Numerical Reconstruction of the kinetic chemotaxis kernel from macroscopic
measurement, wellposedness and illposedness

Kathrin Hellmuth
Department of Mathematics
University of Würzburg
97074 Würzburg, Germany
<EMAIL_ADDRESS>
& Christian Klingenberg
Department of Mathematics
University of Würzburg
97074 Würzburg, Germany
<EMAIL_ADDRESS>
& Qin Li
Department of Mathematics
University of Wisconsin-Madison
Madison, WI, 53706, USA
<EMAIL_ADDRESS>
& Min Tang
School of Mathematics and Institute of Natural Sciences, MOE-LSC
Shanghai Jiaotong University
Shanghai, 200240, China
<EMAIL_ADDRESS>
###### Abstract

Directed bacterial motion due to external stimuli (chemotaxis) can, on the
mesoscopic phase space, be described by a velocity change parameter $K$. The
numerical reconstruction for $K$ from experimental data provides useful
insights and plays a crucial role in model fitting, verification and
prediction. In this article, the PDE-constrained optimization framework is
deployed to perform the reconstruction of $K$ from velocity-averaged,
localized data taken in the interior of a 1D domain. Depending on the data
preparation and experimental setup, this problem can either be well- or ill-
posed. We analyze these situations, and propose a very specific design that
guarantees local convergence. The design is adapted to the discretization of
$K$ and decouples the reconstruction of local values into smaller cell
problem, opening up opportunities for parallelization. We further provide
numerical evidence as a showcase for the theoretical results.

_Keywords_ inverse problems in PDEs; numerical methods for inverse porblems
with PDEs; PDE-constrained optimization; kinetic chemotaxis equation;
numerical analysis; well- and ill-posedness; mathematical biology

### Funding

K.H. acknowledges support by the German Academic Scholarship Foundation
(Studienstiftung des deutschen Volkes) and the Marianne-Plehn-Program..
Q.L. is partially supported by Vice Chancellor for Research and Graduate
Education, DMS-2308440 and ONR-N000142112140.
M.T. is partially supported by the Strategic Priority Research Program of
Chinese Academy of Sciences, XDA25010401 and NSFC12031013.

##  1 Introduction

Kinetic chemotaxis equation is one of the classical equations describing the
collective behavior of bacteria motion. Presented on the phase space, the
equation describes the “run-and-tumble" bacteria motion. The solution
$f(t,x,v)$ represents the density of bacteria at any given time $t$ for any
location $x$ moving with velocity $v$. Since it contains more detailed phase-
space information, compared to macroscopic models at the population level,
such as the Keller Segel model, the equation has the greater potential to
capture the fine motion of the bacteria. Indeed, it is observed that the
dynamics predicted by the model is in high agreement with real measurements,
see Berg (1993); Emako et al. (2016); Saragosti et al. (2011, 2010).

It is noteworthy that these comparisons are conducted in the forward-
simulation setting. Guesses are made about parameters, and simulations are run
to be compared with experimental measurements. To fully reveal the bacteria’s
motion and its interaction with the environment, inverse perspectives have to
be taken. The measurement data can be at the individual or the population
level, i.e., biophysicists can use a high-resolution camera and trace each
single bacterium for a long time or they can take photos and record the
evolution of the density of bacteria on a cell cultural dish. These collected
data should be used to unveil the true interaction between particles Li et al.
(2019). This framework necessitates the application of numerical inversion
algorithms. To be specific, we frame this problem into a PDE-constrained
optimization and study the well-posedness and the ill-posedness of the
numerical reconstruction when different types of initial condition and
measurement schemes are provided.

As more first-principle based physics get involved in applications, kinetic
models are becoming more important in scientific domains, see modeling of
neutrons Davison and Sykes (1958), photons or electrons Rybicki and Lightman
(1986) and rarefied gas Cercignani (2012). The applications on biological and
social science have also been put forward in Othmer et al. (1988) for cell
motion, in Taylor-King et al. (2014) for animal (birds) migration or in Albi
et al. (2023); Carrillo et al. (2009); Chu et al. (2022); Motsch and Tadmor
(2014); Toscani (2006) for opinion formation. In most, if not all of these
models, parameters are included to characterize the interactions between
agents or with the media. The applications in which the interactions are hard
to be measured experimentally naturally prompts the use of inverse solvers.

The most prominent application of inverse problem confined to the domain of
kinetic-equation governed systems is optical tomography from medical imaging,
where non-intrusive boundary data maps out the relation between optical
properties of interior bio-tissue and the measured light intensity on the
surface of the domain. Mathematically the problem is framed to evaluate the
richness of data in the albedo operator. Singular decomposition is deployed as
a specific mathematical technique to conduct such investigation Bal et al.
(2008); Choulli and Stefanov (1996); Lai et al. (2019); Li and Sun (2020), and
these studies have their numerical counterparts in Arridge and Schotland
(2009); Chen et al. (2018); Egger and Schlottbom (2013); Prieto and Dorn
(2016); Ren (2010), just to mention a few references.

Since tracing every single bacterium is much more difficult than measuring the
density evolution and is sometimes not possible in some extreme environments,
one natural question is whether it is possible to unveil how the bacteria
interact with the environment by the measurement at the population level. Due
to the specific biological question at hand, the biggest difference between
our problem setup and the previous ones is the fact that our measurements are
taken in the interior of the domain, but are macroscopic. The kind of data
preparation is intrusive in the sense that photos are taken over the entire
cultural dish but not only on the boundary (domain surface), so it enriches
the available dataset. While optical tomography equipment can read off the
velocity information, the photos usually only provide density information,
except for very special cases Jeckel et al. (2019); Zhang et al. (2010) that
have very high requirements on the lab equipements. Since the measurement is
macroscopic, this reduces the richness of data.

In Hellmuth et al. (2022) the authors examined the theoretical aspect of this
reconstruction problem with macroscopic interior data. It was shown that
trading off the microscopic information for the interior data still gives us
sufficient information to recover the transition kernel, but the experiments
need to be accordingly designed. However, in the theoretical paper we assumed
that the transition kernel is an unknown function, and thus an infinitely
dimensional object, and the available data is the full map (from initial
condition to density for all time and space), and thus an infinite dimensional
object as well. This infinite-to-infinite setup is hard to be implemented
numerically, so the theoretical results only provide a guidance but not a
direct guarantee. The current paper can be seen as the numerical counterpart
of Hellmuth et al. (2022). In particular, we study, on the discrete level, if
measurement data are finite in size, and the to-be-reconstructed transition
kernel is also represented by a finite dimensional vector, can one still
successfully recover the unknowns.

It turns out that the numerical issue is significantly more convoluted. When
the dimension of $K$, the transition kernel, is changed from infinite to
finite, we expect the amount of data needed to recover this finite-dimentional
parameter should also be reduced. However, by how much and in what way is far
from being clear. We will present below two different scenarios to argue that
when data is prepared well, a stable reconstruction is expected, but when the
data “degenerates," it loses information for a full recovery. Such well-
posedness and ill-posedness are separately presented in two subsections of
Section 3. Then in Section 5 we present the numerical evidence to showcase the
theoretical prediction.

It should be noted that it is well within anticipation that different data
preparation gives different conditioning for parameter reconstruction. This
further prompts the study of experimental design. In the context of
reconstructing the transition kernel in the chemotaxis equation, in Section 4
we will design a particular experimental setup that guarantees a unique
reconstruction.

We should further note that reconstructing parameters for bacterial motion
using the inversion perspective is not entirely new. In literature, there
exist two different approaches: the first involves the utilization of
statistical information at the individual level to extrapolate the microscopic
transition kernel, whereas the second entails employing density data at a
macroscopic scale to reconstruct certain parameters associated with a
parametrized model through an optimization framework Ford and Lauffenburger
(1991); Giometto et al. (2015); Salek et al. (2019); Tranquillo et al. (1988).
To our knowledge, these available studies focus on either microscopic or
macroscopic models with a very limited number of unknowns to be recovered, and
data of the corresponding scale are used to construct model parameters of the
corresponding scale. For instance, in Pohl et al. (2017); Seyrich et al.
(2018), the tumbling behavior is inferred statistically on a microscopic
level, i.e. the tumbling, as an individual random process, is described by a
few moments of its probability distribution that are recovered from data. In
Egger et al. (2015); Fister and Mccarthy (2008), the macroscopic problem was
considered where parameterization emerged from discretization, and
regularization was used to counter the noise. Moreover, the viewpoint of
constructing the optimization problem in this article significantly differs
from the existing literature. Similar as in Egger et al. (2015); Fister and
Mccarthy (2008), we recover the discretized version of the kinetic parameter,
as this framework brings more flexibility. Our focus, however, lies on the
study of well- and ill-posedness of the optimization problem related to the
parameter reconstruction. To observe these effects, no regularization is
applied and numerical examples are presented in a noise-free setting. This
demonstrates the necessity for well-designed experimental setups, which are
adapted to the fineness of the parameter discretization.

##  2 Framing a PDE-constrained optimization problem

We frame the problem as a PDE-constrained optimization, which is to
reconstruct $K$ that fits data as much as possible conditioned on the fact
that the kinetic chemotaxis model is satisfied.

To start off, we first present the kinetic chemotaxis model. Denoting
$f(t,x,v)$ the probability density distribution of bacteria in space
$x\in\mathbb{R}^{1}$, time $t>0$ and velocity $v\in V$, the equation writes:

$\displaystyle\partial_{t}f+v\cdot\nabla_{x}f$
$\displaystyle=\mathcal{K}(f):=\int_{V}K(x,v,v^{\prime})f(x,t,v^{\prime})-K(x,v^{\prime},v)f(x,t,v)\,\text{d}v^{\prime},$
(1) $\displaystyle f(t=0,x,v)$
$\displaystyle=\phi(x,v)\in{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L^{\infty}}_{+,c}(\mathbb{R}\times
V)$ (2)

where $v\cdot\nabla_{x}f$ characterizes the “run"-part where bacteria move
straight forward with velocity $v$, and the terms on the right characterize
the “tumble"-part, with bacteria changing from having velocity $v^{\prime}$ to
$v$ using the transitional rate $K(x,v,v^{\prime})\geq 0$ and
$K(x,v,v^{\prime})$ is called the tumbling kernel. Initial data is given at
$t=0$ and is denoted by $\phi(x,v)$. We reduce the original problem for
$(x,v)\in\mathbb{R}^{3}\otimes\mathbb{S}^{2}$ to
$(x,v)\in\mathbb{R}^{1}\otimes\\{\pm 1\\}$ Giometto et al. (2015); Saragosti
et al. (2011, 2010),i.e. the bacteria either moves to the left or to the
right, and $x$ is 1D in space. This simple setting on the one hand applies to
the case when experiments are conducted in a bacteria culture tube, thus is
biologically meaningful, on the other hand, it includes the difficulties of
our setting of inversion. More details will be discussed in the subsequent
part. Moreover, in some applications, the environment changes with time then
the tumbling kernel $K$ may depend on time as well, we focus on the time-
independent case here when the outside signaling does not change.

To understand the the particle interaction with the environment, one needs to
determine $K$ and data is collected to infer it. Typically, it is unnecessary
to recover it as a function, but some fine-discretization of it would suffice.
To do so, we assume that $K$ can be well represented by a list of finite many
parameters:

$K(x,v,v^{\prime})=\sum_{r=1}^{R}K_{r}(v,v^{\prime})\mathds{1}_{I_{r}}(x)\,,$
(3)

meaning in the interval of $I_{r}=[a_{r-1},a_{r})$, $r=2,...,R-1$ (with
$a_{r-1}<a_{r}$ and $I_{1}=(-\infty,a_{1})$, $I_{R}=[a_{R-1},\infty)$),
$K(x,v,v^{\prime})$ can be well approximated by a function independent of the
spatial variable $x$. Since $V=\\{\pm 1\\}$, there are only two choices for
the velocity change encrypted by $K_{r}(v,v^{\prime})$: $K_{r}(1,-1)$ or
$K_{r}(-1,1)$, and thus there are in total $2R$ free values for $K$.
Throughout the paper we abuse the notation and denote $K\in\mathbb{R}^{2R}$ as
the unknown vector to be reconstructed. Moreover, we set:

$K_{r}=[K_{r,1},K_{r,2}]\,,\quad\text{with}\quad K_{r,1}=K_{r}(1,-1)\,\quad
K_{r,2}=K_{r}(-1,1)\,.$ (4)

The dataset is also finite in size. In particular, we mathematically represent
the local pixel reading of the photo by a test function
$\mu_{l}\in{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L^{1}}(\mathbb{R})$
for some $l$, then the data takes the form of

$M_{l}(K)=\int_{\mathbb{R}}\int_{V}f_{K}(x,T,v)\,\text{d}v\
\mu_{l}(x)\,\text{d}x,\qquad l=1,...,L\,,$ (5)

where $f_{K}$ denotes the solution to (1) with kernel $K$. Denote the ground-
truth transition kernel to be $K_{\star}$, then the true data is:

$\displaystyle y_{l}=M_{l}(K_{\star}),\qquad l=1,...,L\,.$ (6)

As discussed in Section 1, when $K$ is reduced to be represented by a finite
dimensional vector we expect the amount of data needed is also finite, but how
to do the reduction for a stable reconstruction is still unknown.
Mathematically, this amounts to studying the _intricate relation between $R$
and $L$ and $\\{\mu_{l}\\}$_.

The numerical inversion is presented as a PDE-constrained optimization. We aim
to minimize the square loss between the simulated data $M(K)$ and the data
$y$:

$\displaystyle\min_{K}\quad\mathcal{C}(K)$
$\displaystyle=\min\frac{1}{2L}\sum_{l=1}^{L}\left(M_{l}(K)-y_{l}\right)^{2}$
(7) subject to $\displaystyle\quad\text{\eqref{eq:chemotaxis}, and
\eqref{eq:chemotaxis_init}}.$

There are many algorithms that can be deployed to solve this minimization
problem, and we are particularly interested in calling the simple gradient-
descent (GD) algorithm. The update is given by:

$K^{(n+1)}=K^{(n)}-\eta_{n}\nabla_{K}\mathcal{C}(K^{(n)})\,,$ (8)

with a suitable step size $\eta_{n}\in\mathbb{R}_{+}$. It is a standard
application of calculus-of-variation, as detailed in Appendix A, to derive
that the $(r,i)$-th ($i=1,2$, $r=1,\cdots,R$) entry of the gradient
$\nabla_{K}\mathcal{C}$:

${\frac{\partial\mathcal{C}}{\partial
K_{r,i}}=\int_{0}^{T}\int_{I_{r}}f(t,x,v_{i}^{\prime})(g(t,x,v_{i}^{\prime})-g(t,x,v_{i}))\,\text{d}x\,\text{d}t\,,}$
(9)

where $(v_{i},v_{i}^{\prime})=((-1)^{i},(-1)^{i+1})$ in analogy to notation
(4) for $K$ and $g$ is the adjoint state that solves the adjoint equation

$\displaystyle-\partial_{t}g-v\cdot\nabla g$
$\displaystyle=\tilde{\mathcal{K}}(g):=\int_{V}K(x,v^{\prime},v)(g(x,t,v^{\prime})-g(x,t,v))\,\text{d}v^{\prime},$
(10) $\displaystyle g(x,t=T,v)$
$\displaystyle=-\frac{1}{L}\sum_{l=1}^{L}\mu_{l}(x)\left(M_{l}(K)-y_{l}\right).$
(11)

Notice that by definition of the measurement procedure (5), the final
condition of $g$ in (11) is independent of $v$ and contains the spatial test
functions $\mu_{l}$.

