# FAST BLOCK LINEAR SYSTEM SOLVER USING Q-LEARNING SCHEDULING FOR UNIFIED
DYNAMIC POWER SYSTEM SIMULATIONS

Yingshi Chen
Giga Design Automation Co., Ltd
Shenzhen 518055, China
<EMAIL_ADDRESS>
&Xinli Song , HanYang Dai, Tao Liu, Wuzhi Zhong, Guoyang Wu
Power system Dept.
Electric Power Research Institute
Beijing, 100192, China
<EMAIL_ADDRESS>
Co-first authors with equal contribution

###### Abstract

We present a fast block direct solver for the unified dynamic simulations of
power systems. This solver uses a novel Q-learning based method for task
scheduling. Unified dynamic simulations of power systems represent a method in
which the electric-mechanical transient, medium-term and long-term dynamic
phenomena are organically united. Due to the high rank and large numbers in
solving, fast solution of these equations is the key to speeding up the
simulation. The sparse systems of simulation contain complex nested block
structure, which could be used by the solver to speed up. For the scheduling
of blocks and frontals in the solver, we use a learning based task-tree
scheduling technique in the framework of Markov Decision Process. That is, we
could learn optimal scheduling strategies by offline training on many sample
matrices. Then for any systems, the solver would get optimal task partition
and scheduling on the learned model. Our learning-based algorithm could help
improve the performance of sparse solver, which has been verified in some
numerical experiments. The simulation on some large power systems shows that
our solver is 2-6 times faster than KLU, which is the state-of-the-art sparse
solver for circuit simulation problems.

_Keywords_ Sparse Solver $\cdot$ Adaptive scheduling $\cdot$ Q-Learning
$\cdot$ Block matrix $\cdot$ Unified dynamic power system simulation

##  1 Introduction

Unified dynamic simulations of power systems represent a method in which the
electric-mechanical transient, medium-term and long-term dynamic phenomena are
organically united. Facilitated by the use of implicit integral algorithms,
they can simulate fast as well as slow phenomena in the long-term dynamic
courses. During the simulation process, it is a one of key steps to solve
sparse systems composed of two kinds of linear equations: difference equations
from dynamic equipments and algebraic equations from power network. Due to the
high rank and large numbers in solving, fast solution of these equations play
an important role in unified dynamic simulations.

To speed up the simulation, a parallel algorithm of large linear system solver
based on block matrix is proposed, in which a learning-based scheduler is used
for parallel programming. This scheduler algorithm is actually derived from
the Markov Decision Process (MDP in short) [1, 2, 3]. MDP based learning
methods have recently been in great glory in academia and industry. For
example, Google’s AlphaGO[4] defeated the go master Lee Sedol. The engine of
AlphaGO is based on deep reinforcement learning, whose theoretical basis is
MDP. It’s time to use these powerful modern weapons to improve the solution of
large power systems.

##  2 Block Structure of Power System Equations

As shown in Fig.1, the overall structure of sparse matrix in linear equations
can be divided into 4 big sub-matrices. The sub-matrix A and D is formed by
the equations from dynamic equipments and power networks respectively. As the
coupling parts between A and D, the sub-matrix B and C represents the
influences from power network and dynamic equipments respectively, i.e., bus
voltages and input currents.

Figure 1:  Overall structure of sparse matrix from dynamic simulations of
power systems

The rank of sub-matrix A is 3-5 times larger that of sub-matrix D. Sub-matrix
B and C have a few elements. Therefore, most computation time is spent on A
and D. Because both A and D have good structures for parallelism, we can get
good speed-up. Like other general direct sparse solvers, this solver includes
3 phases, which are symbolic analysis, numeric factorization (LU
decomposition) and solve (forward and backward substitution).

For sub-matrix A, a new block-reuse parallel method is proposed, which is
based on the facts that A is composed of many small independent diagonal
blocks $A_{i}$, and many blocks possess the same symbolic structure, such as
those buses on which the same type of loads or generators are located.

##  3 Q-Learning based Adaptive Scheduler

In this section, we first introduce the background of Q-Learning, which is a
powerful technique from Markov Decision Process and Reinforcement Learning.
Then give the detail of Q-Learning based scheduler for the sparse solver.

