# AI-assisted Optimization of the ECCE Tracking System at the Electron Ion
Collider

C. Fanelli  Z. Papandreou  K. Suresh  J. K. Adkins  Y. Akiba  A. Albataineh
M. Amaryan  I. C. Arsene  C. Ayerbe Gayoso  J. Bae  X. Bai  M.D. Baker  M.
Bashkanov  R. Bellwied  F. Benmokhtar  V. Berdnikov  J. C. Bernauer  F. Bock
W. Boeglin  M. Borysova  E. Brash  P. Brindza  W. J. Briscoe  M. Brooks  S.
Bueltmann  M. H. S. Bukhari  A. Bylinkin  R. Capobianco  W.-C. Chang  Y. Cheon
K. Chen  K.-F. Chen  K.-Y. Cheng  M. Chiu  T. Chujo  Z. Citron  E. Cline  E.
Cohen  T. Cormier  Y. Corrales Morales  C. Cotton  J. Crafts  C. Crawford  S.
Creekmore  C.Cuevas  J. Cunningham  G. David  C. T. Dean  M. Demarteau  S.
Diehl  N. Doshita  R. Dupré  J. M. Durham  R. Dzhygadlo  R. Ehlers  L. El
Fassi  A. Emmert  R. Ent  R. Fatemi  S. Fegan  M. Finger  M. Finger Jr  J.
Frantz  M. Friedman  I. Friscic  D. Gangadharan  S. Gardner  K. Gates  F.
Geurts  R. Gilman  D. Glazier  E. Glimos  Y. Goto  N. Grau  S. V. Greene  A.
Q. Guo  L. Guo  S. K. Ha  J. Haggerty  T. Hayward  X. He  O. Hen  D. W.
Higinbotham  M. Hoballah  T. Horn  A. Hoghmrtsyan  P.-h. J. Hsu  J. Huang  G.
Huber  A. Hutson  K. Y. Hwang  C. Hyde  M. Inaba  T. Iwata  H.S. Jo  K. Joo
N. Kalantarians  G. Kalicy  K. Kawade  S. J. D. Kay  A. Kim  B. Kim  C. Kim
M. Kim  Y. Kim  Y. Kim  E. Kistenev  V. Klimenko  S. H. Ko  I. Korover  W.
Korsch  G. Krintiras  S. Kuhn  C.-M. Kuo  T. Kutz  J. Lajoie  D. Lawrence  S.
Lebedev  H. Lee  J. S. H. Lee  S. W. Lee  Y.-J. Lee  W. Li  W.B. Li  X. Li  X.
Li  X. Li  X. Li  Y. T. Liang  S. Lim  C.-h. Lin  D. X. Lin  K. Liu  M. X. Liu
K. Livingston  N. Liyanage  W.J. Llope  C. Loizides  E. Long  R.-S. Lu  Z. Lu
W. Lynch  D. Marchand  M. Marcisovsky  P. Markowitz  H. Marukyan  P. McGaughey
M. Mihovilovic  R. G. Milner  A. Milov  Y. Miyachi  A. Mkrtchyan  P. Monaghan
R. Montgomery  D. Morrison  A. Movsisyan  H. Mkrtchyan  A. Mkrtchyan  C. Munoz
Camacho  M. Murray  K. Nagai  J. Nagle  I. Nakagawa  C. Nattrass  D. Nguyen
S. Niccolai  R. Nouicer  G. Nukazuka  M. Nycz  V. A. Okorokov  S. Orešić  J.D.
Osborn  C. O’Shaughnessy  S. Paganis  S. F. Pate  M. Patel  C. Paus  G. Penman
M. G. Perdekamp  D. V. Perepelitsa  H. Periera da Costa  K. Peters  W. Phelps
E. Piasetzky  C. Pinkenburg  I. Prochazka  T. Protzman  M. L. Purschke  J.
Putschke  J. R. Pybus  R. Rajput-Ghoshal  J. Rasson  B. Raue  K.F. Read  K.
Røed  R. Reed  J. Reinhold  E. L. Renner  J. Richards  C. Riedl  T. Rinn  J.
Roche  G. M. Roland  G. Ron  M. Rosati  C. Royon  J. Ryu  S. Salur  N.
Santiesteban  R. Santos  M. Sarsour  J. Schambach  A. Schmidt  N. Schmidt  C.
Schwarz  J. Schwiening  R. Seidl  A. Sickles  P. Simmerling  S. Sirca  D.
Sharma  Z. Shi  T.-A. Shibata  C.-W. Shih  S. Shimizu  U. Shrestha  K. Slifer
K. Smith  D. Sokhan  R. Soltz  W. Sondheim  J. Song  J. Song  I. I. Strakovsky
P. Steinberg  P. Stepanov  J. Stevens  J. Strube  P. Sun  X. Sun  V.
Tadevosyan  W.-C. Tang  S. Tapia Araya  S. Tarafdar  L. Teodorescu  A. Timmins
L. Tomasek  N. Trotta  R. Trotta  T. S. Tveter  E. Umaka  A. Usman  H. W. van
Hecke  C. Van Hulse  J. Velkovska  E. Voutier  P.K. Wang  Q. Wang  Y. Wang  Y.
Wang  D. P. Watts  N. Wickramaarachchi  L. Weinstein  M. Williams  C.-P. Wong
L. Wood  M. H. Wood  C. Woody  B. Wyslouch  Z. Xiao  Y. Yamazaki  Y. Yang  Z.
Ye  H. D. Yoo  M. Yurov  N. Zachariou  W.A. Zajc  W. Zha  J. Zhang  Y. Zhang
Y. X. Zhao  X. Zheng  P. Zhuang

###### Abstract

The Electron-Ion Collider (EIC) is a cutting-edge accelerator facility that
will study the nature of the “glue” that binds the building blocks of the
visible matter in the universe. The proposed experiment will be realized at
Brookhaven National Laboratory in approximately 10 years from now, with
detector design and R&D currently ongoing. Notably, EIC is one of the first
large-scale facilities to leverage Artificial Intelligence (AI) already
starting from the design and R&D phases. The EIC Comprehensive Chromodynamics
Experiment (ECCE) is a consortium that proposed a detector design based on a
1.5T solenoid. The EIC detector proposal review concluded that the ECCE design
will serve as the reference design for an EIC detector. Herein we describe a
comprehensive optimization of the ECCE tracker using AI. The work required a
complex parametrization of the simulated detector system. Our approach dealt
with an optimization problem in a multidimensional design space driven by
multiple objectives that encode the detector performance, while satisfying
several mechanical constraints. We describe our strategy and show results
obtained for the ECCE tracking system. The AI-assisted design is agnostic to
the simulation framework and can be extended to other sub-detectors or to a
system of sub-detectors to further optimize the performance of the EIC
detector.