The convergence of GD in (8) is guaranteed for a suitable step size if the
objective function is convex. Denoting $H_{K}\mathcal{C}$ the Hessian function
of the loss function, we need $H_{K}\mathcal{C}>0$ at least in a small
neighborhood around $K_{\star}$. If so, a constant step size
$\eta_{n}=\eta=\frac{2\lambda_{\min}}{\lambda_{\max}^{2}}$ approximates the
step size suggested in Wright and Recht (2022) for optimal convergence. Here
$\lambda_{\min},\lambda_{\max}$ denote the smallest and largest eigenvalues of
$H_{K}\mathcal{C}(K_{\star})$. More sophisticated methods include line search
for the step size or higher order methods for the update are also possible,
see e.g. Ren (2010); Wright and Recht (2022).

To properly set up the problem, we make some general assumptions and fix some
notations.

######  Assumption 1.

We make assumptions to ensure the wellposedness of the forward problem in a
feasible set, in particular:

  * •

We will work locally in $K$, so we assume in a neighbourhood
$\mathcal{U}_{K_{\star}}$ of $K_{\star}$, there is a constant $C_{K}$ so that
for all $K\in\mathcal{U}_{K_{\star}}$:

$0<\|K\|_{\infty}\leq C_{K}\,.$ (12)

  * •

Assume the initial data $\phi$ be in the space
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L^{\infty}_{+,c}(\mathbb{R}\times
V)}$ of non negative, compactly supported functions with essential bound

$\|\phi\|_{L^{\infty}(\mathbb{R}\times V)}=:C_{\phi}\,.$

  * •

Reciprocally, we assume the test functions $\mu_{l}$, $l=1,...,L$, are in the
space
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L^{1}}(\mathbb{R})$
with uniform $L^{1}$ bound

$\displaystyle\int_{\mathbb{R}}|\mu_{l}|\,\text{d}x\leq C_{\mu},\quad
l=1,...,L\,.$

These assumptions allow us to operate $f$ and $g$ in the right spaces. In
particular, we can give an upper bound for both the forward and adjoint
solution in $L_{\infty}$ sense, see Lemma B.1 and B.2 in Appendix B. In fact,
these assumptions are in line with realistic modelling: the boundedness of the
parameter $K$ emerges from its interpretation as a probability of changing
directions. Non-negativity and boundedness of the initial bacteria density are
physical, as bacteria cannot infinitely aggregate due to volume filling
effects.

##  3 Well-posedness vs. ill-posedness

The well-posedness of the inversion heavily depends on the data preparation.
If a suitable experimental setting is arranged, the optimization problem is
expected to provide local wellposedness around the groundtruth parameter
$K_{\star}$, so the classical GD can reconstruct the groundtruth. However, if
data becomes degenerate, we also expect ill-conditioning and the GD will find
it hard to converge to the global minimum. We spell out the two scenarios in
the two theorems below.

######  Theorem 3.1.

Assume the hessian matrix of the cost function is positive definite at
$K_{\star}$ and let the remaining assumptions of Proposition 3.1 hold, then
there exists a neighbourhood $U$ of $K_{\star}$, in which the optimization
problem (7) is Tykhonov well-posed. In particular, the gradient descent
algorithm (8) with initial value $K_{0}\in U$ converges.

This theorem provides the well-posedness of the problem. To be specific, it
spells out the sufficient condition for GD to find the global minimizer
$K_{\star}$. The condition of the hessian being positive definite at
$K_{\star}$ may seem strong, but, paying attention to certain restrictions
such as the minimal of measurements number $L\geq 2R$, we can carefully craft
an experiment so to make sure it holds true. This line of study is in essence
experimental design, as we will be more specific in Section 4.

On contrary to the previous wellposedness discussion, we also provide a
negative result below on ill-conditioning.

######  Theorem 3.2.

Let $L=2R$ and let Assumption 1 hold for all considered quantities. Consider a
sequence $(\mu_{1}^{(m)})_{m}$ of test functions for the first measurement
$M_{1}(K)$ for which one of the following scenarios holds:

  1. 1.

$\mu_{1}^{(m)}\to\mu_{2}$ in $L^{1}$ as $m\to\infty$.

  2. 2.

$(\mu_{1}^{(m)})_{m}$ and $\mu_{2}$ are mollifications of singular point-
measurements in measurement points $\\{(x_{1}^{(m)})_{m},x_{2}\\}$ such that
$x_{1}^{(m)}\to x_{2}$ as $m\to\infty$. Furthermore, let the assumptions of
Proposition 3.3 hold.

Then, as $m\to\infty$, the loss function cannot be strongly convex, and the
convergence of the gradient descent algorithm (8) to $K_{\star}$ cannot be
guaranteed. In scenario 2, this holds independently of the mollification
parameter.

The two theorems, to be proved in detail in Section 3.1 and 3.2 respectively,
hold vast contrast to each other. The core of the difference between the two
theorems is the data selection, with the former guaranteeing the convexity of
the objective function, and the latter does not. To evaluate the convexity of
the loss function amounts to the study of the hessian, a $2R\times 2R$ matrix:

$H_{K}\mathcal{C}(K)=\frac{1}{L}\sum_{l=1}^{L}\left(\nabla_{K}M_{l}(K)\otimes\nabla_{K}M_{l}(K)+(M_{l}(K)-y_{l})H_{K}M_{l}(K)\right)\,.$
(13)

It is a well-known fact Polyak and Shcherbakov (2017) that a positive definite
hessian provides the strong convexity of the loss function, and is a
sufficient criterion that permits the convergence in the parameter space. If
$H_{K}\mathcal{C}(K_{\star})$ is known to be positive, given in a small
neighborhood, the hessian matrix does not change much, the convexity is
guaranteed. Such boundedness of perturbation in the hessian is spelled out in
Proposition 3.1, and Theorem 3.1 naturally follows. Theorem 3.2 is to look at
the opposite side of the problem. In particular, it examines the degeneracy
when two data collection points get very close. The degeneracy is reflected
mathematically by the deficient rank structure in the hessian (13), prompting
the collapse of the landscape of the objective function. The two scenarios of
deficient ranks are presented in Proposition 3.3 and 3.2 respectively, and
then Theorem 3.2 naturally follows.

###  3.1 Local well-posedness of the optimization problem

Generally speaking, it would not be easy to characterize the landscape of the
distribution and thus hard to prescribe conditions for obtaining global
convergence. However, suppose the data is prepared well enough that guarantees
the positive definiteness for the Hessian $H_{K}\mathcal{C}(K_{\star})$
evaluated at the groundtruth $K_{\star}$, there is a good chance that in a
small neighborhood of this groundtruth, positive-definiteness persists and GD,
if starts within this neighborhood, finds the global minimum to (7). This
gives us a local well-posedness.

This local behavior is characterized in the following proposition.

######  Proposition 3.1.

Let Assumption 1 hold. Assume the Hessian $H_{K}\mathcal{C}(K_{\star})$ is
positive definite at $K_{\star}$, and that there is a uniform bound for the
Hessian of the measurements in the neighborhood $\mathcal{U}_{K_{\star}}$ in
the sense that $\|H_{K}M_{l}(K)(v,v^{\prime})\|_{F}\leq C_{H_{K}M}$ for all
$l=1,...,L$ and $K\in\mathcal{U}_{K}$ in the Frobenius norm. Then there exists
a (bounded) neighbourhood $U\subset\mathcal{U}_{K_{\star}}$ of $K_{\star}$,
where $H_{K}\mathcal{C}(K)$ is positive definite for all $K\in U$. Moreover,
the minimal eigenvalues $\lambda_{\min}(H_{K}\mathcal{C})$ satisfies

$|\lambda_{\min}(H_{K}\mathcal{C}(K_{\star}))-\lambda_{\min}(H_{K}\mathcal{C}(K))|\leq\|K_{\star}-K\|_{\infty}C^{\prime},$
(14)

where the constant $C^{\prime}$ depends on the measurement time $T$, $R$, and
the bounds $C_{\mu}$, $C_{\phi}$, $C_{K}$ in Assumption 1 and $C_{H_{K}M}$. As
a consequence, the radius of $U$ can be chosen as
${\lambda_{\min}(H_{K}\mathcal{C}(K_{\star}))}/{C^{\prime}}.$

The proposition is hardly surprising. Essentially it suggests the hessian term
is Lipschitz continuous with respect to its argument. This is expected if the
solution to the equation is somewhat smooth. Such strategy will be spelled out
in detail in the proof. With this proposition in hand, Theorem 3.1 is
immediate.

###### Proof for Theorem 3.1.

By Proposition 3.1, there exists a neighbourhood $U$ in which the Hessian is
positive definite, $H_{K}\mathcal{C}(K)>0$ for all $K\in U$. Without loss of
generality, we can assume that $U$ is a convex set. By the strong convexity of
$\mathcal{C}$ in $U$, the minimizer $K_{\star}\in U$ of $\mathcal{C}$ is
unique and thus the finite dimension of the parameter space
$K\in\mathbb{R}^{2R}$ guarantees Tykhonov well-posed of the optimization
problem (7) (Ferrentino and Boniello, 2019, Prop.3.1). ∎

Now we give the proof for Proposition 3.1. It mostly relies on the matrix
perturbation theory (Horn and Johnson, 1985, Cor. 6.3.8) and continuity of the
equation.

###### Proof for Proposition 3.1.

According to the matrix perturbation theory, the minimal eigenvalue is
continuous with respect to a perturbation to the matrix, we have

$\displaystyle|\lambda_{\min}(H_{K}\mathcal{C}(K_{\star}))-\lambda_{\min}(H_{K}\mathcal{C}(K))|\leq\|H_{K}\mathcal{C}(K_{\star})-H_{K}\mathcal{C}(K)\|_{F}$
$\displaystyle\leq\frac{1}{L}\sum_{l}\bigg{(}\|(\nabla_{K}M_{l}\otimes\nabla_{K}M_{l})(K_{\star})-(\nabla_{K}M_{l}\otimes\nabla_{K}M_{l})(K)\|_{F}$
$\displaystyle\hskip 36.98866pt+\|(M_{l}(K)-y_{l})H_{K}M_{l}(K)\|_{F}\bigg{)}$
(15)
$\displaystyle\leq\frac{1}{L}\sum_{l}\bigg{(}\|\nabla_{K}M_{l}(K_{\star})-\nabla_{K}M_{l}(K)\|_{F}\left(\|\nabla_{K}M_{l}(K_{\star})\|_{F}+\|\nabla_{K}M_{l}(K)\|_{F}\right)$
$\displaystyle\hskip 36.98866pt+|M_{l}(K)-y_{l}|\|H_{K}M_{l}(K)\|_{F}\bigg{)}$

where we used the hessian form (13), triangle inequality and sub-
multiplicativity for Frobenius norms. To obtain the bound (14) now amounts to
quantifying each term on the right hand side of (3.1) and bounding them by
$\|K_{\star}-K\|_{\infty}$. This is respectively achieved in Lemmas 3.3, 3.5
and 3.6 that give controls to $M_{l}(K)-y_{l}$, $\|\nabla_{K}M_{l}(K)\|_{F}$
and $\|\nabla_{K}M_{l}(K_{\star})-\nabla_{K}M_{l}(K)\|_{F}$. Putting these
results together, we have:

$\displaystyle|\lambda_{\min}(H_{K}\mathcal{C}(K_{\star}))-\lambda_{\min}(H_{K}\mathcal{C}(K))|\leq\|H_{K}\mathcal{C}(K_{\star})-H_{K}\mathcal{C}(K)\|_{F}$
$\displaystyle\leq
2\|K_{\star}-K\|_{\infty}C_{\mu}C_{\phi}e^{2C_{K}|V|T}\Bigg{[}8RC_{\phi}C_{\mu}e^{2|V|C_{K}T}T\left(|V|T^{2}+\frac{1}{C_{K}}\left(\frac{e^{2C_{K}|V|T}-1}{2C_{K}|V|}-T\right)\right)$
$\displaystyle\hskip 130.88284pt+|V|^{2}TC_{H_{K}M}\Bigg{]}$
$\displaystyle=:\|K_{\star}-K\|_{\infty}C^{\prime}.$

The positive definiteness in a small neighborhood of $K_{\star}$ now follows.
Finally, given
$\|K_{\star}-K\|_{\infty}<\lambda_{\min}(H_{K}\mathcal{C}(K_{\star}))/C^{\prime}$,
the triangle inequality shows

$\lambda_{\min}(H_{K}\mathcal{C}(K))\geq\lambda_{\min}(H_{K}\mathcal{C}(K_{\star}))-|\lambda_{\min}(H_{K}\mathcal{C}(K_{\star}))-\lambda_{\min}(H_{K}\mathcal{C}(K))|>0.$

We note the form of $C^{\prime}$ is complicated but the dependence is spelled
out in the following lemmas and summarized in the theorem statement.

∎

As can be seen from the proof, Proposition 3.1 strongly relies on the
boundedness of the terms in (3.1). We present the estimates below.