###  3.1 Q-Learning technique

####  3.1.1 Markov Decision Process (MDP)

The Markov decision process (MDP in short) [1, 2] is a smart and powerful
model. As its name suggests, this model could find the solution (decision
process) on the Markov property. The standard MDP includes four elements
$\left(E_{n},O,P_{o},R_{o}\right)$[1]:

  * •

$E_{n}$ is the state space of MDP.

  * •

$O$ is all actions. We also use $O_{s}$ to denote the set of actions available
from state $s$,

  * •

${P_{o}(s,s^{\prime})=\Pr(s_{t+1}=s^{\prime}\mid s_{t}=s,o_{t}=o)}$ is the
probability that action $o$ in state $s$ at time $t$,

  * •

$R_{o}(s,s^{\prime})$ is the immediate reward (or expected immediate reward)
received after transitioning from state $s$ to state $s^{\prime}$, due to
action $o$.

A transition matrix $M$ is a square matrix used to describe the transitions of
state space.

$M=\begin{bmatrix}M_{1,1}&\cdots&M_{1,j}&\cdots&M_{1,n}\\\ &&\vdots\\\
M_{i,1}&\cdots&M_{i,j}&\cdots&M_{i,n}\\\ &&\vdots\\\
M_{n,1}&\cdots&M_{n,j}&\cdots&M_{n,n}\\\ \end{bmatrix}$ (1)

Each element $M_{i,j}$ is a non-negative real number representing a
probability $Pr(j|i)$ of moving from state $i$ to state $j$. Each row summing
of $M$ is 1: $\sum_{j}M_{i,j}=1$.

MDP is the theoretical foundation of Reinforcement Learning[5]. A widely-used
Reinforcement Learning technique is Q-Learning.

####  3.1.2 Q-Learning

The MDP model presents a general paradigm and an abstract mathematical
framework. For practical problems, there are many powerful techniques, such as
Q-Learning[6, 7]. "Q" refers to the expected rewards for an action taken in a
given state[6]. At each step, Q-Learning would take some action on the
estimate of Q-values. If we know "Q" value of any action in a state, then it’s
simple to find the optimal solution of MDP.

The following is a typical framework of Q-Learning[8].

Algorithm 1 A general framework of Q-Learning

Init Parameters: Learning rate $\alpha\in\left(0,1\right]$, discount rate
$\gamma$ and reward function $R$

1:Initialize $Q\left(s,a\right)$ for all state $s$ and action $a$. For the
terminal state, $Q(terminal,\cdot)=0$

2:for each episode do

3: Pick a initial state $s$

4: while true do

5: Choose action $a$ for the current state $s$, using policy derived from $Q$

6: Take action $a$ from some policy $\pi$ then enter the next state
$s^{\prime}$

7: Get reward $r=R\left(s,a\right)$

8: Update Q by some method, for example:

9: $Q\left(s,a\right)\leftarrow Q\left(s,a\right)+\alpha\left[r+\gamma
Q\left(s^{\prime},a^{\prime}\right)-Q\left(s,a\right)\right]$

10: Set $s=s^{\prime}$

11: if $s$ is terminal then

12: break

This framework shows a key advantage of Q-Learning - avoiding the usage of
transition matrix. In the case of sparse solver, the probability in the
transition matrix is hard to estimate. Or the transition matrix needs huge
memory The number of states of a million-order matrix is astronomical!).
Instead, we could always get the reward or Q-value. So in practice, Q-Learning
is more suitable for studying various combinatorial problems in GE. Watkins[7]
prove that it would converge to the optimum Q-values with probability 1. So
it’s a reliable technique for the combinatorial optimization problems in
sparse solver.

Solution time is one of the most important metrics of sparse solver. We could
avoid the training time in Q-Learning by offline technique. That is, first
collect many matrices and train the model on these sample matrices. Then apply
the learned Q-value to solve a matrix. This offline Q-Learning (or batch
Q-Learning) has recently regained popularity as a viable path towards
effective real-world application. For example, Conservative Q-learning[9, 10].

###  3.2 Q-Learning based task tree scheduling technique

We first take a symmetric $11\times 11$ matrix as an example. Figure 2.(a)
shows its pattern (only shows the lower triangle since it’s symmetric). In
this figure, non-zero element is represented by black dots $\bullet$ and
$\otimes$ is the extra fill-in. In the elimination process, there are some
continuous rows(columns) with nearly same non-zero patterns, which is called
frontal [11, 12] or supernodal. These frontals are actually dense matrices,
which can apply high performance BLAS operators. For the detail of frontal
based method, please see [11, 12, 13, 14, 15, 16].