###### keywords:

ECCE , Electron Ion Collider , Tracking , Artificial Intelligence ,
Evolutionary Algorithms , Bayesian Optimization.

††journal: Nuclear Instruments and Methods A

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###### Contents

  1. 1 Introduction
  2. 2 AI-assisted Detector Design
  3. 3 ECCE Tracking System Simulation
  4. 4 Analysis Workflow
  5. 5 Computing Resources
  6. 6 Summary
  7. A Details on Parametrization
  8. B Baseline and R&D designs

##  1 Introduction

The Electron Ion Collider (EIC) [1] is a future cutting-edge discovery machine
that will unlock the secrets of the gluonic force binding the building blocks
of the visibile matter in the universe. The EIC will consist of two
intersecting accelerators, one producing an intense beam of electrons and the
other a beam of protons or heavier atomic nuclei; it will be the only
electron-nucleus collider operating in the world. The EIC Comprehensive
Chromodynamics Experiment (ECCE) [2] is an international consortium assembled
to develop a detector that can offer full energy coverage and an optimized far
forward detection region. ECCE has investigated a detector design based on the
existing BABAR 1.5T magnet; this detector will be ready for the beginning of
EIC operations. More details on the ECCE detector design and what is described
in the following can be found in [3].

ECCE is an integrated detector that extends for about 40 m, and includes a
central detector built around the interaction point and far-forward (hadron-
going direction) and far-backward (electron-going direction) regions [1]. To
fulfill the physics goals of the EIC, the central detector needs to be
hermetic and provide good particle identification (PID) over a large phase
space. The central detector itself consists of multiple sub-detectors: a
tracking system made by inner and outer tracker stations allows the
reconstruction of charged particles moving in the magnetic field; a system of
PID sub-detectors will cover the barrel and the electron-going and hadron-
going directions; electromagnetic and hadronic calorimeters are used to detect
showers and provide complete information on the particle flow which is
essential for certain event topologies, e.g., those containing jets.

As outlined in [1], Artificial Intelligence (AI) can provide dedicated
strategies for complex combinatorial searches and can handle multi-objective
problems characterized by a multidimensional design space, allowing the
identification of hidden correlations among the design parameters. ECCE
included these techniques in the design workflow during the detector proposal.
At first this AI-assisted design strategy was used to steer the design. After
the base technology is selected using insights provided by AI, its detector
parameters can be further fine-tuned using AI. During the ECCE detector
proposal stage, the design of the detector underwent a continual optimization
process [4].

The article is structured as follows: in Sec. 2 we provide an overview of
design optimization and describe the AI-assisted strategy; in Sec. 3 we
introduce the ECCE tracker and describe the software stack utilized in this
work to which AI is coupled for the optimization; in Sec. 4 we describe the
implemented pipeline that results in a sequential strategy, fostering the
interplay between the different working groups in a post hoc decision making
process; in Sec. 5 we present perspectives and planned activities.

The ECCE detector at the EIC will be one of the first examples of detectors
that will be realized leveraging AI during the design and R&D phases.

##  2 AI-assisted Detector Design

Detector optimization with AI is anticipated to continue in the months
following the detector proposal towards CD-2 and CD-3. Optimizing the design
of large-scale detectors such as ECCE—that are made of multiple sub-detector
systems—is a complex problem. Each sub-detector system is characterized by a
multi-dimensional design parameter space. In addition, detector simulations
are typically computationally intensive, and rely on advanced simulation
platforms used in our community such as Geant4 [5] to simulate the interaction
of radiation with matter. Additional computationally expensive steps are
present along the data reconstruction and analysis pipeline. The software
stack that is utilized in the detector design process involves three main
steps: (i) generation of events, (ii) detector simulations and (iii)
reconstruction and analysis.

As pointed out in [6], the above bottlenecks render the generation and
exploration of mutliple design points cumbersome. This in turn represents an
obstacle for deep learning (DL)-based approaches that learn the mapping
between the design space and the functional space [7, 8, 9], which could
facilitate the identification of optimal design points. In principle fast
simulations with DL can reduce the most CPU-intensive parts of the simulation
and provide accurate results [10], although several design points need to be
produced with Geant4 before injection in any DL architecture. Similar
considerations exist in deploying DL for reconstruction during the design
optimization process.

In this context, a workflow for detector design that has gained popularity in
recent years [11] is represented by the schematic in Fig. 1.

Figure 1: Workflow of detector design assisted by AI: physics events are
injected in a detector characterized by given design parameters. Reconstructed
events are analyzed and figures of merit are quantified and passed to an AI-
based strategy, which in turn suggests the next design point in this
sequential approach; note that AI can also intervene in the simulation and
reconstruction steps.

It consists of a sequential AI-based strategy that collects information
associated to previously generated design points, in the form of figures of
merit (called objectives in the following) that quantify the goodness of the
design, and which suggests promising new design points for the next iteration.

The ECCE AI Working Group achieved a continual multi-objective optimization
(MOO) of the tracker design. Our approach deals with a complex optimization in
a multidimensional design space (describing, e.g., geometry, mechanics,
optics, etc) driven by multiple objectives that encode the detector
performance, while satisfying several mechanical constraints. This framework
has been developed in a way that can be easily extended to other sub-detectors
or to a system of sub-detectors.

The definition of a generic MOO problem can be formulated as follows:

$\displaystyle min\ \mathbf{f_{m}}(\mathbf{x})$ $\displaystyle\ \ \ \ \ \
m=1,\cdots,M$ (1) $\displaystyle s.t.\ \ \ \mathbf{g_{j}}(\mathbf{x})\leq 0,$
$\displaystyle\ \ \ \ \ \ j=1,\cdots,J$ $\displaystyle\ \ \ \ \
\mathbf{h_{k}}(\mathbf{x})=0,$ $\displaystyle\ \ \ \ \ \ k=1,\cdots,K$
$\displaystyle x_{i}^{L}\leq x_{i}\leq x_{i}^{U},$ $\displaystyle\ \ \ \ \ \
i=1,\cdots,N$

where one has $M$ objective functions $f_{m}$ to optimize (e.g., detector
resolution, efficiency, costs), subject to $J$ inequalities $g_{j}(x)$ and $K$
equality constraints $h_{k}(z)$ (e.g., mechanical constraints), in a design
space of $N$ dimensions (e.g., geometry parameters that change the Geant4
design) with lower and upper bounds on each dimension.111Constraints are
described later in Table 3. Notice that overlaps in the design are checked
before and during the optimization and are excluded by the constraints and
ranges of the parameters. In solving these problems, one can come up with a
set of non-dominated or trade-off solutions [12], popularly known as Pareto-
optimal solutions (see also Fig. 2).