######  Lemma 3.3.

Let Assumptions 1 holds, then the measurement difference is upper bounded by:

$\displaystyle|M_{l}(K)-y_{l}|\leq|V|C_{\mu}\|(f_{K_{\star}}-f_{K})(T)\|_{L^{\infty}(\mathbb{R}\times
V)}\leq\|K_{\star}-K\|_{\infty}2|V|^{2}C_{\mu}C_{\phi}Te^{2C_{K}|V|T}.$

###### Proof.

Apply Lemma B.1 to the difference equation for $\bar{f}:=f_{K_{\star}}-f_{K}$

$\displaystyle\partial_{t}\bar{f}+v\cdot\nabla_{x}\bar{f}=\mathcal{K}_{K}(\bar{f})+\mathcal{K}_{(K_{\star}-K)}(f_{K_{\star}})$
(16)

with initial condition $0$ and source
$h=\mathcal{K}_{(K_{\star}-K)}(f_{K_{\star}})\in{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L^{1}}((0,T);L^{\infty}(\mathbb{R}\times
V))$ by the regularity (39) of $f_{K_{\star}}$. This leads to

$\displaystyle\operatorname*{ess\sup}_{v,x}|\bar{f}|(x,t,v)\leq$
$\displaystyle\int_{0}^{t}e^{2|V|C_{K}(t-s)}\operatorname*{ess\sup}_{v,x}|\mathcal{K}_{(K_{\star}-K)}(f_{K_{\star}})(s)|\,\text{d}s$
$\displaystyle\leq$ $\displaystyle
2|V|\|K_{\star}-K\|_{\infty}e^{2|V|C_{K}t}C_{\phi}t,$ (17)

where we used the estimate $\|f_{K_{\star}}(s)\|_{L^{\infty}(\mathbb{R}\times
V)}\leq e^{2|V|C_{K}s}\|\phi\|_{L^{\infty}(\mathbb{R}\times V)}$ from Lemma
B.1 in the last step. ∎

To estimate the gradient $\nabla_{K}M_{l}(K)$ and its difference, we first
recall the form in (9) with $\mathcal{C}$ changed to $M_{l}$ here.
Analogously, we can use the adjoint equation to explicitly represent the
gradient:

######  Lemma 3.4.

Let Assumption 1 hold. Denote by $f_{K}$ the mild solution of (1) and by
$g_{l}\in
C^{0}\left([0,T];L^{\infty}(V;{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L^{1}}(\mathbb{R}))\right)$
the mild solution of

$\displaystyle-\partial_{t}g_{l}-v\cdot\nabla g_{l}$
$\displaystyle=\tilde{\mathcal{K}}(g_{l}):=\int_{V}K(x,v^{\prime},v)(g_{l}(x,t,v^{\prime})-g_{l}(x,t,v))\,\text{d}v^{\prime},$
(18) $\displaystyle g_{l}(t=T,x,v)$ $\displaystyle=-\mu_{l}(x)\,.$

Then

${\frac{\partial M_{l}(K)}{\partial
K_{r,i}}=\int_{0}^{T}\int_{I_{r}}f^{\prime}(g_{l}^{\prime}-g_{l})\,\text{d}x\,\text{d}t\,,}$
(19)

where we used the abbreviated notation $h:=h(t,x,v_{i})$ and
$h^{\prime}:=h(t,x,v_{i}^{\prime})$ for $h=f,g_{l}$, with
$(v_{i},v_{i}^{\prime})$ defined as in (9).

We omit explicitly writing down the $x,t$ dependence when it is not
controversial. The proof for this lemma is the application of calculus-of-
variation and will be omitted from here. We are now in the position to derive
the estimates of the gradient norms.

######  Lemma 3.5.

Under Assumption 1, the gradient is uniformly bounded

$\displaystyle\|\nabla_{K}M_{l}(K)\|_{F}\leq\sqrt{2R}2C_{\phi}C_{\mu}e^{2C_{K}|V|T}T,\qquad\text{
for all }K\in\mathcal{U}_{K}.$

###### Proof.

The Frobenius norm is bounded by the entries $\|\nabla
M_{l}(K)\|_{F}\leq\sqrt{2R}\max_{r,i}|\frac{\,\text{d}M_{l}(K)}{\,\text{d}K_{r,i}}|$.
Representation (19) together with (40) then gives the bound

$\displaystyle\left|\frac{\,\text{d}M_{l}}{\,\text{d}K_{r,i}}\right|\leq
2C_{\phi}\int_{0}^{T}e^{2|V|C_{K}t}\max_{v}\left(\int_{\mathbb{R}}\left|g_{l}\right|\,\text{d}x\right)\,\text{d}t,$
(20)

Application of lemma B.2 to $g=g_{l},h=0$ and $\psi=-\mu_{l}$ yields

$\displaystyle\max_{v}\int_{\mathbb{R}}\left|g_{l}\right|\,\text{d}x\
(t)\leq\int_{\mathbb{R}}|-\mu_{l}(x)|\,\text{d}x\ e^{2C_{K}|V|(T-t)}\leq
C_{\mu}e^{2C_{K}|V|(T-t)},$ (21)

which, when plugged into (20), gives

$\displaystyle\left|\frac{\partial M_{l}}{\partial K_{r,i}}\right|\leq
2C_{\phi}C_{\mu}e^{2C_{K}|V|T}T\,.$

∎

######  Lemma 3.6.

In the setting of Theorem 3.1 and under Assumption 1, the gradient difference
is uniformly bounded in $K\in\mathcal{U}_{K}$ by

$\displaystyle\|\nabla M_{l}(K_{\star})-\nabla M_{l}(K)\|_{F}$
$\displaystyle\leq\sqrt{2R}\|K_{\star}-K\|_{\infty}2C_{\phi}C_{\mu}e^{2C_{K}|V|T}\left(|V|T^{2}+\frac{1}{C_{K}}\left(\frac{e^{2C_{K}|V|T}-1}{2C_{K}|V|}-T\right)\right)\,.$

###### Proof.

Now consider the entries of $\nabla M_{l}(K_{\star})-\nabla M_{l}(K)$ to show
smallness of $\|\nabla M_{l}(K_{\star})-\nabla M_{l}(K)\|_{F}$. Rewrite, using
lemma 3.4 and (40)

$\displaystyle\left|\frac{\partial M_{l}(K_{\star})}{\partial
K_{r,i}}-\frac{\partial M_{l}(K)}{\partial K_{r,i}}\right|=$
$\displaystyle\left|\int_{0}^{T}\int_{I_{r}}f_{K_{\star}}({g^{\prime}}_{l,K_{\star}}-g_{l,K_{\star}})-f_{K}({g^{\prime}}_{l,K}-g_{l,K})\,\text{d}x\,\text{d}t\right|$
$\displaystyle\leq$
$\displaystyle\int_{0}^{T}\|(f_{K_{\star}}-f_{K})(t)\|_{L^{\infty}(\mathbb{R}\times
V)}2\max_{v}\int_{\mathbb{R}}|{g}_{l,K_{\star}}(t)|\,\text{d}x\,\text{d}t$
$\displaystyle+2C_{\phi}\int_{0}^{T}e^{2|V|C_{K}t}\max_{v}\int_{\mathbb{R}}|(g_{l,K_{\star}}-g_{l,K})(t)|\,\text{d}x\,\text{d}t.$

The first summand can be bounded by (17) and (21). To estimate the second
summand, apply Lemma B.2 to $\bar{g}:=g_{l,K_{\star}}-g_{l,K}$ with evolution
equation

$\displaystyle-\partial_{t}\bar{g}-v\cdot\nabla_{x}\bar{g}$
$\displaystyle=\tilde{\mathcal{K}}_{K_{\star}}(\bar{g})+\tilde{\mathcal{K}}_{(K_{\star}-K)}(g_{l,K}),$
$\displaystyle\bar{g}(t=T)$ $\displaystyle=0,$

and
$h=\tilde{\mathcal{K}}_{(K_{\star}-K)}(g_{l,K})\in{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L^{1}}((0,T);L^{\infty}(V;L^{1}(\mathbb{R})))$
by the regularity (44) of
$g_{l,K}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\in{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}C^{0}\left((0,T);L^{\infty}(V;L^{1}(\mathbb{R}))\right)}$.
This leads to

$\displaystyle\max_{v}\int_{\mathbb{R}}|\bar{g}|\,\text{d}x$
$\displaystyle\leq
e^{2|V|C_{K}(T-t)}\int_{0}^{T-t}\max_{v}\|\tilde{\mathcal{K}}_{(K_{\star}-K)}(g_{l,K})(T-s,v)\|_{L^{1}(\mathbb{R})}\,\text{d}s$
$\displaystyle\leq
2|V|\|K_{\star}-K\|_{\infty}e^{2|V|C_{K}(T-t)}\int_{0}^{T-t}\max_{v}\|g_{l,K}(T-s,v)\|_{L^{1}(\mathbb{R})}\,\text{d}s$
$\displaystyle\leq\|K_{\star}-K\|_{\infty}\frac{C_{\mu}}{C_{K}}e^{2|V|C_{K}(T-t)}(e^{2C_{K}|V|(T-t)}-1),$

where we used (21) in the last line. In summary, one obtains

$\displaystyle\left|\frac{\partial M_{l}(K_{\star})}{\partial
K_{r,i}}-\frac{\,\text{d}M_{l}(K)}{\,\text{d}K_{r,i}}\right|$
$\displaystyle\leq\|K_{\star}-K\|_{\infty}\bigg{[}\int_{0}^{T}2|V|C_{\phi}te^{2C_{K}|V|t}\cdot
2C_{\mu}e^{2C_{K}|V|(T-t)}\,\text{d}t$ $\displaystyle\hskip
56.9055pt+2C_{\phi}\int_{0}^{T}e^{2|V|C_{K}t}\frac{C_{\mu}}{C_{K}}e^{2C_{K}|V|(T-t)}(e^{2C_{K}|V|(T-t)}-1)\,\text{d}t\bigg{]}$
$\displaystyle\leq\|K_{\star}-K\|_{\infty}2C_{\phi}C_{\mu}e^{2C_{K}|V|T}\left(|V|T^{2}+\frac{1}{C_{K}}\left(\frac{e^{2C_{K}|V|T}-1}{2C_{K}|V|}-T\right)\right).$

∎

Together with the boundedness of the gradient (20), this shows that the first
summands in (3.1) are Lipschitz continuous in $K$ around $K_{\star}$ which
concludes the proof of Proposition 3.1.

###  3.2 Ill-conditioning for close measurements

While the positive hessian at $K_{\ast}$ guarantees local convergence, such
positive-definiteness will disappear when data are not prepared well.
Especially, when a minimal number of measurements is considered and two
measurements, $M_{1}(K)$ and $M_{2}(K)$ for example, become close, we will
show that the hessian degenerates, and the strongly convexity is lost, and
hence the convergence to $K_{\star}$ is no longer guaranteed.

The closeness of two measurements can be quantified through different manners.
For example, we can argue that the two measurements are close when the two
test functions $\mu_{1},\mu_{2}$ are close in $L^{1}$ sense. Or they can be
close if the reading of the measurements are taken at two locations closeby.
In this case, $\mu_{1}$ and $\mu_{2}$ can be taken as mollifiers from direct
Dirac-$\delta$ readings of the density at $x_{1}$ and $x_{2}$, and the
closeness is quantified by $|x_{1}-x_{2}|$.

We will study how the hessian degenerates in these two scenarios. In both
cases, we examine the two parts in (13) and evaluate their change as two
measurements get close. In particular, the application of Lemma 3.3 already
suggests the second part in (13) is negligible for $K$ is close to $K_{\star}$
and the rank structure of the hessian is predominantly controlled by the first
part, which reads as the summation of many rank 1 matrices
$\nabla_{K}M_{l}(K)\otimes\nabla_{K}M_{l}(K)$. When two measurements
($\mu_{1}$ and $\mu_{2}$) get close, we will argue that $\nabla_{K}M_{1}(K)$
is almost parallel to $\nabla_{K}M_{2}(K)$, making the hessian lacking at
least one rank, and the strong convexity is lost. Mathematically, this means
we need to show $\|\nabla_{K}M_{1}(K)-\nabla_{K}M_{2}(K)\|_{2}\approx 0$ when
$\mu_{1}\approx\mu_{2}$ in the two senses spelled out above.

Recalling (19), we have for every $r\in\\{1,\cdots,R\\}$ and $i\in\\{1,2\\}$

$\displaystyle\frac{\partial M_{1}(K)}{\partial K_{r,i}}-\frac{\partial
M_{2}(K)}{\partial K_{r,i}}$
$\displaystyle=\int_{0}^{T}\int_{I_{r}}f^{\prime}((g_{1}-g_{2})^{\prime}-(g_{1}-g_{2}))\,\text{d}x\,\text{d}t$
$\displaystyle=\int_{0}^{T}\int_{I_{r}}f^{\prime}(\bar{g}^{\prime}-\bar{g})\,\text{d}x\,\text{d}t\,,$
(22)

where $\bar{g}:=g_{1}-g_{2}$ solves (10) with final condition
$\bar{g}(t=T,x,v)=\mu_{2}(x)-\mu_{1}(x)$. So the bulk of the analysis in the
two subsections below is to quantify the smallness of (22) in terms of the
smallness of $\mu_{1}(x)-\mu_{2}(x)$.

####  3.2.1 $L^{1}$ measurement closeness

The following proposition states the loss of strong convexity as
$\mu_{2}-\mu_{1}\to 0$ in $L^{1}(\mathbb{R})$. In particular, the requirement
of Proposition 3.1 that $H_{K}\mathcal{C}(K_{\star})$ is positive definite is
no longer satisfied, so local well-posedness of the optimization problem and
thus the convergence of the algorithm can no longer be guaranteed.

######  Proposition 3.2.

Let Assumption 1 hold. Then, as $\mu_{1}^{(m)}\xrightarrow{m\to\infty}\mu_{2}$
in $L^{1}(\mathbb{R})$, one eigenvalue of the Hessian
$H_{K}\mathcal{C}(K_{\star})$ vanishes.

This proposition immediately allows us to prove scenario 1 in Theorem 3.2:

###### Proof of Theorem 3.2.

Propositions 3.2 establishes one eigenvalue of $H_{K}\mathcal{C}(K_{\star})$
vanishes as $m\to\infty$. This lack of positive definiteness and thus strong
convexity of $\mathcal{C}$ around $K_{\star}$ means that it cannot be
guaranteed that the minimizing sequences of $\mathcal{C}$ converge to
$K_{\star}$. ∎

We now give the proof of the proposition.