There are many greedy algorithms to find the frontal structure. It’s not the
focus of this paper. We mainly study how to schedule these frontals to improve
the efficiency of parallelism. For a frontal partition
$[\left\\{1\right\\},\left\\{2,3\right\\},\left\\{4\right\\},\left\\{5\right\\},\left\\{6\right\\},\left\\{7,9\right\\},\left\\{8\right\\},\left\\{11\right\\}]$,
figure 2.(b) shows its frontal elimination tree[17, 18]. Each node of tree is
a frontal, and the edges represent dependencies between frontals. We call this
tree as the initial task-tree $T_{0}$, then study how to get different
parallel task tree from $T_{0}$.

|
---|---
Figure 2: (a) Pattern of a $11\times 11$ matrix (b) Its frontal elimination
tree or task-tree $T_{0}$

If two nodes have no dependence (neither parent node nor child node of each
other), their elimination can be carried out at the same time. We could
further extend it to all independent nodes. if there is no dependency between
these nodes, their elimination can be carried out at the same time. We could
divide these independent frontals into independent tasks.Then run these tasks
on different cores to get high parallelism. As shown in Figure 3, there are
three different task trees $T_{1},T_{2},T_{3}$ with different task partition.

  * •

$T_{1}$: Delete edge $a,b$, get two parallel
tasks=$[\\{2,3\\},\\{5\\},\\{8\\}];[\\{1\\},\\{4\\},\\{6\\},\\{7,9\\}]$

  * •

$T_{2}$: Delete edge $c,d,b$, get three parallel
tasks=$[\\{2,3\\}];[\\{5\\}];[\\{1\\},\\{4\\},\\{6\\},\\{7,9\\}]$

  * •

$T_{3}$: Delete edge $c,d,e,f$, get four parallel
tasks=$[\\{2,3\\}];[\\{5\\}];[\\{1\\}];[\\{4\\}]$

The three trees $T_{1},T_{2},T_{3}$ come from different delete actions on the
initial task-tree $T_{0}$. We could also add some edges to get a new task-
tree. For example, at the state of $T_{2}$, we could add edge $c,d$ and re-
delete edge $a$ to get $T_{1}$. We can try delete/add more edges to get more
partitions! How to get the optimal tree? We propose a novel Q-Learning based
algorithm for this problem. Compared to the standard Q-Learning algorithm 1 in
section 3.1.2, different trees correspond to different states of MDP, and the
action at each state is Delete/Add some edges. That is, the Q-Learning based
task tree scheduling technique has three basic modules
$\left(T_{n},\mathscr{A},R\right)$ :

  1. 1.

State space: $T_{n}$ is a set includes all possible split of task-tree.

  2. 2.

Action space: $\mathscr{A}$=$\\{\textrm{Delete},\textrm{Add},\textrm{SKIP}\\}$

     * •

Delete: - Delete some edges to create more tasks. For example, at state
$T_{2}$, if some cores are idle, we could delete edge $e,f$ to get more tasks,
then assign $[\\{1\\}];[\\{4\\}]$ to idle cores.

     * •

Add: - Add some edges to union more tasks. For example, at state $T_{3}$, we
could delete edge $e,f$ to create a large task [{ 1 },{ 4 },{ 6 }].

     * •

SKIP: The workloads of all computing cores are full. No need to change current
task-tree.

  3. 3.

Reward function: $R$ represents the target of scheduling. The scheduling
module needs to consider many factors. The main factors are listed below

     * •

TIME - the wall time spent by the solver

     * •

MEMORY - memory consumption [19]

     * •

BALANCE - workload balance balancing between the resources

     * •

OVERLAP - Efficient Communication/Computation Overlap [20, 21]

     * •

POWER - Reduce the power and energy consumed[22].

So the total reward is a weighted sum of these metrics
$R=\alpha\textrm{TIME}+\beta\textrm{MEM}+\gamma\textrm{BALANCE}+\cdots$.