Figure 2: Example of Pareto frontier in a two-dimensional objective space: The
point C is not on the frontier and is dominated by both point A and point B.
All the other points which are dominated by the Pareto frontier and that
satisfy any constraints in the optimization problem are called feasible
solutions; The hypervolume is used as a metric for convergence, and is
calculated with respect to a reference point $r$.

In this setting, we used a recently developed framework for MOO called pymoo
[13] which supports evolutionary MOO algorithms such as Non-Dominated Sorting
Genetic Algorithm (or NSGA-II, [14]).222The pymoo framework also supports
other MOO approaches and a full list is documented in [13]. The rationale
behind this choice instead of, for example, principled approaches such as
Bayesian Optimization [11], emanates from the ECCE needs at the time of the
detector proposal, such as the capability to quickly implement and run
multiple parallel optimization pipelines implementing different technology
choices and the possibility of dealing with non-differentiable objectives at
the exploratory stage.

The NSGA workflow is described in Fig. 3.

Figure 3: The NSGA Workflow: At time $t$, an offspring $Q(t)$ is created
through a genetic algorithm [15] from an $N-$sized population of design points
$P(t)$. The two populations are combined into an augmented population which is
classified into different non-dominated classes $F_{i}$, starting from the
first front $F_{1}$. To restore the initial size of the population, the
augmented space of solutions is trimmed. A metric called crowding distance is
used to reject solutions and eventually provide an updated population of size
$N$ at time $t+1$.

The main features of NSGA-II are (i) the usage of an elitist principle, (ii)
an explicit diversity preserving mechanism, and (iii) ability of determining
non-dominated solutions. The latter feature is of great importance for
problems where objectives are of conflict with each other: that is an improved
performance in an objective results in worse performance in another objective.
For our purposes, we also tested NSGA-III which is suitable for the
optimization of large number of objectives [16].333For $\geq$ 4 objectives,
NSGA-III is expected to perform better than NSGA-II.

During the design optimization process of the tracking system, we used full
Geant4 simulations of the entire ECCE detector. AI played a crucial role in
helping choose a combination of technologies for the inner tracker and was
used as input to multiple iterations of the ECCE tracker design, which led to
the current tracker layout. This was the result of a continual optimization
process that evolved in time: results were validated by looking at figures of
merit that do not enter as objective functions in the optimization process
(more details can be found in Sec. B); the decision making is left post hoc
and discussed among the Computing, Detector and Physics teams. A flowchart
describing this continual optimization process is shown in Fig. 4.

Figure 4:  Flowchart of continual optimization during proposal: AI assists the
design optimization process by providing insights and capturing hidden
correlations among the design parameters. This has been used during the entire
detector proposal process to steer the design. At a given instant in time, N
alternative candidate configurations are studied. For each we create an
optimization pipeline, which results in a Pareto front of solutions. This new
information helps steering the design: some configurations are rejected, while
other ones (also dubbed ‘new references’) are identified to potentially
improve the design. New optimization pipelines are defined inspired by the new
results and the process is iterated. During the design process, AI propelled
the fundamental interplay between the ECCE Teams working on Physics, Detector
and Computing.

Ultimately this continual AI-assisted optimization led to a projective design
after having extended the parametrized design to include the support structure
of the inner tracker. The latter represents an ongoing R&D project that is
discussed in the next sections.

##  3 ECCE Tracking System Simulation

Figure 5: Tracking and PID system in the non-projective (left) and the ongoing
R&D projective (right) designs: the two figures show the different geometry
and parametrization of the ECCE non-projective design (left) and of the
ongoing R&D projective design to optimize the support structure (right).
Labels in red indicate the sub-detector systems that were optimized, while the
labels in blue are the sub-detector systems that were kept fixed due to
geometrical constraint. The non-projective geometry (left) is a result of an
optimization on the inner tracker layers (labeled in red) while keeping the
support structure fixed, The angle made by the support structure to the IP is
fixed at about $36.5\degree$. The projective geometry (right) is the result of
an ongoing project R&D to reduce the impact of readout and services on
tracking resolution.

The simulation and detector response shown in this document is based on Geant4
[17] and was carried out using the Fun4All framework [18, 19].

The optimization pipelines are based on particle gun samples of pions, where
we used $\pi^{-}$ and tested that the performance with $\pi^{+}$ were
consistent. Performance in the electron-going direction was also checked post-
hoc with particle gun samples of electrons. The improved performance is
further validated with physics analyses, using the datasets generated during
the ECCE simulation campaigns; in Sec. 4 we show in particular results based
on semi-inclusive deep inelastic scattering (SIDIS) events.

The ECCE tracking detector [20], represented in Fig. 5 (left), consists of
different layers in the barrel and the two end-caps, and is tightly integrated
with the PID detectors:

(i) The silicon vertex/tracking detector is an ALICE ITS-3 type high precision
cylindrical/disk vertex tracker [21, 22]) based on the new Monolithic Active
Pixel Sensor (MAPS); the barrel detector consists of 5 MAPS layers; the
silicon hadron endcap consists of 5 MAPS disks; and the silicon electron
endcap has 4 MAPS disks.

(ii) A gas tracking system is based on $\mu$Rwell technology, that is a
single-stage amplification Micro Pattern Gaseous Detector (MPGD) that is a
derivative of the Gas Electron Multiplier (GEM) technology. In ECCE $\mu$Rwell
layers will form three barrel tracking layers further out from the beam-pipe
than the silicon layers; namely, two inner-barrel layers and a single outer-
barrel $\mu$Rwell layer. All $\mu$Rwell detectors will have 2D strip based
readout. The strip pitch for all three layers will be 400 $\mu$m.

(iii) The tracking system is completed by AC-LGAD-based time of flight (TOF)
detectors providing additional hit information for track reconstruction as
well. In the central region a TOF (dubbed CTTL) is placed behind the high-
performance DIRC (hpDIRC); in the hadron-going side a TOF (dubbed FTTL) is
placed before the dual RICH (dRICH) and a $\mu$Rwell placed after the dRICH;
in the electron-going direction a $\mu$Rwell layer is placed before the
modular RICH (mRICH), which is followed by a TOF later (dubbed ETTL).

An important consideration for all large-scale detectors is the provision of
readout (power and signal cables) and other services (e.g., cooling). Clearly
the aim is to minimize the impact of readout and services in terms of
affecting the detector’s acceptance or tracking resolution, for example. This
effort is ongoing R&D for the project.

In the following sections, the reader can find more details on the
implementation of the optimization pipelines and utilized computing resources.

##  4 Analysis Workflow

The optimization of the ECCE-tracking system [3, 20] has been characterized by
two main phases during which the sub-detectors composing the tracker evolved
into more advanced renditions.