###### Proof.

As argued above, we show
$\|\nabla_{K}M_{1}^{(m)}(K)-\nabla_{K}M_{2}(K)\|_{2}\to 0$ as $m\to\infty$.
Recall (22), we need to show:

$\displaystyle\frac{\partial M^{(m)}_{1}(K)}{\partial K_{r,i}}-\frac{\partial
M_{2}(K)}{\partial
K_{r,i}}\xrightarrow{m\to\infty}0\quad\forall(r,i)\in\\{1,\cdots,R\\}\times\\{1,2\\}\,.$
(23)

where $\bar{g}:=g_{1}-g_{2}$ solves (10) with final condition
$\bar{g}(t=T,x,v)=\mu_{2}(x)-\mu^{(m)}_{1}(x)$. Application of Lemma B.2 gives

$\displaystyle\|\bar{g}(t)\|_{L^{\infty}(V;L^{1}(\mathbb{R}))}\leq
e^{2C_{K}|V|(T-t)}\|\mu_{2}-\mu^{(m)}_{1}\|_{L^{\infty}(V;L^{1}(\mathbb{R}))}=e^{2C_{K}|V|(T-t)}\|\mu_{2}-\mu^{(m)}_{1}\|_{L^{1}(\mathbb{R})}.$

Plug this into (22) and estimate $f$ by (40) to obtain

$\displaystyle\left|\frac{\partial(M^{(m)}_{1}-M_{2})(K)}{\partial
K_{r,i}}\right|$ $\displaystyle\leq
2C_{\phi}\int_{0}^{T}e^{2C_{K}|V|t}\|\bar{g}(t)\|_{L^{\infty}(V;L^{1}(\mathbb{R}))}\,\text{d}t$
$\displaystyle\leq
2C_{\phi}e^{2C_{K}|V|T}T\|\mu_{2}-\mu^{(m)}_{1}\|_{L^{1}(\mathbb{R})}.$

Since every entry $(r,i)$ converges, the gradient difference vanishes
$\|\nabla_{K}M_{1}^{(m)}(K)-\nabla_{K}M_{2}(K)\|_{2}\to 0$ as $m\to\infty$.

We utilize this fact to show the degeneracy of the Hessian. Noting:

$\displaystyle
H_{K}\mathcal{C}(K_{\star})=\underbrace{\left[\sum_{l=3}^{2R}\nabla
M_{l}\otimes\nabla M_{l}+2\nabla M_{2}\otimes\nabla
M_{2}\right]}_{A}+\underbrace{\left[\nabla M_{1}^{(m)}\otimes\nabla
M_{1}^{(m)}-\nabla M_{2}\otimes\nabla M_{2}\right]}_{B^{(m)}}\,.$

It is straightforward that the rank of $A$ is at most $2R-1$, so  the $j$-th
largest eigenvalue $\lambda_{j}(A)=0$ vanishes for some $j$. Moreover, since
$\|\nabla_{K}M_{1}^{(m)}(K)-\nabla_{K}M_{2}(K)\|_{2}\to 0$, we have
$\|B^{(m)}\|_{F}\to 0$. Using the continuity of the minimal eigenvalue with
respect to a perturbation of the matrix, the $j$-th largest eigenvalue of
$H_{K}\mathcal{C}(K_{\star})$ vanishes

$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}|\lambda_{j}(H_{K}\mathcal{C}(K_{\star}))|=|\lambda_{j}(H_{K}\mathcal{C}(K_{\star}))-\lambda_{j}(A)|}\leq\|B^{(m)}\|_{F}\to
0,\quad\text{as }m\to\infty\,.$

∎

####  3.2.2 Pointwise measurement closeness

We now study the second scenario of Theorem 3.2 and consider $\mu_{1}$,
$\mu_{2}$ as mollifications of a singular pointwise testing. For this purpose,
let $\xi\in C_{c}^{\infty}(\mathbb{R})$ be a smooth function, compactly
supported in the unit ball $B_{1}(0)$ with $0\leq\xi\leq 1$ and $\xi(0)=1$. In
the following, we consider the measurement test functions

$\mu_{i}^{\eta}(x)=\frac{1}{\eta}\xi\left(\frac{x-x_{i}}{\eta}\right),\quad
i=1,2.$ (24)

Our aim is to show that the assertion of Theorem 3.2 is true independently of
the mollification parameter $\eta>0$. This shows that in the limit as $\eta\to
0$, i.e. in the pointwise measurement case, we still lose strong convexity
around $K_{\star}$.

######  Proposition 3.3.

Let $\mu_{1}^{\eta},\mu_{2}^{\eta}$ be of the form (24) with measurement
locations $x_{2}\notin\\{a_{r}\\}_{r=1,...,R}$ for the partition of
$\mathbb{R}$ from (3). Consider a small neighbourhood of $K_{\star}$ and let
Assumption 1 hold. Additionally, let the measurement time $T$ and locations be
chosen such that

$(e^{T|V|C_{K}}-1)<1,\qquad\min_{r}|x_{2}-a_{r}|-T>\eta_{0}>0.$

If the initial condition $\phi$ is uniformly continuous in $x$, uniformly in
$v$, then $\nabla_{K}M_{1}(K)\to\nabla_{K}M_{2}(K)$ as $x_{1}\to x_{2}$ in the
standard Euclidean norm, and the convergence is independent of
$\eta\leq\eta_{0}$.

This proposition explains the breakdown of well-posedness presented in Theorem
3.2 in the second scenario. Since the proof for the theorem is rather similar
to that of the first scenario, we omit it from here.

Similar to the previous scenario, we need to show smallness of the gradient
difference (22). This time, we have to distinguish two sources of smallness:
For singular parts of the adjoint $\bar{g}$, the smallness of the
corresponding gradient difference is generated by testing it on a sufficiently
regular $f$ at close measuring locations. So it is small in the weak sense.
The regular parts $\bar{g}_{>N}$ of $\bar{g}$ represent the difference of
$\bar{g}$ and its singular parts and evolve form the integral operator on the
right hand side of (10), which exhibits a diffusive effect. Smallness is
obtained by adjusting the cut off regularity $N$.

Let us mention, however, that the time constraint is mostly induced for a
technical reason. In order to bound the size of the regular parts of the
adjoint solution, we use the plain Grönwall inequality which leads to an
exponential growth that we counterbalance by a small measuring time $T$. The
spatial requirement $\min_{r}|x_{2}-a_{r}|-T>\eta_{0}>0$ is a reflection of
the fact that we need the measuring blob (support of $\mu$) to be somewhat
centered in the constant pieces of the piecewise-constant function $K$. This
helps to force the measuring to precisely pick up only the information from
that particular piece. This specific design will later be discussed in Section
4 as well.

To put the above considerations into a mathematical framework, we deploy the
singular decomposition approach, and we are to decompose

$\bar{g}=\sum_{n=0}^{N}\bar{g}_{n}+\bar{g}_{>N},$ (25)

where the regularity of $\bar{g}_{n}$ increases with $n$. Here, we define
$\bar{g}_{0}$ as the solution to

$\displaystyle-\partial_{t}\bar{g}_{0}-v\cdot\nabla_{x}\bar{g}_{0}$
$\displaystyle=-\sigma\bar{g}_{0}\,,$ $\displaystyle\bar{g}_{0}(t=T,x,v)$
$\displaystyle=\mu_{2}^{\eta}(x)-\mu_{1}^{\eta}(x)\,,$

for $\sigma(x,v):=\int_{V}K(x,v^{\prime},v)\,\text{d}v^{\prime}$, and
$\bar{g}_{n}$ are inductively defined by

$\displaystyle-\partial_{t}\bar{g}_{n}-v\cdot\nabla_{x}\bar{g}_{n}$
$\displaystyle=-\sigma\bar{g}_{n}+\tilde{\mathcal{L}}(\bar{g}_{n-1})\,,$ (26)
$\displaystyle\bar{g}_{n}(t=T,x,v)$ $\displaystyle=0\,,$

where we used the notation $\tilde{\mathcal{L}}(\bar{g}):=\int
K(x,v^{\prime},v)\bar{g}(x,t,v^{\prime})\,\text{d}v^{\prime}$. The remainder
$\bar{g}_{>N}$ satisfies

$\displaystyle-\partial_{t}\bar{g}_{>N}-v\cdot\nabla_{x}\bar{g}_{>N}$
$\displaystyle=-\sigma\bar{g}_{>N}+\tilde{\mathcal{L}}(\bar{g}_{N}+\bar{g}_{>N})\,,$
(27) $\displaystyle\bar{g}_{>N}(t=T,x,v)$ $\displaystyle=0\,.$

It is a straightforward calculation that

$\eqref{eq:gradMdifference}={\sum_{n=0}^{N}\int_{0}^{T}\int_{I_{r}}f^{\prime}(\bar{g}_{n}^{\prime}-\bar{g}_{n})\,\text{d}x\,\text{d}t}+\int_{0}^{T}\int_{I_{r}}f^{\prime}(\bar{g}_{>N}^{\prime}-\bar{g}_{>N})\,\text{d}x\,\text{d}t\,.$
(28)

We are to show, in the two lemmas below, that both terms are small when
$x_{1}\to x_{2}$. To be more specific:

######  Lemma 3.7.

Let the assumptions of Proposition 3.3 be satisfied. For any $\varepsilon>0$,
and any $n\in\mathbb{N}_{0}$, there exists a $\delta_{n}(\varepsilon)>0$ such
that

$\left|\int_{0}^{T}\int_{I_{r}}f^{\prime}\bar{g}_{n}\,\text{d}x\,\text{d}t\right|\leq\varepsilon\,,{\quad\text{if
}\quad|x_{1}-x_{2}|<\delta_{n}(\varepsilon)\,.}$ (29)

The remainder can be bounded similarly.

######  Lemma 3.8.

Under the assumptions of Proposition 3.3, one has

$\displaystyle\left|\int_{0}^{T}\int_{I_{r}}f^{\prime}\bar{g}_{>N}\,\text{d}x\,\text{d}t\right|\leq
T^{2}|V|C_{K}C_{\phi}e^{2|V|C_{K}T}(e^{C_{K}|V|T}-1)^{N}C_{\mu},$

which becomes arbitrarily small for large $N$.

The proofs for both lemmas exploit the continuity of $f$ by choice of $\phi$,
and the smallness of the higher regularity components of the $g$ term. Since
it is not keen to the core of the paper, we leave the details to Appendix C.
The application of the two lemmas gives Proposition 3.3:

###### Proof of Proposition 3.3.

Let $\varepsilon>0$. Because $e^{C_{K}|V|T}-1<1$ by assumption, we can choose
$N\in\mathbb{N}$ large enough such that
$2T^{2}|V|C_{K}C_{\phi}e^{2|V|C_{K}T}(e^{C_{K}|V|T}-1)^{N}<\frac{\varepsilon}{2}$.
Furthermore, let $|x_{1}-x_{2}|<\min_{n\leq
N}\delta_{n}(\frac{\varepsilon}{4(N+1)})$. Then with the triangle inequality
and Lemmas 3.7 and 3.8, we obtain from (28)

$\displaystyle\left|\frac{\partial(M_{1}-M_{2})(K)}{\partial
K_{r,i}}\right|\leq$
$\displaystyle\sum_{n=0}^{N}\left|\int_{0}^{T}\int_{I_{r}}f^{\prime}(\bar{g}_{n}^{\prime}-\bar{g}_{n})\,\text{d}x\,\text{d}t\right|+\left|\int_{0}^{T}\int_{I_{r}}f^{\prime}(\bar{g}_{>N}^{\prime}-\bar{g}_{>N})\,\text{d}x\,\text{d}t\right|$
$\displaystyle\leq$ $\displaystyle
2N\frac{\varepsilon}{4{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(N+1)}}+2T^{2}|V|C_{K}C_{\phi}e^{2|V|C_{K}T}(e^{C_{K}|V|T}-1)^{N}C_{\mu}$
$\displaystyle\leq$ $\displaystyle\varepsilon\,.$

∎

##  4 Experimental Design

As discussed in the previous sections, it is clear that different setups bring
different conditioning to the inverse problem. We are to study a particular
design where the well-posedness can be ensured. To be more specific, in
Proposition 3.1 we require the positive-definiteness of the Hessian term at
$K_{\star}$. This is a strong assumption and is typically not true unless
certain initial condition and the measuring setups are in place. We propose to
use the following:

######  Design (D).

We divide the domain $I=[a_{0},a_{R})$ into $R$ intervals
$I=\bigcupdot_{r=1}^{R}I_{r}$ with $I_{r}=[a_{r-1},a_{r})$, and the center for
each interval is denoted by
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}a_{r-1/2}}:=\frac{a_{r-1}+a_{r}}{2}$.
The spatial supports of the values $K_{r}(v,v^{\prime})$ takes on the form of
(3). The design is:

  * •

_initial condition_ $\phi(x,v)=\sum_{r=1}^{R}\phi_{r}(x)$ is a sum of $R$
positive functions $\phi_{r}$ that are compactly supported in
$a_{r-1/2}+[-d,d]$ with $d<\min\left(\frac{a_{r}-a_{r-1}}{4}\right)$,
symmetric and monotonously decreasing in $|x-a_{r-1/2}|$ (for instance, a
centered Gaussian with a cut-off tail);

  * •

_measurement test functions_
$\mu_{l_{i}^{r}}=\bar{C}_{\mu}\mathds{1}_{[(-1)^{i}T-{d_{\mu}},(-1)^{i}T+{d_{\mu}}]+a_{r-1/2}}$,
$i=1,2$, for some $\bar{C}_{\mu}>0$, centered around $a_{r-1/2}\pm T$ with
$d_{\mu}\leq d$;

  * •

_measurement time_ $T$ such that

$\displaystyle
T<\min\left((1-\delta)\frac{0.09}{C_{K}|V|},\min_{r}\left(\frac{a_{r}-a_{r-1}}{4}-\frac{d}{2}\right)\right)$
(30) $\displaystyle\text{for }\quad\delta=(d+d_{\mu})/T<e^{-TC_{K}|V|}.$ (31)

######  Remark 4.1.

Note that this design requires a delicate balancing between $T$ and $d$ and
$d_{\mu}$. Requirement (30) prescribes that $T$ must not be too large. On the
other hand, (31) requires that it must not be too small compared to
$d,d_{\mu}$. An exemplary choice of $d=d_{\mu}=cT^{2}$ for some $c>0$, for
instance, automatically verifies requirement (31) for small enough $T$.

This particular design of initial data and measurement is to respond to the
fact that the equation has a characteristic and particles moves along the
trajectories. The measurement is set up to single out the information we would
like to reconstruct along the propagation. The visualization of this design is
plotted in Figure 1.

$a_{r-1}$$x_{1}$$a_{r-1/2}$$x_{2}$$a_{r}$$x$$2d_{\mu}$$2d$ Figure 1: Motion of
the ballistic parts $f^{(0)}(t=0,v)$ (cyan, dashdotted) to $f^{(0)}(t=T,v=+1)$
(blue, dotted) and $f^{(0)}(t=T,v=-1)$ (blue, dashed) and
$g_{1}^{(0)}(t=0,v=+1)$ (orange, dotted) and $g_{1}^{(0)}(t=0,v=-1)$ (orange,
dashed) to $g_{1}^{(0)}(t=T,v)$ (red, dashdotted), compare also (34).

Under this design, we have the following proposition:

######  Proposition 4.1.

The design Design (D) decouples the reconstruction of $K_{r}$. To be more
specific, recall (4)

$K=[K_{r}]\,,\quad\text{with}\quad K_{r}=[K_{r,1},K_{r,2}]\,.$

The Hessian $H_{K}\mathcal{C}$ has a block diagonal structure with each of the
blocks is a $2\times 2$ matrix given by the Hessian $H_{K_{r}}\mathcal{C}$.