(a) $T_{1}$: Delete edge $a,b$, get two parallel
tasks=$[\\{2,3\\},\\{5\\},\\{8\\}];[\\{1\\},\\{4\\},\\{6\\},\\{7,9\\}]$

(b) $T_{2}$: Delete edge $c,d,b$, get three parallel
tasks=$[\\{2,3\\}];[\\{5\\}];[\\{1\\},\\{4\\},\\{6\\},\\{7,9\\}]$

(c) $T_{3}$: Delete edge $c,d,e,f$, get four parallel
tasks=$[\\{2,3\\}];[\\{5\\}];[\\{1\\}];[\\{4\\}]$

Figure 3:  Different task partition on different split of task-tree

The following is the detailed algorithm to learn $Q$ (the expected rewards for
an action taken in a given state $s$):

Algorithm 2 An offline Q-Learning based task-tree scheduling technique

Training stage:

Learn $Q\left(s,a\right)$ from the scheduling of many matrices(task-trees)

1:Initialize $Q\left(s,a\right)$. For the terminal state,
$Q(terminal,\cdot)=0$

2:for each episode do

3: Pick a initial state $s$. It is just $T_{0}$(Elimination-Tree) of some
matrix

4: while true do

5: Choose action $a$ for the current state $s$, using policy derived from $Q$

6: Take action $a$ then enter the next state $s^{\prime}$

7: Get reward $r=R\left(s,a\right)$

8: Update Q by some method, for example:

9: $Q\left(s,a\right)\leftarrow Q\left(s,a\right)+\alpha\left[r+\gamma
Q\left(s^{\prime},a^{\prime}\right)-Q\left(s,a\right)\right]$

10: Set $s=s^{\prime}$

11: if $s$ is terminal then

12: break

Inference stage:

1:The initial state $s$ is just $T_{0}$(Elimination-Tree)

2:while un-eliminated frontals do

3: For the current state $T_{k}$, take action $a$ from learned $Q$ table

4: Run one or more task, eliminate some frontals, update task-tree

5: Get reward $r=R\left(T_{k},p\right)$

6: k=k+1

##  4 Experimental Result

A new solver named ESS is developed using the above methods. We have
implemented Q-Learning based scheduling algorithm in ESS. It uses Q-Learning
based task-tree split technique to create an optimal task partition. Each task
is assigned to a different computing core. In the actual elimination process,
the task partition is based on pretrained $Q$ table. That is, if some cores
have high computing power, then it will run more tasks automatically.

To test the performance of ESS, 2 large equations with the ranks of 54603 and
126869 from power system unified dynamic simulations are taken for numeric
factorization and solving. The CPU for testing is Intel i7-4770(3.4Ghz). The
solver for comparison is KLU [23], a direct serial sparse solver for circuit
simulations, which is developed in Florida University. From the test results
with 2 4 threads in parallel shown in table 2 and table 3, we can see that the
speed-up of LU is about 1.9, 2.5 and 3.0 respectively and the speed-up of
solving is about 1.5, 1.8 and 2.0. Also, the new algorithm in series is much
faster than KLU.

Table 4: Time of numerical factorization(compared with KLU) Table 5: Time of
solve(compared with KLU)

The test cases show that, with the Q-Learning adaptive scheduling technique,
the new parallel algorithm can achieve higher computational efficiency on
multi-core computers.

##  5 Conclusion

We present a novel sparse solver with Q-Learning scheduler for the dynamic
simulations of power system. Numerical experiments on some large power systems
shows that our solver is 2-6 times faster than classical sparse solver(KLU -
the state-of-the-art sparse solver for circuit simulation problems). We would
try more machine learning based method for this problem.