#### Phase I optimization

444Phase I corresponds to a timeline between June-2021 to Sept-2021.
Preliminary studies done between March-2021 to May-2021 are not reported here.

The Geant4 implementation of the detectors were at first simplified, e.g.,
detector modules were mounted on a simplified conical support structure made
of aluminum. The optimization pipelines consisted of symmetric arrangement of
detectors in the electron-going and hadron-going directions (5 disks on each
side). The DIRC detector for PID in the barrel region was modelled with a
simple geometry made by a cylinder and conical mirrors. AC-LGAD-based TOF
detectors were modelled as simplified silicon disks at first; the outer
trackers had more fine-grained simulations implemented, with realistic support
structures and services implemented. The optimization pipelines included
various combinations of detector technologies for the inner trackers. At the
end of this phase, a decision on the choice of the barrel technology and the
disk technologies was made using the AI results.

#### Phase II optimization

555Phase II corresponds to optimization pipelines that run from Sept-2021 to
Nov-2021.

These pipelines had a more realistic implementation of the support structure
incorporating cabling, support carbon fiber, cooling system, etc. More
detailed simulation of the PID Detectors (e.g., DIRC bars and dRICH sub-
systems) were integrated as well as fine-grained simulations of TTL layers
(CTTL, ETTL, FTTL) previously simulated as simple silicon layers modules. More
stringent engineering constraints were considered such as the sensor size for
MAPS detector (ITS3). This phase also considered an asymmetric arrangement of
the detectors in the endcap regions, with a maximum of 4 EST disks in the
electron-going end-cap and 5 FST disks in the hadron-going endcap: due to this
asymmetric spatial arrangement, the angle subtended by detectors in the two
endcap regions could be varied. This eventually developed into the idea of a
projective geometry in a pipeline that characterizes an ongoing R&D project
for optimizing the design of the support structure.

A detailed description of the most recent parametrization used for the
detector proposal can be found in A, along with the parametrization used in an
ongoing R&D project to optimize the support structure of the inner tracker.

Fig. 5 shows a comparison of the ECCE reference non-projective design and the
projective design from the ongoing R&D, both of which resulted from the AI-
assisted procedure described in this paper.

###  4.1 Encoding of Design Criteria

Design criteria need to be encoded to steer the design during the optimization
process. For each design point we need to compute the corresponding objectives
$f_{m}$, namely the momentum resolution, angular resolution, and Kalman filter
efficiency.

We will refer in the following only to the more recent Phase II
optimization.666Similar considerations apply also for Phase I optimization.
Phase II has been characterized by two types of optimization pipelines: the
first used a parametrization of the inner tracker during the optimization
process and led to the ECCE tracker non-projective design; the second branched
off the first as an independent R&D effort that included the parametrization
of the support structure and led to a projective design.

Details on the two types of optimization pipelines can be found in the
following tables: Table 1 describes the main hyperparameters and the
dimensionality of the optimization problem, in particular of the design space
and the objective space; Table 2 reports the range of each design
parameter777The design points are normalized in the range [0-1], using a min-
max scaler $x_{i}=x(x_{max}-x_{min})+x_{min}$, where $x_{i}$ is the normalized
design point with a un-normalized design point $x$ generated between the range
$[x_{min},x_{max}]$.; Table 3 summarizes the constraints for both the non-
projective and projective geometries. We also considered in our design a safe
minimum distance between the disks of 10 cm and include a constraint on the
difference between the outer and inner radii of each disk, namely Rout \- Rin,
to be a multiple of the sensor cell size (17.8 mm $\times$ 30.0 mm), see Table
3. These constraints are common to the non-projective and the projective
designs. For more details on the parametrizations and on the corresponding
detector performance the reader can refer to A and B, respectively.

description |  symbol |  value
---|---|---
population size |  N |  100
# objectives |  M |  3
offspring |  O |  30
design size |  D |  11 (9)
# calls (tot. budget) |  $-$ |  200
# cores |  $-$ |  same as offspring
# charged $\pi$ tracks |  N${}_{\textup{trk}}$ |  120k
# bins in $\eta$ |  Nη |  5
# bins in p |  N${}_{\textup{p}}$ |  10
Table 1: Summary of the hyperparameters of the design optimization: the values
reported in the table have been used during the optimization of the non-
projective design of the ECCE tracker. For completeness and when they differ
from the non-projective case, we also report in parentheses the values
corresponding to the ongoing R&D project for the projective design of the
support structure.

ECCE design (non-projective) Design Parameter Range $\mu$RWELL 1 (Inner) ($r$)
Radius [17.0, 51.0 cm] $\mu$RWELL 2 (Inner) ($r$) Radius [18.0, 51.0 cm] EST 4
$z$ position [-110.0, -50.0 cm] EST 3 $z$ position [-110.0, -40.0 cm] EST 2
$z$ position [-80.0, -30.0 cm] EST 1 $z$ position [-50.0, -20.0 cm] FST 1 $z$
position [20.0, 50.0 cm] FST 2 $z$ position [30.0, 80.0 cm] FST 3 $z$ position
[40.0, 110.0 cm] FST 4 $z$ position [50.0, 125.0 cm] FST 5 $z$ position [60.0,
125.0 cm] ECCE ongoing R&D (projective) Design Parameter Range Angle (Support
Cone) [25.0$\degree$, 30.0$\degree$] $\mu$RWELL 1 (Inner) Radius [25.0, 45.0
cm] ETTL $z$ position [-171.0, -161.0 cm] EST 2 $z$ position [45, 100 cm] EST
1 $z$ position [35, 50 cm] FST 1 $z$ position [35, 50 cm] FST 2 $z$ position
[45, 100 cm] FST 5 $z$ position [100, 150 cm] FTTL $z$ postion [156, 183 cm]

Table 2: Ranges of the design parameters: the table summarises the design
points that are optimized for the non-projective and the projective case. The
optimization range is also given for each parameter. The parameters
corresponding to the non-projective case were optimized sequentially over at
least three iterations, with each iteration having a set of parameters fixed.
For instance, the EST/FST disks were optimized during the first iteration
assuming symmetric design, and for the subsequent iterations the EST disks
were optimized further in the electron endcap region. Along with the design
parameters the design comes with geometrical constraints too. These are
implemented as strong and soft constraints in the Table 3.

sub-detector |  constraint |  description
---|---|---
EST/FST disks |  min { ∑_i^disks — Riout\- Riind - ⌊Riout\- Riind⌋ — } |  soft constraint: sum of residuals in sensor coverage for disks; sensor dimensions: $d$ = 17.8 (30.0) mm
EST/FST disks |  $z_{n+1}-z_{n}>=10.0$ cm |  strong constraint: minimum distance between 2 consecutive disks
sagitta layers |  min{ — 2 πrsagittaw - ⌊2 πrsagittaw⌋ — } |  soft constraint: residual in sensor coverage for every layer; sensor strip width: $w$ = 17.8 mm
$\mu$RWELL |  $r_{n+1}-r_{n}>=5.0$ cm |  strong constraint: minimum distance between $\mu$Rwell barrel layers