###### Proof.

By the linearity of (1), (10), their solutions $f=\sum_{r=1}^{R}f_{r}$ and
$g=\sum_{r=1}^{R}\sum_{i=1}^{2}g_{l_{i}^{r}}$ decompose into solutions $f_{r}$
of (1) with initial conditions $\phi_{r}$ and $g_{l_{i}^{r}}$ with final
condition $-(M_{l_{i}^{r}}-y_{l_{i}^{r}})\mu_{l_{i}^{r}}/2R$, the summands of
the final condition (11), correspondingly. By construction of $T$ and the
constant speed of propagation $|v|=1$, the spatial supports of $f_{r}$ and
$g_{l_{1}^{r}}$, $g_{l_{2}^{r}}$ are is fully contained only in $I_{r}$ for
all $t\in[0,T],v\in V$. As such, only $f_{r}$ and $g_{l^{r}_{j}}$ carry
information about $K_{r}$, and no information for other $K_{s}$ with $s\neq
r$. ∎

This not only makes boundary conditions superfluous, but also translates the
problem of finding a $2R$ valued vector $K$ into $R$ individual smaller
problems of finding the two-constant pair $(K_{r,1},K_{r,2})$ within $I_{r}$.
This comes with the cost of prescribing very detailed measurements depending
on the experimental scales $I_{r}$ and $d$, but opens the door for
parallelized computation.

Furthermore, under mild conditions, this design ensures the local
reconstructability of the inverse problem.

######  Theorem 4.2.

Let Assumption 1 hold. Given the Hessian $H_{K}M_{l}(K)$ is bounded in
Frobenius norm in a neighbourhood of $K_{\star}$, Design Design (D) generates
a locally well-posed optimization problem (7).

The proof is layed out in the subsequent subsection 4.1.

######  Remark 4.3.

Let us mention that the bounds for $T$ in Design Design (D) are not optimal.
In the proof of theorem 4.2 we used crude estimates, and we believe these
estimates can potentially be relaxed to allow for longer measurement times
$T$. Furthermore, the setup can easily be modified to use different
measurement times for different intervals $I_{r}$, for instance. In this case,
again, the bounds on $T$ can be relaxed.

######  Remark 4.4.

Design Design (D) shares similarities with the theoretical reconstruction
setting in Hellmuth et al. (2022), where a pointwise reconstruction of a
continuous kernel $\tilde{K}$ was obtained by a sequence of experiments where
the measurement time $T$ became small and the measurement location gets close
to the initial location. The situation is also seen here. As we refine the
discretization for the underlying $K$-function using higher dimensional
vector, measurement time has to be shortened to honor the refined
discretization. However, we should also note the difference. In Hellmuth et
al. (2022), we studied the problem in higher dimension and thus explicitly
excluded the ballistic part of the data from the measurement

###  4.1 Proof of Theorem 4.2

Given Theorem 3.1, it remains to prove $H_{K}\mathcal{C}(K_{\star})>0$. As the
Hessian attains a block diagonal structure (Proposition 4.1), we are to study
the $2\times 2$-blocks

$H_{K_{r}}\mathcal{C}(K_{\star})=\nabla_{K_{r}}M_{l^{r}_{1}}(K_{\star})\otimes\nabla_{K_{r}}M_{l^{r}_{1}}(K_{\star})+\nabla_{K_{r}}M_{l^{r}_{2}}(K_{\star})\otimes\nabla_{K_{r}}M_{l^{r}_{2}}(K_{\star}).$
(32)

Here the two measurements $M_{l^{r}_{1}}$, $M_{l^{r}_{2}}$ are inside $I_{r}$,
and $\nabla_{K_{r}}=[\partial_{K_{r,1}},\partial_{K_{r,2}}]$. The positive
definiteness of the full $H_{K}\mathcal{C}(K_{\star})$ is equivalent to the
positive definiteness of each individual $H_{K_{r}}\mathcal{C}(K_{\star})$.
This is established in the subsequent proposition.

######  Proposition 4.2.

Let Assumption 1 hold. If the Hessian $H_{K}M_{l}(K)$ is bounded in Frobenius
norm in a neighbourhood of $K_{\star}$, then the Design Design (D) produces a
positive-definite hessian $H_{K}\mathcal{C}(K_{\star})$.

According to (32), $H_{K_{1}}\mathcal{C}(K_{\star})$ is positive definite if

$\left|\frac{\partial M_{1}(K_{\star})}{\partial
K_{1,1}}\right|>\left|\frac{\partial M_{1}(K_{\star})}{\partial
K_{1,2}}\right|\quad\text{ and }\quad\left|\frac{\partial
M_{2}(K_{\star})}{\partial K_{1,1}}\right|<\left|\frac{\partial
M_{2}(K_{\star})}{\partial K_{1,2}}\right|$ (33)

holds true for the measurements $M_{1},M_{2}$ corresponding to $K_{1}$. Due to
design symmetry, it is sufficient to study the first inequality. Consider the
difference $\frac{\partial M_{1}(K_{\star})}{\partial K_{1,1}}-\frac{\partial
M_{1}(K_{\star})}{\partial K_{1,2}}$. Similar to (25) and (28), we are to
decompose the equation for $f$ and $g$ ((1) and (18) respectively, with
$K=K_{\star}$) into the ballistic parts $g_{1}^{(0)}$ and $f^{(0)}$ and the
remainder terms. Namely, let $g^{(0)}_{1}$ and $f^{(0)}$ satisfy

$\displaystyle\begin{cases}-\partial_{t}g_{1}^{(0)}-v\cdot\nabla_{x}g_{1}^{(0)}&=-\sigma
g_{1}^{(0)}\\\
g_{1}^{(0)}(t=T,x,v)&=\mu_{1}(x)\end{cases}\quad\text{and}\quad\begin{cases}\partial_{t}f^{(0)}-v\cdot\nabla_{x}f^{(0)}&=-\sigma
f^{(0)}\\\ f^{(0)}(t=0,x,v)&=\phi(x,v).\end{cases}$ (34)

Then the following two lemmas are in place with $\mu_{1}(x)$ and $\phi(x,v)$
as in Design (D).

######  Lemma 4.5.

In the setting of Proposition 4.2, for $(v,v^{\prime})=(+1,-1)$, the ballistic
part

$\displaystyle B:=$
$\displaystyle\left|\int_{0}^{T}\int_{I_{1}}f^{(0)}(v^{\prime})(g_{1}^{(0)}(v^{\prime})-g_{1}^{(0)}(v))\,\text{d}x\,\text{d}t\right|$
(35)
$\displaystyle-\left|\int_{0}^{T}\int_{I_{1}}f^{(0)}(v)(g_{1}^{(0)}(v)-g_{1}^{(0)}(v^{\prime}))\,\text{d}x\,\text{d}t\right|$

satisfies

$\displaystyle B\geq C_{\phi\mu}\left(e^{-TC_{K}|V|}T-(d_{\mu}+d)\right)>0,$
(36)

where
$C_{\phi\mu}=\int_{I_{1}}\phi_{1}(x)\mu_{1}(-T+x)\,\text{d}x=\max_{a,b}\int_{I_{1}}\phi_{1}(x+a)\mu_{1}(-T+x+b)\,\text{d}x$
by construction of $\phi_{1},\mu_{1}$.

At the same time, the remainder term is small.

######  Lemma 4.6.

In the setting of Proposition 4.2, the remaining scattering term

$\displaystyle S:=$
$\displaystyle\int_{0}^{T}\int_{I_{1}}f(v^{\prime})(g_{1}(v^{\prime})-g_{1}(v))\,\text{d}x\,\text{d}t-\int_{0}^{T}\int_{I_{1}}f^{(0)}(v^{\prime})(g_{1}^{(0)}(v^{\prime})-g_{1}^{(0)}(v))\,\text{d}x\,\text{d}t$

is bounded uniformly in $(v,v^{\prime})$ by

$\displaystyle|S|\leq 4C_{\phi\mu}T\frac{C_{K}|V|T}{(1-C_{K}|V|T)^{2}}.$ (37)

Proposition 4.2 is a corollary of Lemmas 4.5, 4.6.

###### Proof of Proposition 4.2.

By the bounds obtained in lemmas 4.5, 4.6, one has

$\displaystyle\left|\frac{\partial M_{1}(K_{\star})}{\partial
K_{1,1}}\right|-\left|\frac{\partial M_{1}(K_{\star})}{\partial
K_{1,2}}\right|\geq B-2|S|$ $\displaystyle\geq
C_{\phi\mu}\left(e^{-TC_{K}|V|}T-{(d_{\mu}+d)}\right)-8C_{\phi\mu}T\frac{C_{K}|V|T}{(1-C_{K}|V|T)^{2}}$
$\displaystyle\geq
C_{\phi\mu}T\left(1-TC_{K}|V|-\delta-8\frac{0.09(1-\delta)}{(1-0.09)^{2}}\right).$

By assumption $0<T<(1-\delta)\frac{0.09}{C_{K}|V|}$ with
$\delta=\frac{d+d_{\mu}}{T}<1$, the last line is positive. In total, this
shows the first part of inequality (33). As the second part can be treated in
analogy, it follows that $H_{K_{1}}\mathcal{C}(K_{\star})$ is positive
definite. ∎

Finally, Theorem 4.2 is a direct consequence of Proposition 4.2.

###### Proof of Theorem 4.2.

Repeated application of the arguments to all
$H_{K_{r}}\mathcal{C}(K_{\star})$, $r=1,...,R$ shows that
$H_{K}\mathcal{C}(K_{\star})>0$. Assuming boundedness of the Hessian
$H_{K}M_{l}(K)$ in a neighbourhood of $K_{\star}$, theorem 3.1 proves local
well-posedness of the inverse problem. ∎

The proofs for the Lemmas 4.5-4.6 are rather technical and we leave them to
Appendix D. Here we only briefly present the intuition. According to Figure 1,
$f^{(0)}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}v^{\prime}=-1})$
and $g_{1}^{(0)}({v^{\prime}}=-1)$ have a fairly large overlapping support,
whereas $g_{1}^{(0)}(v=+1)$ overlaps with $f^{(0)}(v^{\prime}=-1)$ and
$g_{1}^{(0)}(v^{\prime}=-1)$ with $f^{(0)}(v=+1)$ only for a short time spans
$T\approx T$ and $T\approx 0$ respectively. Recalling (35), we see the
negative components of the term $B$ are small, making $B$ positive overall.
The smallness of $S$ is a result of small measurement time $T$.

##  5 Numerical experiments

As a proof of concept for the prediction given by the theoretical results in
Section 3, we present some numerical evidence.

An explicit finite difference scheme is used for the discretization of (1) and
(10). In particular, the transport operator is discretized by the Lax-Wendroff
method and the operator $\mathcal{K}$ is treated explicitly in time. The
scheme is consistent and stable when $\Delta t\leq\min(\Delta x,C_{K}^{-1})$,
and thus it converges according to the Lax-Equivalence theorem. More
sophisticated solvers for the forward model Filbet and Yang (2014) can be
deployed when necessary. Also, when a compatible solver Apel and Flaig (2012)
for the adjoint equation exists, these pairs of solvers can readily be
incorporated in the inversion setting.

All subsequent experiments were conducted with noise free synthetic data
$y_{l}=M_{l}(K_{\star})$ that was generated by a forward computation with the
true underlying parameter $K_{\star}$.

###  5.1 Illustration of well-posedness

In Section 4, it was suggested a specific design of initial data and
measurement mechanism can provide a successful reconstruction of the kernel
$K$, and that the loss function is expected to be strongly convex. We observe
it numerically as well. In particular, we set $R=20$ and use Gaussian initial
data, and plot the (marginal) loss function in Figure 2.

Figure 2: (Marginal) loss functions $\mathcal{C}(K)$ for $R=20$: For a fixed
$r\in\\{2,9,13,15\\}$, we plot $\mathcal{C}$ as a function of $K_{r}$ with all
$K_{s\neq r}$ set to be the groundtruth $(K_{\star})_{s}$.

Figure 3 depicts the convergence of some parameter values
$K_{r}(v,v^{\prime})$ in this scenario against the corresponding loss function
value. An exponential decay of the loss function, as expected from theory
(Polyak and Shcherbakov, 2017, Th.3), can be observed.

Figure 3: Convergence of the parameter values $K_{r}(v,v^{\prime})$ from (3)
for $r=2,9,13,15$ to the ground truth as the cost function converges.

The strictly positive-definiteness feature persists in a small neighborhood of
the optimal solution $K_{\star}$. This means adding a small perturbation to
$K_{\star}$, the minimal eigenvalue of the Hessian matrix
$H_{K}\mathcal{C}(K)$ stays above zero. In Figure 4 we present, for two
distinct experimental setups, the minimum eigenvalue as a function of the
perturbation to $K_{r}(v,v^{\prime})$. In both cases, the green spot (the
groundtruth) is positive, and it enjoys a small neighborhood where the minimum
eigenvalue is also positive, as predicted by Theorem 3.1. In the right panel,
we do observe, as one moves away from the groundtruth, the minimal eigenvalue
takes on a negative value, suggesting the loss of convexity. This numerically
verifies that the well-posedness result in Theorem 3.1 is local in nature. The
panel on the left deploys the experiment design provided by Section 4. The
simulation is ran over the entire domain of $[0,1]^{2}$ and the positive-
definiteness stays throughout the domain, hinting the proposed experimental
design Design (D) can potentially be globally well-posed.To generate the
plots, a simplified setup with $R=2$ and constant initial data was considered.

Figure 4: Minimal eigenvalues of the Hessian $H_{K}\mathcal{C}(K)$ around the
true parameter $K_{\star}$ for two experimental designs. We perturb $K$ by
changing values in $K_{1}(1,-1)$ and $K_{2}(-1,1)$. The groundtruth is marked
green in both plots.

###  5.2 Ill-conditioning for close measurement locations

We now provide numerical evidence to reflect the assertion from 3.2. In
particular, the strong convexity of the loss function would be lost if
measurement location $x_{1}$ becomes close to $x_{2}$.