## References

  * [1] Richard Bellman.  A markovian decision process.  Journal of mathematics and mechanics, 6(5):679–684, 1957.
  * [2] Abhijit Gosavi.  Solving markov decision processes via simulation.  In Handbook of Simulation Optimization, pages 341–379. Springer, 2015.
  * [3] Yingshi Chen.  Learning the markov decision process in the sparse gaussian elimination.  arXiv preprint arXiv:2109.14929, 2021.
  * [4] David Silver, Aja Huang, Chris J Maddison, Arthur Guez, Laurent Sifre, George Van Den Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, et al.  Mastering the game of go with deep neural networks and tree search.  nature, 529(7587):484–489, 2016.
  * [5] Leslie Pack Kaelbling, Michael L Littman, and Andrew W Moore.  Reinforcement learning: A survey.  Journal of artificial intelligence research, 4:237–285, 1996.
  * [6] Christopher John Cornish Hellaby Watkins.  Learning from delayed rewards.  1989\.
  * [7] Christopher JCH Watkins and Peter Dayan.  Q-learning.  Machine learning, 8(3-4):279–292, 1992.
  * [8] Richard S Sutton and Andrew G Barto.  Reinforcement learning: An introduction.  MIT press, 2018.
  * [9] Aviral Kumar, Aurick Zhou, George Tucker, and Sergey Levine.  Conservative q-learning for offline reinforcement learning.  arXiv preprint arXiv:2006.04779, 2020.
  * [10] Sergey Levine, Aviral Kumar, George Tucker, and Justin Fu.  Offline reinforcement learning: Tutorial, review, and perspectives on open problems.  arXiv preprint arXiv:2005.01643, 2020.
  * [11] Iain S Duff, Albert Maurice Erisman, and John Ker Reid.  Direct methods for sparse matrices.  Oxford University Press, 2017.
  * [12] Bruce M Irons.  A frontal solution program for finite element analysis.  International Journal for Numerical Methods in Engineering, 2(1):5–32, 1970.
  * [13] Olaf Schenk and Klaus Gärtner.  Solving unsymmetric sparse systems of linear equations with pardiso.  Future Generation Computer Systems, 20(3):475–487, 2004.
  * [14] Xiaoye S Li and James W Demmel.  Superlu_dist: A scalable distributed-memory sparse direct solver for unsymmetric linear systems.  ACM Transactions on Mathematical Software (TOMS), 29(2):110–140, 2003.
  * [15] Patrick R Amestoy, Iain S Duff, Jean-Yves L’Excellent, and Jacko Koster.  Mumps: a general purpose distributed memory sparse solver.  In International Workshop on Applied Parallel Computing, pages 121–130. Springer, 2000.
  * [16] Yingshi Chen.  Gss — fast cpu/gpu sparse solver for large sparse matrices on q-learning scheduler.  https://github.com/closest-git/GSS, 2018.
  * [17] John R Gilbert and Joseph WH Liu.  Elimination structures for unsymmetric sparse lu factors.  SIAM Journal on Matrix Analysis and Applications, 14(2):334–352, 1993.
  * [18] Joseph WH Liu.  The role of elimination trees in sparse factorization.  SIAM journal on matrix analysis and applications, 11(1):134–172, 1990.
  * [19] Xavier Lacoste.  Scheduling and memory optimizations for sparse direct solver on multi-core/multi-gpu duster systems.  PhD thesis, Université de Bordeaux, 2015.
  * [20] Vladimir Marjanović, Jesús Labarta, Eduard Ayguadé, and Mateo Valero.  Overlapping communication and computation by using a hybrid mpi/smpss approach.  In Proceedings of the 24th acm International Conference on Supercomputing, pages 5–16, 2010.
  * [21] Piyush Sao, Xiaoye Sherry Li, and Richard Vuduc.  A communication-avoiding 3d lu factorization algorithm for sparse matrices.  In 2018 IEEE International Parallel and Distributed Processing Symposium (IPDPS), pages 908–919. IEEE, 2018.
  * [22] Xavier Aguilar.  Performance Monitoring, Analysis, and Real-Time Introspection on Large-Scale Parallel Systems.  PhD thesis, KTH Royal Institute of Technology, 2020.
  * [23] Timothy A Davis and Ekanathan Palamadai Natarajan.  Algorithm 907: Klu, a direct sparse solver for circuit simulation problems.  ACM Transactions on Mathematical Software (TOMS), 37(3):1–17, 2010\.