Table 3: Constraints in the design optimization: the table summarises
constraints for both the projective and the non-projective designs. Soft
constraints are constraints that can be violated to a certain degree by the
MOO and then one can quantify the degree of unfeasibility (see [13]). Also,
minor adjustments on tiling up the pixels can be done post hoc optimization
such that the detector geometry is realisable. Strong constraints during the
optimization designs rest in high penalties since these constraints cannot be
violated. Two additional constraints are implicitly taken into account by
internal parametrization and reflected in the Geant4 design: the inner radii
of the disks in the endcap and the inner vertex layer in the barrel have to be
compatible with the beam envelope dimensions; the barrel layers lengths and
the outer radii of the disks have to be compatible with an tracking support
structure [23]. Potential overlaps among modules are checked before and during
the optimization.

The objectives depend on the kinematics and are calculated in 5 main bins in
pseudorapidity ($\eta$): (i) -3.5 $\leq\eta<$ -2.0 (corresponding to the
electron-going direction), (ii) -2.0 $\leq\eta<$ -1.0 (the transition region
in electron-going direction), (iii) -1 $\leq\eta<$ 1 (the central barrel),
(iv) 1 $\leq\eta<$ 2.0 (the transition region in the hadron-going direction)
and (v) 2.0 $\leq\eta<$ 3.5 (the hadron-going direction). The rationale behind
this binning is a combination of different aspects: the correspondence with
the binning in the EIC Yellow Report [1], the asymmetric arrangement of
detectors in electron-going and hadron-going directions and the division in
pseudorapidity between the barrel region and the endcap. Particular attention
is given to the transition region between barrel and endcaps as well as at
large $|\eta|\sim$ 3.5 close to the beamline.

Charged pions are generated uniformly in the phase-space that covers the range
in momentum magnitude $p$ $\in$ [0,20] GeV/c and the range in pseudorapidity
$\eta$ $\in$ (-3.5,3.5). Each bin in $\eta$ is further subdivided in 15 bins
in momentum $p$. For each design point we simulated N${}_{trk}=$120k charged
pions.888From Phase I to Phase II, the design became asymmetric in the two
endcaps, therefore we needed to extend the $\eta$-coverage and increase the
statistics. The momentum range was reduced to [0,20] GeV/c to optimize the
computing budget. This number ensured large enough statistics over the entire
phase space and the stability of the fits in all of the bins of Eqs. (4).

In order to calculate the relative momentum (cf. Fig. 13) and absolute angular
resolution (cf. Fig 14) we fit the following objectives:

$\begin{split}\frac{\Delta p}{p}&=\frac{p_{reco}-p_{true}}{p_{true}}\\\
\Delta\theta&=\theta_{reco}-\theta_{true}\end{split}$ (2)

Following the definitions of Eq. (2), histograms of the relative momentum
resolution and the absolute angular resolution are produced for each bin in
$\eta$ and $p$ and the corresponding fits are calculated. Using single-
Gaussian (SG) fits (also utilized in the Yellow Report [1]) implies
systematically better resolutions but worse reduced $\chi^{2}$: therefore we
decided to utilize double-Gaussian (DG) fits, as shown in Fig. 6. This
provided a more robust fit strategy. The reduced $\chi^{2}$ range with DG fits
ranges from 1.2 to 2.8 at most, with the majority of the fits stable at lower
values. The largest numbers correspond either to the transition between the
barrel and endcaps—where tracks cross more material in the non-projective
design—or to large pseudorapidity, particularly close to the inner radii of
the disks. By using SG fits, the reduced $\chi^{2}$ values can be as large as
10-20 in the transition region. A detailed study comparing SG to DG fits is
shown in Fig. 13.

Figure 6: Fit strategy:  a double-Gaussian fit function is utilized to extract
the resolutions. Such a fit function provided good reduced $\chi^{2}$ and more
stable extractions compared to single-Gaussian fits. The resolution is
obtained as an average of the two $\sigma$’s weighted by the relative areas of
the two Gaussians according to Eq. (3). The figure represents the results
corresponding to a particular bin in $\eta$ and $p$.

The final DG resolution has been defined as an average of the two $\sigma$’s
weighted by the relative areas of the two Gaussians:999A different definition
could be based on the weighted average of the variances $\sigma_{1,2}^{2}$ to
obtain the final variance $\sigma_{DG}^{2}$. This typically implied a few %
relative difference on the final value of $\sigma_{DG}$ which has been
considered a negligible effect.

$\sigma_{DG}\sim\frac{A_{1}\sigma_{1}+A_{2}\sigma_{2}}{A_{1}+A_{2}}$ (3)

The results obtained for the resolutions in each bin corresponding to each new
design point are divided by the values corresponding to the baseline design,
so that in each bin a ratio $R(f)_{p,\eta}$ is provided. Finally a weighted
sum of these ratios is performed to build a global figure of merit (for both
the relative momentum and the angular resolutions):

$R(f)=\frac{1}{N_{\eta}}\sum_{\eta}\left(\frac{\sum_{p}w_{p,\eta}\cdot
R(f)_{p,\eta}}{\sum_{p}w_{p,\eta}}\right)$ (4)

where the objective function $f$ is either the momentum or the angular
resolution described by Eq. (2), and the weight
$w_{p,\eta}=1/\sigma^{2}(f_{p,\eta})$ is calculated in each $\eta,p$ bin and
it is proportional to the inverse of the variance corresponding to the
objective functions $f$.

An additional objective function has been included in the optimization
problem: this is a global objective function corresponding to the fraction of
tracks that are not reconstructed by the Kalman filter (KF [24]), or
equivalently the KF inefficiency:

$R(\textup{KF})=\frac{\textup{N(tracks \ not \
reconstructed)}}{\textup{N(total number of tracks)}}$ (5)

Following the above definitions, the design problem corresponds to minimizing
the objective functions defined in Eqs. (4), (5): a weighted average ratio
smaller than one and a KF inefficiency as close as possible to zero correspond
to an improvement in performance compared to the baseline design.

###  4.2 Convergence and Performance at Pareto Front

We remind the reader that the Pareto front is the set of trade-off solutions
of our problem. Fig. 7 shows the convergence plot obtained utilizing the
hypervolume as metric in the objective space.101010Early stopping can occur if
no change in the hypervolume is observed after a certain number of
evaluations. A petal diagram is used to visualize the values of three
objectives corresponding to one of the solutions extracted from the Pareto
front.