We summarize the numerical evidence in Figure 5. Here we still use $R=20$ and
constant initial data but vary the detector positions. To be specific, we
assign values to $x_{1}$ using
$\\{x_{1}^{(0)}=c_{1}-T\,,x_{1}^{(1)}=c_{1}+\frac{T}{2}\,,x_{1}^{(2)}=c_{1}+\frac{4}{5}T\,,x_{1}^{(3)}=x_{2}=c_{1}+T\\}$.
As the superindex grows, $x_{1}\to x_{2}$ with $x_{1}^{(3)}=x_{2}$ when the
two measurements exactly coincide. For $x_{1}=x_{2}$, the cost function is no
longer strongly convex around the ground truth $K_{\star}$, as its hessian is
singular. The thus induced vanishing learning rate
$\eta=\frac{2\lambda_{\min}}{\lambda_{\max}^{2}}$ was exchanged by the
learning rate for $x_{1}=x_{1}^{(2)}$ in this case to observe the effect of
the gradient.

In the first, third and fourth panel of Figure 5, we observe that the cost
function as well as the parameter reconstructions for $K_{9}$ and $K_{15}$
nevertheless converge,but convergence rates that slow down significantly
comparing purple (for $x_{1}^{(0)}$), blue (for $x_{1}^{(1)}$), green (for
$x_{1}^{(2)}$) and orange (for $x_{1}^{(3)}$) due to smaller learning rates.
The overlap of the parameter reconstructions for
$x_{1}\in\\{x_{1}^{(2)}\,,x_{1}^{(3)}\\}$ is due to the coinciding choice of
the learning rate and a very similar gradient for parameters $K_{9},K_{15}$
whose information is not reflected in the measurement in $x_{1}$.

As parameter $K_{1}$ directly affects the measurement at $x_{1}$, Panel 2
showcases the degenerating effect of the different choices of $x_{1}$ on the
reconstruction. Whereas convergence is still obtained in the blue curve (for
$x_{1}^{(1)}$), the reconstructions of $K_{1}$ from measurements at
$x_{1}^{(2)}$ (green) and $x_{1}^{(3)}$ (orange) clearly fail to converge to
the true parameter value in black. This offset seems to grow with stronger
degeneracy in the measurements.

Figure 5: Cost function and reconstructions of $K_{r}(+1,-1)$ (solid lines)
and $K_{r}(-1,+1)$ (dotted lines) for $r=1,9,15$ and $R=20$ under different
measurement locations for $x_{1}$. $x_{1}$ takes the values of
$\\{x_{1}^{(0)}=c_{1}-T\,,x_{1}^{(1)}=c_{1}+\frac{T}{2}\,,x_{1}^{(2)}=c_{1}+\frac{4}{5}T\,,x_{1}^{(3)}=c_{1}+T\\}$
with $x_{1}^{(3)}=x_{2}$.

##  6 Discussion

In this paper we present an optimization framework for the reconstruction of
the velocity jump parameter $K$ in the chemotaxis equation (1) using velocity
averaged measurements (5) from the interior domain. In the numerical setting
when PDE-constrained optimization is deployed, depending on the experimental
setup, the problem is can be either locally well-posedness or ill-conditioned.
We further propose a specific experimental design that is adaptive to the
discretization of $K$. This design decouples the reconstruction of local
values of the parameter $K$ using the corresponding measurements. The design
thus opens up opportunities to parallelization. As a proof of concept,
numerical evidence were presented. They are in good agreement with the
theoretical predictions

A natural extension of the results presented in the current paper is the
algorithmic performance in higher dimensions. The theoretical findings seem to
apply in a straightforward manner, but details need to be evaluated.
Numerically one can certainly also refine the solver implementation. For
example, it is possible that higher order numerical PDE solvers that preserve
structures bring extra benefit. More sophisticated optimization methods such
as the (Quasi-)Newton method or Sequential Quadratic Programming are possible
alternatives for conducting the inversion Burger and Mühlhuber (2002); Haber
et al. (2000); Ren (2010); Smyl et al. (2021). Furthermore, we adopted a first
optimize, then discretize approach in this article. Suggested in Apel and
Flaig (2012); Gunzburger (2002); Liu and Wang (2019), a first discretize, then
optimize framework could be bring automatic compatibility of forward and
adjoint solvers, but extra difficulties Hinze et al. (2008) need to be
resolved.

Our ultimate goal is to form a collaboration between practitioners to solve
the real-world problem relalted to bacteria motion reconstruction Le (2002).
To that end, experimental design is non avoidable. A class of criteria
proposed under the Bayesian shed light, see Alexanderian (2021) and references
therein. In our context, it translates to whether the design proposed in
Section 4 satisfies these established optimality criteria.

##  Appendix A Derivation of the gradient (9)

This section justifies formula (9) for the gradient of the cost function
$\mathcal{C}$ with respect to $K$. Let us first introduce some notation:
Denote by

$\displaystyle\mathcal{J}(f):=\frac{1}{2L}\sum_{l=1}^{L}\left(\int_{\mathbb{R}}\int_{V}f(T,x,v)\,\text{d}v\
\mu_{l}(x)\,\text{d}x-y_{l}\right)^{2}$

the loss for $f\in\mathcal{Y}=\\{h\mid h,\partial_{t}h+v\cdot\nabla h\in
C^{0}([0,T];L^{\infty}(\mathbb{R}\times V))\\}$. Note that mild solutions of
(1) are contained in $\mathcal{Y}$, since $\mathcal{K}(f)\in
C^{0}([0,T];L^{\infty}(\mathbb{R}\times V))$ by regularity of $f$ from Lemma
B.1. Then $\mathcal{C}(K):=\mathcal{J}(f_{K})$ in the notation of (5). The
Lagrangian function for the PDE constrained optimization problem (7) reads

$\mathcal{L}(K,f,g,\lambda)=\mathcal{J}(f)+\langle
g,\partial_{t}f+v\cdot\nabla
f-\mathcal{K}(f)\rangle_{x,v,t}+\langle\lambda,f(t=0)-\phi\rangle_{x,v},$

for $f\in\mathcal{Y}$ and $g\in\mathcal{Z}=\\{h\mid
h,\partial_{t}h+v\cdot\nabla h\in
C^{0}([0,T];L^{\infty}(V;L^{1}(\mathbb{R})))\\}.$ For $f=f_{K}$, our cost
function $\mathcal{C}(K)=\mathcal{J}(f_{K})=\mathcal{L}(K,f_{K},g,\lambda)$
and

$\frac{\,\text{d}\mathcal{C}(\hat{K})}{\,\text{d}K}=\frac{\partial\mathcal{L}}{\partial
K}\bigg{|}_{\begin{subarray}{c}K=\hat{K},\\\
f=f_{\hat{K}}\end{subarray}}+\frac{\partial\mathcal{L}}{\partial
f}\bigg{|}_{\begin{subarray}{c}K=\hat{K},\\\
f=f_{\hat{K}}\end{subarray}}\frac{\partial f_{K}}{\partial
K}\bigg{|}_{K=\hat{K}}$

To avoid the calculation of $\frac{\partial f_{K}}{\partial K}$, choose the
Lagrange multipliers $g,\lambda$ such that
$\frac{\partial\mathcal{L}}{\partial f}|_{\begin{subarray}{c}K=\hat{K},\\\
f=f_{\hat{K}}\end{subarray}}=0$. Then

$\displaystyle\frac{\,\text{d}\mathcal{C}(\hat{K})}{\,\text{d}K_{r}}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial
K_{r}}\bigg{|}_{\begin{subarray}{c}K=\hat{K},\\\
f=f_{\hat{K}}\end{subarray}}=-\frac{\partial\langle
g,\mathcal{K}_{K}(f)\rangle_{x,t,v}}{\partial
K_{r}}\bigg{|}_{\begin{subarray}{c}K=\hat{K},\\\ f=f_{\hat{K}}\end{subarray}}$
$\displaystyle=\int_{0}^{T}\int_{I_{r}}f_{\hat{K}}(x,t,v^{\prime})\big{(}g(x,t,v^{\prime})-g(x,t,v)\big{)}\,\text{d}x\,\text{d}t.$

To compute the gradient, $g$ has to be specified. Recall the requirement

$\displaystyle 0$ $\displaystyle=\frac{\partial L}{\partial
f}\bigg{|}_{\begin{subarray}{c}K=\hat{K},\\\ f=f_{\hat{K}}\end{subarray}}$
$\displaystyle=\frac{1}{L}\sum_{l=1}^{L}\left(\int_{\mathbb{R}}\int_{V}f(T)\,\text{d}v\
\mu_{l}\,\text{d}x-y_{l}\right)\frac{\partial}{\partial
f}\left\langle\mu_{l},f(T)\right\rangle_{x,v}\Bigg{|}_{\begin{subarray}{c}K=\hat{K},\\\
f=f_{\hat{K}}\end{subarray}}$ (38) $\displaystyle\hskip
28.45274pt+\frac{\partial}{\partial f}\Bigg{[}\langle
g,\partial_{t}f+v\cdot\nabla
f-\mathcal{K}_{K}(f)\rangle_{x,t,v}+\langle\lambda,f(t=0)\rangle_{x,v}\Bigg{]}\Bigg{|}_{\begin{subarray}{c}K=\hat{K},\\\
f=f_{\hat{K}}\end{subarray}}$

We will motivate the choice of $g$ such that the derivatives cancel out each
other. Because we are dealing with mild solutions where integration by parts
in time and space cannot be used right away, we approximate $f$ and $g$ by
sequences of functions

  * •

$(f^{n})_{n}\subset C^{1}([0,T];L^{\infty}(\mathbb{R}\times V))\cap
C^{0}([0,T];W^{1,\infty}(\mathbb{R};L^{\infty}(V)))$ that converge $f_{n}\to
f$ with $\partial_{t}f_{n}+v\cdot\nabla f_{n}\to\partial_{t}f+v\cdot\nabla f$
in $C^{0}([0,T];L^{\infty}(\mathbb{R}\times V))$ and

  * •

$(g^{n})_{n}\subset C^{1}([0,T];L^{\infty}(V;L^{1}(\mathbb{R})))\cap
C^{0}([0,T];L^{\infty}(V;W^{1,1}(\mathbb{R})))$ with $g_{n}\to g$ with
$-\partial_{t}g_{n}-v\cdot\nabla g_{n}\to-\partial_{t}g-v\cdot\nabla g$ in
$C^{0}([0,T];L^{\infty}(V;L^{1}(\mathbb{R})))$.

This is possible, because the respective spaces for $f_{n}$ and $g_{n}$ are
dense in $\mathcal{Y}$ and $\mathcal{Z}$. Replacing $f$ by $f_{n}$ and $g$ by
$g_{n}$ in $\langle g,\partial_{t}f+v\cdot\nabla
f-\mathcal{K}(f)\rangle_{x,t,v}$, we obtain

$\displaystyle\langle g,\partial_{t}f+v\cdot\nabla
f-\mathcal{K}(f)\rangle_{x,t,v}=\lim_{n}\langle
g_{n},\partial_{t}f_{n}+v\cdot\nabla f_{n}-\mathcal{K}(f_{n})\rangle_{x,t,v}$
$\displaystyle=\lim_{n}(\langle-\partial_{t}g_{n}-v\cdot\nabla
g_{n}-\tilde{\mathcal{K}}(g_{n}),f_{n}\rangle_{x,t,v}+\langle
f_{n}(t=T),g_{n}(t=T)\rangle_{x,v}-\langle
f_{n}(t=0),g_{n}(t=0)\rangle_{x,v})$
$\displaystyle=\langle-\partial_{t}g-v\cdot\nabla
g-\tilde{\mathcal{K}}(g),f\rangle_{x,t,v}+\langle
f(t=T),g(t=T)\rangle_{x,v}-\langle f(t=0),g(t=0)\rangle_{x,v},$

where

$\displaystyle\tilde{\mathcal{K}}_{K}(g):=\int_{V}K(x,v^{\prime},v)(g(x,t,v^{\prime})-g(x,t,v))\,\text{d}v^{\prime}.$

Now, collect all terms in (A) with the same integration domain and equate them
to $0$. This leads to

$\displaystyle-\partial_{t}g-v\cdot\nabla g-\tilde{\mathcal{K}}_{K}(g)=0$
$\displaystyle\text{in }x\in\mathbb{R},v\in V,t\in(0,T)$ $\displaystyle
g(x,t=T,v)=-\frac{1}{L}\sum_{l=1}^{L}\left(\int_{\mathbb{R}}\int_{V}f(T,x,v)\,\text{d}v\
\mu_{l}(x)\,\text{d}x-y_{l}\right)\mu_{l}(x)$ $\displaystyle\text{in
}x\in\mathbb{R},(v\in V)$ $\displaystyle\lambda=g(t=0)$ $\displaystyle\text{in
}x\in\mathbb{R},v\in V.$

Note that since $g$ reflects the measurement procedure, it makes sense that
$g(t=T)$ is isotropic in $v$. For computation of
$\frac{\,\text{d}\mathcal{C}(\hat{K})}{\,\text{d}K_{r}}$, use the solution $g$
to the first two equations with kernel $K=\hat{K}$ and $f=f_{\hat{K}}$.

##  Appendix B Some a-priori estimates

By Assumption 1, semigroup theory yields the existence of a mild solution to
(1)–(2).

######  Lemma B.1.

Let Assumption 1 hold and assume
$h\in{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L^{1}}((0,T);L^{\infty}(\mathbb{R}\times
V))$. Then there exists a mild solution

$\displaystyle f\in
C^{0}\left([0,T];L^{\infty}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mathbb{R}\times}V)\right)$
(39)

to

$\displaystyle\partial_{t}f+v\cdot\nabla_{x}f$
$\displaystyle=\mathcal{K}(f)+h,$ $\displaystyle f(t=0,x,v)$
$\displaystyle=\phi(x,v)\in{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L^{\infty}}_{+}(\mathbb{R}\times
V)$

that is bounded

$\max_{v}\|f(t)\|_{L^{\infty}(\mathbb{R})}\leq
e^{2|V|C_{K}t}C_{\phi}+\int_{0}^{t}e^{2|V|C_{K}(t-s)}\|h(s)\|_{L^{\infty}(\mathbb{R}\times
V)}\,\text{d}s.$

We carry out the proof once to make clear, how the constant in the bound is
derived.

###### Proof.

Rewrite (1) as

$\partial_{t}f=\mathcal{A}f+\mathcal{B}f+h$

with operators
$\mathcal{A}:\mathcal{D}(\mathcal{A})\to\mathcal{X},f\mapsto-v\cdot\nabla_{x}f$
and $\mathcal{B}:\mathcal{X}\to\mathcal{X},f\mapsto\mathcal{K}(f)$, where the
function spaces
$\mathcal{D}(\mathcal{A}):=W^{1,\infty}(\mathbb{R};L^{\infty}(V))$ and
$\mathcal{X}:=L^{\infty}(\mathbb{R}\times V)$ are used. The transport operator
$\mathcal{A}$ generates a strongly continuous semigroup $T(t)u(x)=u(x-vt)$
with operator norm $\|T(t)\|\leq 1$. Clearly, $\mathcal{B}$ is bounded in
operator norm by $2|V|C_{K}$. The bounded perturbation theorem, see e.g. Engel
and Nagel (2001), shows that $\mathcal{A}+\mathcal{B}$ generates a strongly
continuous semigroup $S$ with $\|S(t)\|\leq e^{2|V|C_{K}t}$. As
$\phi\in\mathcal{X}$, (1) admits a mild solution

$f(t)=S(t)\phi+\int_{0}^{t}S(t-s)h(s)\,\text{d}s.$

∎

The regularity of the solution of (1)–(2) is improved by more regular initial
data. This is exploited in the proof of ill-conditioning for pointwise
measurement closeness in Theorem 3.2.