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Ohlson${}^{\textrm{\scriptsize 35}}$, A. Okatan${}^{\textrm{\scriptsize 70}}$,
T. Okubo${}^{\textrm{\scriptsize 47}}$, L. Olah${}^{\textrm{\scriptsize
139}}$, J. Oleniacz${}^{\textrm{\scriptsize 137}}$, A.C. Oliveira Da
Silva${}^{\textrm{\scriptsize 122}}$, M.H. Oliver${}^{\textrm{\scriptsize
140}}$, J. Onderwaater${}^{\textrm{\scriptsize 99}}$, C.
Oppedisano${}^{\textrm{\scriptsize 112}}$, R. Orava${}^{\textrm{\scriptsize
46}}$, M. Oravec${}^{\textrm{\scriptsize 117}}$, A. Ortiz
Velasquez${}^{\textrm{\scriptsize 63}}$, A. Oskarsson${}^{\textrm{\scriptsize
34}}$, J. Otwinowski${}^{\textrm{\scriptsize 119}}$, K.
Oyama${}^{\textrm{\scriptsize 95}}$,${}^{\textrm{\scriptsize 77}}$, M.
Ozdemir${}^{\textrm{\scriptsize 61}}$, Y. Pachmayer${}^{\textrm{\scriptsize
95}}$, D. Pagano${}^{\textrm{\scriptsize 134}}$, P.
Pagano${}^{\textrm{\scriptsize 30}}$, G. Paić${}^{\textrm{\scriptsize 63}}$,
S.K. Pal${}^{\textrm{\scriptsize 136}}$, P. Palni${}^{\textrm{\scriptsize
7}}$, J. Pan${}^{\textrm{\scriptsize 138}}$, A.K.
Pandey${}^{\textrm{\scriptsize 48}}$, V. Papikyan${}^{\textrm{\scriptsize
1}}$, G.S. Pappalardo${}^{\textrm{\scriptsize 108}}$, P.
Pareek${}^{\textrm{\scriptsize 49}}$, J. Park${}^{\textrm{\scriptsize 51}}$,
W.J. Park${}^{\textrm{\scriptsize 99}}$, S. Parmar${}^{\textrm{\scriptsize
89}}$, A. Passfeld${}^{\textrm{\scriptsize 62}}$, V.
Paticchio${}^{\textrm{\scriptsize 105}}$, R.N. Patra${}^{\textrm{\scriptsize
136}}$, B. Paul${}^{\textrm{\scriptsize 112}}$, H. Pei${}^{\textrm{\scriptsize
7}}$, T. Peitzmann${}^{\textrm{\scriptsize 54}}$, X.
Peng${}^{\textrm{\scriptsize 7}}$, H. Pereira Da Costa${}^{\textrm{\scriptsize
15}}$, D. Peresunko${}^{\textrm{\scriptsize 76}}$,${}^{\textrm{\scriptsize
81}}$, E. Perez Lezama${}^{\textrm{\scriptsize 61}}$, V.
Peskov${}^{\textrm{\scriptsize 61}}$, Y. Pestov${}^{\textrm{\scriptsize 5}}$,
V. Petráček${}^{\textrm{\scriptsize 39}}$, V. Petrov${}^{\textrm{\scriptsize
113}}$, M. Petrovici${}^{\textrm{\scriptsize 79}}$, C.
Petta${}^{\textrm{\scriptsize 28}}$, S. Piano${}^{\textrm{\scriptsize 111}}$,
M. Pikna${}^{\textrm{\scriptsize 38}}$, P. Pillot${}^{\textrm{\scriptsize
115}}$, L.O.D.L. Pimentel${}^{\textrm{\scriptsize 82}}$, O.
Pinazza${}^{\textrm{\scriptsize 106}}$,${}^{\textrm{\scriptsize 35}}$, L.
Pinsky${}^{\textrm{\scriptsize 125}}$, D.B. Piyarathna${}^{\textrm{\scriptsize
125}}$, M. Płoskoń${}^{\textrm{\scriptsize 75}}$, M.
Planinic${}^{\textrm{\scriptsize 132}}$, J. Pluta${}^{\textrm{\scriptsize
137}}$, S. Pochybova${}^{\textrm{\scriptsize 139}}$, P.L.M. Podesta-
Lerma${}^{\textrm{\scriptsize 121}}$, M.G. Poghosyan${}^{\textrm{\scriptsize
86}}$, B. Polichtchouk${}^{\textrm{\scriptsize 113}}$, N.
Poljak${}^{\textrm{\scriptsize 132}}$, W. Poonsawat${}^{\textrm{\scriptsize
116}}$, A. Pop${}^{\textrm{\scriptsize 79}}$, H.