Figure 7: Metric for convergence and performance at Pareto front: (left) The
hypervolume can be used as a metric for convergence. Checkpoints are created
during the optimization and snapshots of the evolving designs are saved.
(right) A petal diagram with the three objectives corresponding to one
solution in the Pareto front. The momentum and angular resolutions are
expressed as ratios with respect to a baseline design to improve; the KF
inefficiency is taken as an absolute value. An optimal design optimizes all of
the above defined objectives.

Checkpoints are created to store the NSGA-II-updated population of design
points. A survey of the detector performance is created after each call to
monitor potential anomaly behavior of the fits. The fitting procedure is quite
stable: if an exception occurs the analysis has been automated to adjust the
fitting parameters and ranges. In case of persistent anomalous behavior a flag
is raised, the critical design point purged from the population and examined.

The improvement obtained with the continual multi-objective optimization
process is summarized in Fig. 8, where the momentum resolution obtained during
phase-I optimization using a preliminary detector concept is compared to both
the non-projective and the projective R&D designs which are instead derived
from fully developed simulations in phase-II optimization.

Figure 8: Evolution of momentum resolution driven by the continued MOO:
Momentum resolutions in four bins of $\eta$. From left to right, respectively:
$|\eta|<$ 1.0 corresponding to the barrel, 1.0 $|\leq\eta<|$ 1.5 corresponding
to the transition region, and 1.5 $\leq|\eta|<$ 2.5 and 2.5 $\leq\eta<$ 3.5
corresponding to the two end-caps. Black points represent the first simulation
campaign, and a preliminary detector concept in phase-I optimization which did
not have a developed support structure; blue points represent the fully
developed simulations for the final ECCE detector proposal concept; red points
the ongoing R&D for the optimization of the support structure. Compared to
black, there is an improvement in performance in all $\eta$ bins with the
exception of the transition region, an artifact that depends on the fact that
black points do not include a realistic simulation of the material budget in
the transition region. In the transition region, it can be also appreciate the
improvement provided by the projective design.

A detailed description of the optimized performance for all the objectives
(momentum, angular resolutions and Kalman Filter efficiency) can be found in
B.

###  4.3 Physics Analysis

To show a comparison in physics performance between the non-projective and
projective designs, we analysed ${D}^{0}$ meson decay into $\pi^{+}K^{-}$.
Data have been produced utilizing SIDIS events generated with Pythia6 [25],
corresponding to $ep$ events with 18 GeV $\times$ 275 GeV and high
Q2.111111production: prop.5/prop.5.1; generator: pythia6; kinematics:
ep-18x275-q2-high. More info can be found in [26].

In Fig. 9, the $\pi^{+}K^{-}$ invariant mass is fitted with a double-sided
Crystal Ball function [27, 28].

Figure 9: Analysis of SIDIS events with AI-assisted detector designs:  The
$\pi^{+}K^{-}$ invariant mass obtained from the SIDIS events reconstructed in
ECCE. A comparison of the resolution on the ${D}^{0}$ meson obtained with the
non-projective and the projective design of the ECCE tracker is shown. In
order to emphasize the improvement in the region between $1.0<|\eta|<1.5$, a
selection was made such that both the tracks $\pi^{+}$ or the $K^{-}$ are
within the region of interest. The resolution with the projective design is
improved by more than 10% relative to the resolution obtained with the non-
projective design.

The decay events are selected in such a way to have at least one particle
(either $\pi$ or $K$, or both) in the pseudorapidity bin 1.0$<$$|\eta|$$<$1.5,
where the projective design is expected to improve the performance by
concentrating all the material in a smaller dead area compared to the non-
projective design.

The analysis shows that the resolution obtained with the projective design is
improved by more than 10% relative to that obtained with the non-projective
design. We also calculate the efficiency, defined as the number of
reconstructed ${D}^{0}$ mesons divided by the number of true ${D}^{0}$ mesons.
The efficiency obtained with the two designs is consistent within the
statistical uncertainties.

##  5 Computing Resources

#### Parallelization

A two-level parallelization has been implemented in the MOO framework: the
first level creates the parallel simulations of design points, the second
level parallelizes each design point (see Fig. 10). The evaluation itself can
be distributed to several workers or a whole cluster with libraries like Dask
[29].

Figure 10: Flowchart of the two-level parallelization implemented for detector
design: Multiple design points are produced in parallel, and each design point
is parallelized. The design is assisted by an AI-strategy, in this case MOO
with evolutionary algorithms supported by pymoo [13].

#### Computing Budget

Computing time studies have been carried out to evaluate the simulation time
of each single design point as a function of the number of tracks generated.
We made this study with simulations that included the tracking system and the
PID system and estimated an effective simulation time of 0.2 s/track after
removing an initial latency time. Similarly we made studies of the computing
time taken by the AI-based algorithm in generating a new population of design
points. Results of these studies are summarized in Fig. 11.

A larger population allows to approximate the Pareto front with larger
accuracy. Extension of the design parameter space and the objective space to
larger dimensionality implies a larger amount of CPU time which is mainly
dominated by simulations if the population size remains smaller than 104–105,
see Fig. 11.

For our goals the optimization pipelines of the ECCE tracking system were
parametrized with 10–20 design parameters and 3–4 objectives; this allowed us
to achieve good convergence with evolutionary MOO using a two-level
parallelization strategy, and deployment on single nodes of 128 CPU cores
available on the sci-comp farm at Jefferson Lab [30].121212Work is in progress
to efficiently distribute the optimization pipeline to multiple nodes..

Figure 11: The computing time taken by the genetic algorithm and the sorting
in NSGA-II:. Performance has been benchmarked with test problems like DTLZ1
[16] (red points) and the scaling $\sim MN^{2}$ (see Table 1) has been
verified with convergence to the Pareto front. The simulation time dominates
the AI time during the optimization process (blue point). A two-level
parallelization has been introduced in the framework to reduce this
bottleneck. The AI contribution typically becomes dominant when very large
population size is needed to accurately approximate the Pareto front
(cf.[31]).

#### Planned Activities

As described in this document, detector optimization with AI is an essential
part of the R&D and design process and it is anticipated to continue after the
detector proposal. The AI-assisted design optimization of the ECCE inner
tracker was based on evolutionary algorithms. During the detector proposal
multiple optimization pipelines were run each with a population size of 100,
representing different detector design configurations. At each iteration, AI
updated the population. The total computing budget for an individual pipeline
amounted to approximately 10k CPU-core hours. This number depends on the
dimensionality of the problem. Larger populations may need to be simulated to
cope with the increased complexity in order to improve the accuracy of the
approximated Pareto front. Different AI-based strategies will be compared.