######  Corollary B.1.

Let Assumption 1 hold.

  1. a)

Equation (1) has a mild solution $f$ is bounded

$\max_{v}\|f(t)\|_{L^{\infty}(\mathbb{R})}\leq e^{2|V|C_{K}t}C_{\phi}\leq
e^{2|V|C_{K}T}C_{\phi}=:C_{f}.$ (40)

  2. b)

If, additionally, the initial data $\phi$ is uniformly continuous in $x$,
uniformly in $v$, then $f$ is uniformly continuous in $x$, uniformly in $v,t$,
i.e. $\max_{v}|f(t,x,v)-f(t,y,v)|<\varepsilon$ for all $t\in[0,T]$, if
$|x-y|<\delta(\varepsilon)$.

###### Proof.

Assertion a) is a direct consequence of lemma B.1. We focus on proving
assertion b). Let $\varepsilon>0$. By uniform continuity of $\phi$ in $x$, one
can choose $\delta^{\prime}$ such that

$\displaystyle\operatorname*{ess\sup}_{|x-y|<\delta^{\prime},v}|\phi(x,v)-\phi(y,v)|<e^{-2C_{K}|V|T}\varepsilon/2.$
(41)

Now consider $\delta:=\min\left(\delta^{\prime},\frac{\varepsilon
e^{-2C_{K}|V|T}}{8C_{f}|V|C_{K}(R-1)}\right)$. Integration along
characteristics yields

$\displaystyle\operatorname*{ess\sup}\limits_{|x-y|<\delta,v}|f(t,x,v)-f(t,y,v)|$
$\displaystyle\leq\operatorname*{ess\sup}_{|x-y|<\delta,v}|\phi(x-vt,v)-\phi(y-vt,v)|$
$\displaystyle\hskip
14.22636pt+\operatorname*{ess\sup}_{|x-y|<\delta,v}\bigg{|}\int_{0}^{t}\mathcal{K}(f)(t-s,x-vs,v)-\mathcal{K}(f)(t-s,y-vs,v)\,\text{d}s\bigg{|}$
$\displaystyle\leq\operatorname*{ess\sup}_{|x-y|<\delta,v}|\phi(x,v)-\phi(y,v)|$
$\displaystyle\hskip
14.22636pt+2C_{K}|V|\int_{0}^{t}\operatorname*{ess\sup}_{|x-y|<\delta,v^{\prime}}|f(s,x,v^{\prime})-f(s,y,v^{\prime})|\,\text{d}s$
$\displaystyle\hskip
14.22636pt+2C_{f}|V|\operatorname*{ess\sup}_{|x-y|<\delta,v}\int_{0}^{t}\max_{v^{\prime},v^{\prime\prime}}|K(x-vs,v^{\prime},v^{\prime\prime})-K(y-vs,v^{\prime},v^{\prime\prime})|\,\text{d}s$
$\displaystyle=:(i)+(ii)+(iii),$

where $(i)\leq\frac{\varepsilon}{2}e^{-2C_{K}|V|T}$ by (41). By symmetry,
$(iii)=2\cdot(iv)$ where $(iv)$ coincides with $(iii)$, but $x\geq y$. As $K$
is a step function in space (3), its difference can only be non zero if a jump
occurred between $x-vs$ and $y-vs$. Boundedness of $K$ in (12) then lead to
the estimate

$\displaystyle(iii)=2\cdot(iv)$ $\displaystyle\leq 2\cdot
2C_{f}|V|\operatorname*{ess\sup}_{|x-y|<\delta,v}\int_{0}^{t}C_{K}\sum_{r=1}^{R-1}\mathds{1}_{(x-vs,y-vs]}(a_{r})\,\text{d}s$
(42) $\displaystyle\leq
4C_{f}|V|C_{K}(R-1)\delta\leq\frac{\varepsilon}{2}e^{-2C_{K}|V|T}.$

In summary, Gronwall’s lemma yields

$\displaystyle\operatorname*{ess\sup}\limits_{|x-y|<\delta,v}|f(t,x,v)-f(t,y,v)|\leq\varepsilon
e^{-2C_{K}|V|(T-t)}\leq\varepsilon.$

∎

Again, semigroup theory shows existence of the adjoint equation (10) with
corresponding bounds.

######  Lemma B.2.

Let
$h\in{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L^{1}}((0,T);L^{\infty}(V;L^{1}(\mathbb{R})))$,
$\psi\in{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L^{1}}(\mathbb{R})$
and let (12) hold. Then the equation

$\displaystyle-\partial_{t}g-v\cdot\nabla_{x}g$
$\displaystyle=\alpha\tilde{\mathcal{L}}(g)-\sigma g+h,$ (43) $\displaystyle
g(t=T)$ $\displaystyle=\psi(x)$

with $\alpha\in\\{0,1\\}$ and $\tilde{\mathcal{L}}(g):=\int
K(x,v^{\prime},v)g(x,t,v^{\prime})\,\text{d}v^{\prime}$ and $\sigma(x,v):=\int
K(x,v^{\prime},v)\,\text{d}v^{\prime}$ has a mild solution

$\displaystyle g\in
C^{0}\left([0,T];L^{\infty}(V;{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}L^{1}}(\mathbb{R}))\right)$
(44)

that satisfies

$\displaystyle\|g(t)\|_{L^{\infty}(V;L^{1}(\mathbb{R}))}\leq
e^{(1+\alpha)|V|C_{K}(T-t)}\left(\|\psi\|_{L^{1}(\mathbb{R})}+\int_{0}^{T-t}\max_{v}\|h(T-s,v)\|_{L^{1}(\mathbb{R})}\,\text{d}s\right).$
(45)

If, additionally, $h\in L^{\infty}([0,T]\times V;L^{1}(\mathbb{R}))$, then

$\displaystyle\|g(t)\|_{L^{\infty}(V;L^{1}(\mathbb{R}))}$ (46)
$\displaystyle\quad\leq
e^{(1+\alpha)|V|C_{K}(T-t)}\|\psi\|_{L^{1}(\mathbb{R})}+\frac{e^{(1+\alpha)|V|C_{K}(T-t)}-1}{(1+\alpha)|V|C_{K}}\operatorname*{ess\sup}_{t,v}\|h(t,v)\|_{L^{1}(\mathbb{R})}.$

###### Proof.

Repeating the arguments in the proof of Lemma B.1, semigroup theory yields the
existence of a mild solution

$\displaystyle g(t)=S(T-t)\psi+\int_{0}^{T-t}S(T-t-s)h(T-s)\,\text{d}s$

for the semigroup $S(t)$ generated by the operator
$v\cdot\nabla_{x}+\alpha\tilde{\mathcal{L}}-\sigma$ with $\|S(t)\|\leq
e^{(1+\alpha)|V|C_{K}t}$. This yields (45) and (46). ∎

##  Appendix C Proof of Lemma 3.7-3.8

In this section, we provide the proof for the two Lemmas in section 3.2. In
particular, Lemma 3.7 discusses the smallness of the first term in (28).

###### Proof for Lemma 3.7.

By the assumption on the initial data and Corollary B.1 b), $f$ is uniformly
continuous in $x$, uniformly in $v,t$. For $n=0$, the boundedness (29) is a
consequence of the explicit representation

$\displaystyle\bar{g}_{0}(t,x,v_{0})=e^{-\int_{0}^{T-t}\sigma(x+v_{0}\tau,v_{0})\,\text{d}\tau}(\mu_{2}^{\eta}-\mu_{1}^{\eta})(x+v_{0}(T-t))$
(47)

together with the step function shape (3) of $K$, the continuity of $f$ and
our assumptions: Write
$p_{0}(t,x,v_{0},v^{\prime}):=f(x,t,v^{\prime})e^{-\int_{0}^{T-t}\sigma(x+v_{0}\tau,v_{0})\,\text{d}\tau}$
and assume without loss of generality $x_{1}\geq x_{2}$, then

$\displaystyle\int_{I_{r}}f^{\prime}\bar{g}_{0}\,\text{d}x$
$\displaystyle=\int_{I_{r}}p_{0}(t,x,v_{0},v^{\prime})(\mu_{2}^{\eta}-\mu_{1}^{\eta})(x+v_{0}(T-t))\,\text{d}x$
$\displaystyle=-\int_{a_{r-1}-(x_{1}-x_{2})}^{a_{r-1}}p_{0}(t,x+(x_{1}-x_{2}),v_{0},v^{\prime})\mu_{2}^{\eta}(x+v_{0}(T-t))\,\text{d}x$
$\displaystyle\quad+\int_{a_{r}-(x_{1}-x_{2})}^{a_{r}}p_{0}(t,x,v_{0},v^{\prime})\mu_{2}^{\eta}(x+v_{0}(T-t))\,\text{d}x$
$\displaystyle\quad+\int_{a_{r-1}}^{a_{r}-(x_{1}-x_{2})}(p_{0}(t,x,v_{0},v^{\prime})-p_{0}(t,x+(x_{1}-x_{2}),v_{0},v^{\prime}))\mu_{2}^{\eta}(x+v_{0}(T-t))\,\text{d}x,$

where we used the substitution $x\to x-(x_{1}-x_{2})$ for the integration
domain of test function $\mu_{1}^{\eta}(x)=\mu_{2}^{\eta}(x-(x_{1}-x_{2}))$.
By uniform continuity and boundedness of $f$ a similar argumentation as in
(42) shows that $p_{0}(t,x,v_{0},v^{\prime})$ is uniformly continuous in $x$,
uniformly in $t,v_{0},v^{\prime}$, as well. The corresponding threshold from
the epsilon-delta criterion is denoted by $\delta_{p_{0}}(\varepsilon)$. Then,
for
$0\leq|x_{1}-x_{2}|<\delta_{0}(\varepsilon):=\min(\min_{r}|a_{r}-x_{2}|-T-\eta_{0},\delta_{p_{0}}(\varepsilon))$,
the first two integrals vanish, because $\mu_{2}^{\eta}(x+v_{0}(T-t))=0$ for
all $x$ in the integration domain. We are left with

$\displaystyle\left|\int_{I_{r}}f^{\prime}\bar{g}_{0}\,\text{d}x\right|$
$\displaystyle\leq\int\limits_{a_{r-1}}^{a_{r}-(x_{1}-x_{2})}|p_{0}(t,x,v_{0},v^{\prime})-p_{0}(t,x+(x_{1}-x_{2}),v_{0},v^{\prime})|\mu_{2}^{\eta}(x+v_{0}(T-t))\,\text{d}x$
$\displaystyle\leq\varepsilon\int_{\mathbb{R}}\mu_{2}^{\eta}(x+v_{0}(T-t))\,\text{d}x=\varepsilon.$

For $n\geq 1$, source iteration shows that the solution to (26) has the form

$\displaystyle\bar{g}_{n}(t,x,v_{0})=$
$\displaystyle\int\limits_{0}^{T-t}\int_{V}...\int\limits_{0}^{T-t-\sum_{j=0}^{n-2}s_{j}}\int_{V}p_{n}(t,x,(v_{i})_{i=0,...,n},(s_{j})_{j=0,...,n-1})\cdot$
$\displaystyle\hskip
28.45274pt(\mu_{2}-\mu_{1})\left(x+\sum_{l=0}^{n-1}v_{l}s_{l}+v_{n}\left(T-t-\sum_{l=0}^{n-1}s_{l}\right)\right)\,\text{d}v_{n}\,\text{d}s_{n-1}...\,\text{d}v_{1}\,\text{d}s_{0}\,.$

The function $p_{n}$ is bounded $0\leq p_{n}\leq C_{K}^{n}$ and satisfies

$\displaystyle\int_{0}^{T}|p_{n}(t,x+v_{n}t,(v_{i})_{i},(s_{j})_{j})-p_{n}(t,y+v_{n}t,(v_{i})_{i},(s_{j})_{j})|\,\text{d}t<\varepsilon$

for $|x-y|<\delta_{p_{n}}(\varepsilon)$, uniformly in
$(v_{i})_{i},(s_{j})_{j}$. The assertion then follows in analogy to the case
$n=0$. ∎

Lemma 3.8 argues the smallness of the second term in (28). We provide the
proof below. It is a consequence of the smallness of $\bar{g}_{>N}$ by Lemma
B.2 and the boundedness of $f$.

###### Proof for Lemma 3.8.

Application of lemma B.2 to
$g=\bar{g}_{>N},h=\tilde{\mathcal{L}}\bar{g}_{N},\alpha=1$ and $\psi=0$ yields

$\displaystyle\max_{v}\int_{\mathbb{R}}|\bar{g}_{>N}(t)|\,\text{d}x$
$\displaystyle\leq
e^{2C_{K}|V|(T-t)}\int_{0}^{T-t}\sup_{v}\|\tilde{\mathcal{L}}(\bar{g}_{N})(T-s,v)\|_{L^{1}(\mathbb{R})}\,\text{d}s$
$\displaystyle\leq|V|C_{K}(T-t)e^{2C_{K}|V|(T-t)}\operatorname*{ess\sup}_{s,v}\|\bar{g}_{N}(s,x,v)\|_{L^{1}(\mathbb{R})}.$

Now, application of the same lemma to the evolution equation (26) for $g_{n}$,
$n=1,...,N$, shows

$\displaystyle\operatorname*{ess\sup}_{t,v}\int_{\mathbb{R}}|\bar{g}_{n}|\,\text{d}x\leq(e^{C_{K}|V|T}-1)\operatorname*{ess\sup}_{s,v}\int_{\mathbb{R}}|\bar{g}_{n-1}(s,x,v)|\,\text{d}x.$

The boundedness of $f$ in (40) and repeated application of the above estimate
lead to

$\displaystyle\left|\int_{0}^{T}\max_{v}\int_{\mathbb{R}}f^{\prime}\bar{g}_{>N}\,\text{d}x\,\text{d}t\right|$
$\displaystyle\leq\frac{T^{2}}{2}|V|C_{K}C_{\phi}e^{2|V|C_{K}T}(e^{C_{K}|V|T}-1)^{N}\operatorname*{ess\sup}_{s,v}\int_{\mathbb{R}}|\bar{g}_{0}(s,x,v)|\,\text{d}x$
$\displaystyle\leq\frac{T^{2}}{2}|V|C_{K}C_{\phi}e^{2|V|C_{K}T}\left(e^{C_{K}|V|T}-1\right)^{N}\operatorname*{ess\sup}_{s,v}\int_{\mathbb{R}}|(\mu_{2}^{\eta}-\mu_{1}^{\eta})(x+vs)|\,\text{d}x$
$\displaystyle\leq
T^{2}|V|C_{K}C_{\phi}e^{2|V|C_{K}T}(e^{C_{K}|V|T}-1)^{N}C_{\mu},$

where $|\bar{g}_{0}(s,x,v)|\leq|(\mu_{2}^{\eta}-\mu_{1}^{\eta})(x+vs)|$ can be
observed from the explicit formula for $\bar{g}_{0}$ in (47). ∎

##  Appendix D Proof of Lemmas in Section 4

We provide proofs for Lemma 4.5-4.6 in this section.