Poppenborg${}^{\textrm{\scriptsize 62}}$, S. Porteboeuf-
Houssais${}^{\textrm{\scriptsize 71}}$, J. Porter${}^{\textrm{\scriptsize
75}}$, J. Pospisil${}^{\textrm{\scriptsize 85}}$, S.K.
Prasad${}^{\textrm{\scriptsize 4}}$, R. Preghenella${}^{\textrm{\scriptsize
35}}$,${}^{\textrm{\scriptsize 106}}$, F. Prino${}^{\textrm{\scriptsize
112}}$, C.A. Pruneau${}^{\textrm{\scriptsize 138}}$, I.
Pshenichnov${}^{\textrm{\scriptsize 53}}$, M. Puccio${}^{\textrm{\scriptsize
26}}$, G. Puddu${}^{\textrm{\scriptsize 24}}$, P.
Pujahari${}^{\textrm{\scriptsize 138}}$, V. Punin${}^{\textrm{\scriptsize
101}}$, J. Putschke${}^{\textrm{\scriptsize 138}}$, H.
Qvigstad${}^{\textrm{\scriptsize 21}}$, A. Rachevski${}^{\textrm{\scriptsize
111}}$, S. Raha${}^{\textrm{\scriptsize 4}}$, S.
Rajput${}^{\textrm{\scriptsize 92}}$, J. Rak${}^{\textrm{\scriptsize 126}}$,
A. Rakotozafindrabe${}^{\textrm{\scriptsize 15}}$, L.
Ramello${}^{\textrm{\scriptsize 32}}$, F. Rami${}^{\textrm{\scriptsize 65}}$,
R. Raniwala${}^{\textrm{\scriptsize 93}}$, S. Raniwala${}^{\textrm{\scriptsize
93}}$, S.S. Räsänen${}^{\textrm{\scriptsize 46}}$, B.T.
Rascanu${}^{\textrm{\scriptsize 61}}$, D. Rathee${}^{\textrm{\scriptsize
89}}$, I. Ravasenga${}^{\textrm{\scriptsize 26}}$, K.F.
Read${}^{\textrm{\scriptsize 86}}$,${}^{\textrm{\scriptsize 128}}$, K.
Redlich${}^{\textrm{\scriptsize 78}}$, R.J. Reed${}^{\textrm{\scriptsize
138}}$, A. Rehman${}^{\textrm{\scriptsize 22}}$, P.
Reichelt${}^{\textrm{\scriptsize 61}}$, F. Reidt${}^{\textrm{\scriptsize
95}}$,${}^{\textrm{\scriptsize 35}}$, X. Ren${}^{\textrm{\scriptsize 7}}$, R.
Renfordt${}^{\textrm{\scriptsize 61}}$, A.R. Reolon${}^{\textrm{\scriptsize
73}}$, A. Reshetin${}^{\textrm{\scriptsize 53}}$, K.
Reygers${}^{\textrm{\scriptsize 95}}$, V. Riabov${}^{\textrm{\scriptsize
87}}$, R.A. Ricci${}^{\textrm{\scriptsize 74}}$, T.
Richert${}^{\textrm{\scriptsize 34}}$, M. Richter${}^{\textrm{\scriptsize
21}}$, P. Riedler${}^{\textrm{\scriptsize 35}}$, W.
Riegler${}^{\textrm{\scriptsize 35}}$, F. Riggi${}^{\textrm{\scriptsize 28}}$,
C. Ristea${}^{\textrm{\scriptsize 59}}$, M. Rodríguez
Cahuantzi${}^{\textrm{\scriptsize 2}}$, A. Rodriguez
Manso${}^{\textrm{\scriptsize 83}}$, K. Røed${}^{\textrm{\scriptsize 21}}$, E.
Rogochaya${}^{\textrm{\scriptsize 67}}$, D. Rohr${}^{\textrm{\scriptsize
42}}$, D. Röhrich${}^{\textrm{\scriptsize 22}}$, F.
Ronchetti${}^{\textrm{\scriptsize 73}}$,${}^{\textrm{\scriptsize 35}}$, L.
Ronflette${}^{\textrm{\scriptsize 115}}$, P. Rosnet${}^{\textrm{\scriptsize
71}}$, A. Rossi${}^{\textrm{\scriptsize 29}}$, F.
Roukoutakis${}^{\textrm{\scriptsize 90}}$, A. Roy${}^{\textrm{\scriptsize
49}}$, C. Roy${}^{\textrm{\scriptsize 65}}$, P. Roy${}^{\textrm{\scriptsize
102}}$, A.J. Rubio Montero${}^{\textrm{\scriptsize 10}}$, R.
Rui${}^{\textrm{\scriptsize 25}}$, R. Russo${}^{\textrm{\scriptsize 26}}$, E.