Activities are planned to continue the detector optimization: new optimization
pipelines can deal with a larger parameter space to include a system of sub-
detectors such or to combine tracking and PID in the optimization process. We
also plan to optimize other sub-detectors like, e.g., the dRICH, leveraging on
the expertise internal to the ECCE collaboration regarding specifically the
design of the dRICH with AI-based techniques [11]. As a future activity we aim
to encode physics-driven objectives in the MOO problem. A thorough comparison
of results obtained with different AI-based strategies (e.g., MOO based on
genetic algorithms or bayesian approaches) can be also studied.

We anticipate for 2022 roughly 1M CPU-core hours for these activities.

##  6 Summary

Large scale experiments in high energy nuclear physics entail unprecedented
computational challenges and the optimization of their complex detector
systems can benefit from AI-based strategies [6].

In this paper we described the successful implementation of a multi-objective
optimization approach to steer the multidimensional design of the ECCE
tracking system, taking into account the constraints from the global detector
design. This work was accomplished during the EIC detector proposal, and was
characterized by a continued optimization process where multiple optimization
pipelines integrating different configurations of sub-detectors were compared
using full Geant4 simulations. The insights provided by AI in such a multi-
dimensional objective space characterizing the detector performance (e.g.,
tracking efficiency, momentum and angular resolutions), combined to other
aspects like risk mitigation and costs reduction, helped selecting the
candidate technology of the ECCE tracker. This approach is being used in an
ongoing R&D project where the design parametrization has been extended to
include the support structure of the tracking system.

The design optimization can be also extended to tune the parameters of a
larger system of sub-detectors. Physics analyses are at the moment done after
the optimization for a given detector design solution candidate, but they can
be encoded during the optimization process as physics-driven objectives in
addition to objectives representing the detector performance.

Detector optimization with AI is anticipated to continue after the detector
proposal, and activities are planned to further optimize the tracking system,
including PID sub-detectors, particularly the dual-RICH [11].

## Acknowledgements

We thank the EIC Silicon Consortium for cost estimate methodologies concerning
silicon tracking systems, technical discussions, and comments. We acknowledge
the important prior work of projects eRD16, eRD18, and eRD25 concerning
research and development of MAPS silicon tracking technologies.

We thank the EIC LGAD Consortium for technical discussions and acknowledge the
prior work of project eRD112.

We thank (list of individuals who are not coauthors) for their useful
discussions, advice, and comments.

We acknowledge support from the Office of Nuclear Physics in the Office of
Science in the Department of Energy, the National Science Foundation, and the
Los Alamos National Laboratory Laboratory Directed Research and Development
(LDRD) 20200022DR.

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##  Appendix A Details on Parametrization

### Tracking System Parametrization

#### Vertex layers

There are three vertex barrel layers in the ECCE tracking system made of MAPS
technology. The vertex cylinder consists of strips which are made of pixels,
where the individual sensor unit cell size is 17.8 mm $\times$ 30.0 mm. The
length of the vertex layers is fixed at 27 cm; the radii of the three vertex
layers are fixed to 3.4, 5.67, 7.93 cm, respectively. For the non-projective
design, the angle of the support structure with respect to the interaction
point is fixed ($\theta=36.5\degree$) and the radius of the support is at 6.3
cm, while the length of it is 17 cm. For the projective design, the radius of
the support structure is the same, while the length is calculated based on the
angle of projection and the radius as shown in Fig. 12.

#### Sagitta layers

There are two sagitta barrel layers in the ECCE tracking system. The sagitta
barrel layers are made of MAPS technology and have fixed length of 54 cm. For
the non-projective design the radii of the sagitta layers are 21.0, 22.68 cm,
respectively. For the projective parameterization,the radius of the sagitta
barrel is calculated such that there are no gaps in the acceptance of the
region enclosed by the barrels, according to the following equation:

$r_{sagitta}=\frac{l_{sagitta}}{2}\tan{\theta}.$

The radius of the sagitta layers is also constrained since the strips have
fixed width $w$ = 17.8 mm; therefore we want to minimize the quantity:

$min\Bigg{\\{}\Bigg{|}\frac{2\pi r_{sagitta}}{w}-\left\lfloor\frac{2\pi
r_{sagitta}}{w}\right\rfloor\Bigg{|}\Bigg{\\}},$

where $\left\lfloor x\right\rfloor$ represents the ceiling of x.

####  $\mu$Rwell layers

In the ECCE tracking system there are three cylindrical $\mu$Rwell layers,
each endowed with a support ring. An extended supporting plateau is included
at either ends of the $\mu$Rwell to rest the entire cylindrical detector on
this platform. This results in a constant shift of the support cone by the
plateau length (5 cm) as shown in Fig. 12. For both the non-projective design
and the projective design the $\mu$Rwell-1 radius is a free parameter. The
length of the $\mu$Rwell-1 is calculated based on the angle of the conical
support structure. In the non-projective design we have the conical support
structure angle fixed ($\theta=36.5\degree$), therefore the length of
$\mu$Rwell-1 depends only on its radius; $\mu$Rwell-2 has its radius as a free
parameter; since the angle of the conical support structure is fixed the
length of $\mu$Rwell-2 depends on its radius. In the projective design instead
the $\mu$Rwell-2 has a fixed radius of 51 cm ($i.e.$, $r_{max}+$ 1 cm). The
length of the $\mu$Rwell-2 is calculated based on the angle of the conical
support structure. The length of the $\mu$Rwell takes into account the
constant shift due to the plateau. The dimensions of $\mu$Rwell-3 are fixed in
both non-projective and projective designs; the $\mu$Rwell-3 is outside of the
inner tracking system and it has radius of 77 cm and a total length is 290 cm.

#### EST/FST disks

For both the non-projective and projective designs, $R_{min}$ of the disks
must be compatible with the beam pipe envelope which increases in radius as a
function of $z$; $R_{out}$ of the disks is parametrized to be compatible with
the support cone structure shown in Fig. 12 which has an angle $\theta$ that
is variable in the projective design and fixed in the non-projective case. For
the non-projective design, the $z$ positions of the disks were all free
parameters in the first optimization pipelines. However, to maximize the hit
efficiency, some disks have been eventually placed within the support cone at
the beginning of every plateau (Fig. 12 with fixed angle
$\theta=36.5\degree$). Therefore, two disks in the electron-going direction
and two disks in the hadron-going direction are not free to vary in $z$. For
instance, consider Fig. 5 (right), where EST3, EST4, FST3, FST4 are placed at
the begin of the pleateau, whereas the disks EST1, EST2, FST1, FST2, FST5 are
free to vary in $z$ position. The same parameterization is extended to the
projective design and made compatible with a varying conical support
structure.