###### Proof of Lemma 4.5.

Use the explicit representations

$\displaystyle g_{1}^{(0)}(t,x,v)$
$\displaystyle=e^{-(T-t)\sigma_{1}(v)}\mu_{1}(x+v(T-t)),$ (48) $\displaystyle
f^{(0)}(t,x,v)$ $\displaystyle=e^{-t\sigma_{1}(v)}\phi(x-vt)$ (49)

with $\sigma_{1}(v)=\int_{V}K_{1}(v^{\prime},v)\,\text{d}v^{\prime}$ and set
without loss of generality $c_{1}=0$. Since $f^{(0)}|_{I_{1}}=f^{(0)}_{1}$ in
the notation of the proof of Proposition 4.1, one obtains for
$(v,v^{\prime})=(+1,-1)$

$\displaystyle\int_{0}^{T}\int_{I_{1}}f^{(0)}(v^{\prime})(g_{1}^{(0)}(v^{\prime})-g_{1}^{(0)}(v))\,\text{d}x\,\text{d}t$
$\displaystyle=\int_{0}^{T}\int_{I_{1}}e^{-t\sigma_{1}(v^{\prime})}\phi_{1}(x-v^{\prime}t)\big{(}e^{-(T-t)\sigma_{1}(v^{\prime})}\mu_{1}(x+v^{\prime}(T-t))$
$\displaystyle\hskip
133.72786pt-e^{-(T-t)\sigma_{1}(v)}\mu_{1}(x+v(T-t))\big{)}\,\text{d}x\,\text{d}t$
$\displaystyle\geq
e^{-T\sigma_{1}(-1)}T\int_{a_{0}+T}^{a_{1}}\phi_{1}(x)\mu_{1}(-T+x)\,\text{d}x-\int_{T-\frac{d_{\mu}+d}{2}}^{T}\int_{I_{1}}\phi_{1}(x)\mu_{1}(-T+x)\,\text{d}x\,\text{d}t$
$\displaystyle\geq e^{-TC_{K}|V|}TC_{\phi\mu}-\frac{d_{\mu}+d}{2}C_{\phi\mu},$

where the first inequality is due to the fact that
$\phi_{1}(x-v^{\prime}t)\mu_{1}(x+v(T-t))=\phi_{1}(x+t)\mu_{1}(x+(T-t))\neq 0$
only for $x\in[-t-d,-t+d]\cap[-2T+t-d_{\mu},-2T+t+d_{\mu}]\subset I_{1}$ which
is empty for $t\leq T-\frac{d_{\mu}+d}{2}$.
For $(v^{\prime},v)=(-1,+1)$, instead, we obtain

$\displaystyle\left|\int_{0}^{T}\int_{I_{1}}f^{(0)}(v)(g_{1}^{(0)}(v)-g_{1}^{(0)}(v^{\prime}))\,\text{d}x\,\text{d}t\right|$
$\displaystyle=\bigg{|}\int_{0}^{T}\int_{I_{1}}e^{-t\sigma_{1}(v)}\phi_{1}(x-vt)\big{(}e^{-(T-t)\sigma_{1}(v)}\mu_{1}(x+v(T-t))$
$\displaystyle\hskip
133.72786pt-e^{-(T-t)\sigma_{1}(v^{\prime})}\mu_{1}(x+v^{\prime}(T-t))\big{)}\,\text{d}x\,\text{d}t\bigg{|}$
$\displaystyle\leq C_{\phi\mu}\frac{d+d_{\mu}}{2}$

since

  * •

$\phi_{1}(x-vt)\mu_{1}(x+v(T-t))=\phi_{1}(x-t)\mu_{1}(x+T-t)$ vanishes, as its
support $[t-d,t+d]\cap[-2T+t-{d_{\mu}}{},-2T+t+{d_{\mu}}{}]=\emptyset$ is
empty by construction of $T>d\geq d_{\mu}$ and

  * •

the support $[t-d,t+d]\cap[-t-{d_{\mu}}{},-t+{d_{\mu}}{}]$ of
$\phi_{1}(x-vt)\mu_{1}(x+v^{\prime}(T-t))=\phi_{1}(x-t)\mu_{1}(x-(T-t))$ is
non-empty only for $t\leq\frac{d+d_{\mu}}{2}$.

Since $e^{-TC_{K}|V|}-\frac{d_{\mu}+d}{T}>0$ by assumption, this proves the
assertion. ∎

To show inequality (37) in Lemma 4.6, decompose for some $N\in\mathbb{N}$ to
be determined later

$\displaystyle S=$ $\displaystyle\sum_{\begin{subarray}{c}n,k=0\\\ n+k\geq
1\end{subarray}}^{N}\int_{0}^{T}\int_{I_{1}}f^{(k)}(v^{\prime})(g_{1}^{(n)}(v^{\prime})-g_{1}^{(n)}(v))\,\text{d}x\,\text{d}t$
$\displaystyle+\int_{0}^{T}\int_{I_{1}}f(v^{\prime})(g_{1}^{(>N)}(v^{\prime})-g_{1}^{(>N)}(v))\,\text{d}x\,\text{d}t$
(50)
$\displaystyle+\sum_{n=0}^{N}\int_{0}^{T}\int_{I_{1}}f^{(>N)}(v^{\prime})(g_{1}^{(n)}(v^{\prime})-g_{1}^{(n)}(v))\,\text{d}x\,\text{d}t\,,$

where $g_{1}^{(n)}$ and $g_{1}^{(>N)}$ solve (26) and (27) respectively and
$f^{(k)}$ are solutions to

$\displaystyle\partial_{t}f^{(k)}-v\cdot\nabla_{x}f^{(k)}$
$\displaystyle=\mathcal{L}(f^{(k-1)})-\sigma f^{(k)},$ $\displaystyle
f^{(k)}(t=0,x,v)$ $\displaystyle=0,$

with
$\mathcal{L}(h):=\int_{V}K(v,v^{\prime})h(t,x,v^{\prime})\,\text{d}v^{\prime}$,
and $f^{(>N)}$ satisfies

$\displaystyle\partial_{t}f^{(>N)}-v\cdot\nabla_{x}f^{(>N)}$
$\displaystyle=\mathcal{L}(f^{(N)}+f^{(>N)})-\sigma f^{(>N)},$ $\displaystyle
f^{(>N)}(t=0,x,v)$ $\displaystyle=0.$

Each part of $S$ in representation (D) is estimated separately in the
subsequent three lemmas.

######  Lemma D.1.

In the setting of proposition 4.2,

$\displaystyle\left|\int_{0}^{T}\int_{I_{1}}f^{(k)}(v^{\prime})(g_{1}^{(n)}(v^{\prime})-g_{1}^{(n)}(v))\,\text{d}x\,\text{d}t\right|\leq
2\max_{v,v^{\prime}}\int_{0}^{T}\int_{I_{1}}f^{(k)}(v^{\prime})g_{1}^{(n)}(v)\,\text{d}x\,\text{d}t$
$\displaystyle\leq 2\left(C_{K}|V|\right)^{n+k}T^{n+k+1}C_{\phi\mu}$

###### Proof.

Source iteration

$\displaystyle g_{1}^{(n)}(t,x,v_{0})$
$\displaystyle=\int_{0}^{T-t}\int_{V}e^{-s_{0}\sigma(v_{0})}K_{1}(\hat{v}_{1},v_{0})g_{1}^{(n-1)}(t+s_{0},x+v_{0}s_{0},\hat{v}_{1})\,\text{d}\hat{v}_{1}\,\text{d}s_{0}$
$\displaystyle\leq|V|\int_{0}^{T-t}e^{-s_{0}\sigma(v_{0})}K_{1}(v_{1},v_{0})g_{1}^{(n-1)}(t+s_{0},x+v_{0}s_{0},v_{1})\,\text{d}s_{0},$
$\displaystyle f^{(k)}(t,x,v_{0})$
$\displaystyle=\int_{0}^{t}\int_{V}e^{-s_{0}\sigma(v_{0})}K(v_{0},\hat{v}_{1})f^{(k-1)}(t-s_{0},x-v_{0}s_{0},\hat{v}_{1})\,\text{d}\hat{v}_{1}\,\text{d}s_{0}$
$\displaystyle\leq|V|\int_{0}^{t}e^{-s_{0}\sigma(v_{0})}K(v_{0},v_{1})f^{(k-1)}(t-s_{0},x-v_{0}s_{0},v_{1})\,\text{d}s_{0},$

where $v_{1}=-v_{0}$, together with the explicit formulas (48)–(49) leads to
estimates

$\displaystyle 0\leq g_{1}^{(n)}(x,t,v_{0})\leq$
$\displaystyle\left(C_{K}|V|\right)^{n}\int\limits_{0}^{T-t}...\int\limits_{0}^{T-t-\sum_{i=0}^{n-2}s_{i}}\mu_{1}\left(x+\sum_{i=0}^{n-1}v_{i}s_{i}+v_{n}\left(T-t-\sum_{i=0}^{n-1}s_{i}\right)\right)$
(51) $\displaystyle\hskip 227.62204pt\,\text{d}s_{n-1}...\,\text{d}s_{0},$
$\displaystyle 0\leq f^{(k)}(x,t,v_{0})\leq$
$\displaystyle\left({C_{K}}|V|\right)^{k}\int\limits_{0}^{t}...\int\limits_{0}^{t-\sum_{i=0}^{k-2}s_{i}}\phi\left(x-\sum_{i=0}^{k-1}v_{i}s_{i}+v_{k}\left(t-\sum_{i=0}^{k-1}s_{i}\right)\right)\,\text{d}s_{k-1}...\,\text{d}s_{0}.$

Using again $f^{(k)}|_{I_{1}}=f_{1}^{(k)}$ with initial condition $\phi_{1}$
in the notation of the proof of Porposition 4.1, this proves

$\displaystyle\left|\int_{0}^{T}\int_{I_{1}}f^{(k)}(v^{\prime})(g_{1}^{(n)}(v^{\prime})-g_{1}^{(n)}(v))\,\text{d}x\,\text{d}t\right|\leq
2\max_{v,v^{\prime}}\int_{0}^{T}\int_{I_{1}}f^{(k)}_{1}(v^{\prime})g_{1}^{(n)}(v)\,\text{d}x\,\text{d}t$
$\displaystyle\leq 2\left({C_{K}}|V|\right)^{n+k}T^{n+k+1}C_{\phi\mu}.$

∎

The following bound for the second summand in (D) is obtained in analogy to
Lemma 3.8.

######  Lemma D.2.

In the setting of Proposition 4.2,

$\displaystyle\max_{v}\left|\iint
f(v^{\prime})(g_{1}^{(>N)}(v^{\prime})-g^{(>N)}_{1}(v))\,\text{d}x\,\text{d}t\right|$
$\displaystyle\leq
4T^{2}|V|C_{K}C_{\phi}e^{2|V|C_{K}T}(e^{C_{K}|V|T}-1)^{N}\bar{C}_{\mu}d_{\mu}=:C^{\prime}(T)(e^{C_{K}|V|T}-1)^{N}$

For the third term in (D), one establishes the following bound.

######  Lemma D.3.

In the setting of Proposition 4.2,

$\displaystyle\max_{v}\left|\iint
f^{(>N)}(v^{\prime})(g^{(n)}(v^{\prime})-g^{(n)}(v))\,\text{d}x\,\text{d}t\right|$
$\displaystyle\leq
4|V|C_{K}T^{2}e^{2|V|C_{K}T}(e^{C_{K}|V|T}-1)^{N}C_{\phi}\left({C_{K}|V|T}\right)^{n}\bar{C}_{\mu}d_{\mu}$
$\displaystyle=:C^{\prime\prime}(T)(e^{C_{K}|V|T}-1)^{N}\left({C_{K}|V|T}\right)^{n}$

###### Proof.

An estimate for $f^{(>N)}$ can be derived analogously as the estimate for
$\bar{g}_{>N}$ in Lemma 3.8 from Lemma B.1

$\displaystyle\|f^{(>N)}\|_{L^{\infty}([0,T]\times\mathbb{R}\times
V)}\leq|V|C_{K}Te^{2|V|C_{K}T}(e^{C_{K}|V|T}-1)^{N}C_{\phi}.$

Together with (51), this proves the lemma. ∎

Lemma 4.6 can now be assembled from the previous lemmas.

###### Proof of Lemma 4.6.

Lemmas D.1, D.2 and D.3 yield the $(v,v^{\prime})$ independent bound

$\displaystyle|S|$ $\displaystyle\leq
2C_{\phi\mu}T\sum_{\begin{subarray}{c}n,k=0\\\ n+k\geq
1\end{subarray}}^{N}\left({C_{K}|V|T}\right)^{n+k}+(e^{C_{K}|V|T}-1)^{N}\left(C^{\prime}(T)+C^{\prime\prime}(T)\sum_{n=0}^{N}\left({C_{K}|V|T}\right)^{n}\right)$
$\displaystyle\leq
4C_{\phi\mu}T\frac{C_{K}|V|T}{(1-C_{K}|V|T)^{2}}+(e^{C_{K}|V|T}-1)^{N}\left(C^{\prime}(T)+C^{\prime\prime}(T)\frac{1}{1-C_{K}|V|T}\right)$
$\displaystyle=:4C_{\phi\mu}T\frac{C_{K}|V|T}{(1-C_{K}|V|T)^{2}}+(e^{C_{K}|V|T}-1)^{N}C(T).$

Because $e^{C_{K}|V|T}-1<1$ due to the assumption
$T<(1-\delta)\frac{0.09}{C_{K}|V|}$, the second term in the last line becomes
arbitrarily small for large $N\in\mathbb{N}$, which shows that $|S|$ is in
fact bounded by the first term. ∎

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