As the disks are tiled up using MAPS pixels, the difference between $R_{min}$
and $R_{out}$ is constrained to optimize the sensor coverage for all disks;
this is implemented by means of two functions, namely:

$min\Bigg{\\{}\sum_{i}^{all\
disks}\Bigg{|}\frac{R^{i}_{out}-R^{i}_{in}}{d}-\left\lfloor\frac{R^{i}_{out}-R^{i}_{in}}{d}\right\rfloor\Bigg{|}\Bigg{\\}},$

where $d=$ 17.8, and 30.0 mm. This limits the amount of violation made by a
design solution.

#### TOF system

The central barrel TOF (CTTL) is an AC-LGAD based TOF detector with a fixed
radius of 64 cm and a fixed length of 280 cm. The TOFs at the electron-going
endcap (ETTL) and the hadron-going endcap (FTLL) are AC-LGAD-based TOF disks.
For the non-projective design the TOF detectors have fixed dimensions. For the
projective design the TOF detectors in the end cap regions have their $z$
positions as free parameters. $R_{in}$ and $R_{out}$ of the ETTL/FTTL disks
depends on the position of the disk $z$. The $R_{in}$ of the disk should be
compatible with the radius of the beam envelope which increases linearly as a
function of $z$; $R_{out}$ of the disks varies as a function of $z$ such that
the acceptance coverage by the ETTL/FTTL is roughly unaltered.

#### PID Detectors

The Detection for Internally Reflected Cherenkov light (DIRC) is a detector
for PID in the barrel region. DIRC system has fixed dimensions and occupies a
radial space from 71.5 cm to 76.6 cm. The modular RICH (mRICH) is a Ring
Imaging Cherenkov detector system in the e-going direction with fixed
dimensions. mRICH has a $z$ position starting at -135 cm extending in $z$ to
-161 cm. The dual Radiator Imaging CHerenkov (dRICH) detector is a detector
system in the forward direction with fixed dimensions. dRICH has a $z$
position starting at 180 cm and extends up to 280 cm.

The thickness of the detectors and support structures are also taken into
account to avoid overlaps between the detectors. The most recent optimization
pipelines were extended to also include in the parametrization the outer
tracking layers in the two endcaps, as explained in Sec. 4.1. An overlap check
is performed each time a new design point is evaluated during the optimizaton
process.

### Support Structure Parametrization

The implementation of the projective geometry of the inner tracker is
described in Fig. 12, which shows the parametrization used for the support
cone structure of the inner tracker. Some parameters have been considered
fixed and other free to vary within their ranges. Parameters that are fixed
typically do not have much room for optimization considering the constraints
of the design and potential overlaps. The non-projective design can be
realised by fixing the support structure angle to ($\theta=36.5\degree$) shown
in 12. Therefore, the non-projective design solutions are a subset of
solutions that can be achieved by this parameterization.

Figure 12: Parametrization of the support cone for the inner tracker: The
inner tracker support is characterised by five variables: $\theta$ (the angle
of projection of the support cone structure), $r_{vtx}$ (radius of vertex
support structure), $r_{\mu rwell-1}$ $\mu$Rwell-1 radius, plateau length,
$r_{max}$ maximum allowed radius of inner tracker).

##  Appendix B Baseline and R&D designs

### Resolutions and Efficiency

A thorough comparison between the non-projective ECCE simulation and the
ongoing R&D was carried out to optimize the support structure through a
projective design. Fig. 13 shows a study of the obtained momentum resolution
in bins of pseudorapidity. Similarly Fig. 14 shows the angular resolution
while Fig. 15 shows the Kalman filter efficiency as defined in Eq. (5).

Figure 13:  Momentum resolution for the ECCE tracker reference design and the
projective design (ongoing R&D): The $\Delta p/p$ are extracted with single-
gaussian (SG) fits (circles) to be consistent with the YR PWG and compared to
double-gaussian (DG) fits (triangles). DG (see Eq. (3)) is used in the AI-
assisted pipelines as it provides more stable fits. DG results are
systematically larger than the SG ones as they capture more realistic
distributions (see Fig. 6). This effect is larger in $1<|\eta|<1.5$ for the
non-projective design (blue points) since more material is traversed by the
tracks; the same effect is moderately present also in $0.5<|\eta|<1$ for the
same reason. In the same bins, the ongoing R&D design shows a significant
reduction of the impact of the readout and services and SG (red circle) and DG
(red triangles) are close to each other. DG points are systematically large in
$3<\eta<3.5$ and in $-3.5<\eta<3.0$, and this corresponds to edge effects at
large $|\eta|$ values close to the inner radii of the disks in the endcaps. In
the region $-1<\eta<3.0$ $\Delta p/p$ is consistent with the YR physics
requirements with SG. Note that physics in the backward region relies on the
EM calorimeter also, and in the proposal it has been demonstrated that larger
resolutions are acceptable.  Figure 14: Angular resolution ($d\theta$) for
ECCE projective geometry and ECCE simulation: The resolution shows an
improvement in the transition region between $1.0<|\eta|<1.5$. These plots
have been produced using double-Gaussian fits.  Figure 15: KF Efficiency for
ECCE projective geometry and ECCE simulation: Reconstruction efficiency shows
an improvement in the transition region between $1.0<|\eta|<1.5$. There is a
significant drop in the transition region caused due to multiple scattering of
the support structure.

### Validation

Validation is performed by looking at figures of merit that are not used
during the optimization process. In Sec. 4.3 we already described a physics
analysis with SIDIS events that further consolidates our conclusions. We show
here additional examples of validation: Fig. 16 and Fig. 17 display the
azimuthal angular resolution and the reconstruction efficiency obtained for
both the non-projective and the projective tracker designs. The azimuthal
resolution looks consistent within the uncertainty while the reconstruction
efficiency looks in general better for the projective design, particularly in
the 1 $<$ $|\eta|$ $<$ 1.5 region where the non-projective design has a larger
dead area that affects the reconstruction of tracks.

Figure 16: Angular resolution ($d\phi$) for ECCE projective geometry and ECCE
simulation: The resolution shows an improvement in the transition region
between $1.0<|\eta|<1.5$. These plots have been produced using double-Gaussian
fits. Figure 17: Reconstruction efficiency for ECCE projective geometry and
ECCE simulation: Reconstruction efficiency shows an improvement in the
transition region between $1.0<|\eta|<1.5$

A comparison between the non-projective and projective designs of the inner
tracker is also shown in Fig. 18, where the projective design concentrates the
material in a smaller dead area resulting in better resolution on a wider
range of the pseudorapidity.

Figure 18: Comparison between non-projective and projective inner tracker: a
comparison in momentum resolution as a function of the pseudorapidity $\eta$
between the non-projective and the projective designs in three bins of
momentum. The projective design concentrates the material in a smaller dead
area resulting in better resolution on a wider range of the pseudorapidity.