Ryabinkin${}^{\textrm{\scriptsize 81}}$, Y. Ryabov${}^{\textrm{\scriptsize
87}}$, A. Rybicki${}^{\textrm{\scriptsize 119}}$, S.
Saarinen${}^{\textrm{\scriptsize 46}}$, S. Sadhu${}^{\textrm{\scriptsize
136}}$, S. Sadovsky${}^{\textrm{\scriptsize 113}}$, K.
Šafařík${}^{\textrm{\scriptsize 35}}$, B. Sahlmuller${}^{\textrm{\scriptsize
61}}$, P. Sahoo${}^{\textrm{\scriptsize 49}}$, R.
Sahoo${}^{\textrm{\scriptsize 49}}$, S. Sahoo${}^{\textrm{\scriptsize 58}}$,
P.K. Sahu${}^{\textrm{\scriptsize 58}}$, J. Saini${}^{\textrm{\scriptsize
136}}$, S. Sakai${}^{\textrm{\scriptsize 73}}$, M.A.
Saleh${}^{\textrm{\scriptsize 138}}$, J. Salzwedel${}^{\textrm{\scriptsize
19}}$, S. Sambyal${}^{\textrm{\scriptsize 92}}$, V.
Samsonov${}^{\textrm{\scriptsize 87}}$,${}^{\textrm{\scriptsize 76}}$, L.
Šándor${}^{\textrm{\scriptsize 56}}$, A. Sandoval${}^{\textrm{\scriptsize
64}}$, M. Sano${}^{\textrm{\scriptsize 131}}$, D.
Sarkar${}^{\textrm{\scriptsize 136}}$, N. Sarkar${}^{\textrm{\scriptsize
136}}$, P. Sarma${}^{\textrm{\scriptsize 44}}$, E.
Scapparone${}^{\textrm{\scriptsize 106}}$, F.
Scarlassara${}^{\textrm{\scriptsize 29}}$, C. Schiaua${}^{\textrm{\scriptsize
79}}$, R. Schicker${}^{\textrm{\scriptsize 95}}$, C.
Schmidt${}^{\textrm{\scriptsize 99}}$, H.R. Schmidt${}^{\textrm{\scriptsize
94}}$, M. Schmidt${}^{\textrm{\scriptsize 94}}$, S.
Schuchmann${}^{\textrm{\scriptsize 95}}$,${}^{\textrm{\scriptsize 61}}$, J.
Schukraft${}^{\textrm{\scriptsize 35}}$, Y. Schutz${}^{\textrm{\scriptsize
115}}$,${}^{\textrm{\scriptsize 35}}$, K. Schwarz${}^{\textrm{\scriptsize
99}}$, K. Schweda${}^{\textrm{\scriptsize 99}}$, G.
Scioli${}^{\textrm{\scriptsize 27}}$, E. Scomparin${}^{\textrm{\scriptsize
112}}$, R. Scott${}^{\textrm{\scriptsize 128}}$, M.
Šefčík${}^{\textrm{\scriptsize 40}}$, J.E. Seger${}^{\textrm{\scriptsize
88}}$, Y. Sekiguchi${}^{\textrm{\scriptsize 130}}$, D.
Sekihata${}^{\textrm{\scriptsize 47}}$, I.
Selyuzhenkov${}^{\textrm{\scriptsize 99}}$, K. Senosi${}^{\textrm{\scriptsize
66}}$, S. Senyukov${}^{\textrm{\scriptsize 3}}$,${}^{\textrm{\scriptsize
35}}$, E. Serradilla${}^{\textrm{\scriptsize 64}}$,${}^{\textrm{\scriptsize
10}}$, A. Sevcenco${}^{\textrm{\scriptsize 59}}$, A.
Shabanov${}^{\textrm{\scriptsize 53}}$, A. Shabetai${}^{\textrm{\scriptsize
115}}$, O. Shadura${}^{\textrm{\scriptsize 3}}$, R.
Shahoyan${}^{\textrm{\scriptsize 35}}$, A. Shangaraev${}^{\textrm{\scriptsize
113}}$, A. Sharma${}^{\textrm{\scriptsize 92}}$, M.
Sharma${}^{\textrm{\scriptsize 92}}$, M. Sharma${}^{\textrm{\scriptsize 92}}$,
N. Sharma${}^{\textrm{\scriptsize 128}}$, A.I. Sheikh${}^{\textrm{\scriptsize
136}}$, K. Shigaki${}^{\textrm{\scriptsize 47}}$, Q.
Shou${}^{\textrm{\scriptsize 7}}$, K. Shtejer${}^{\textrm{\scriptsize
26}}$,${}^{\textrm{\scriptsize 9}}$, Y. Sibiriak${}^{\textrm{\scriptsize
81}}$, S. Siddhanta${}^{\textrm{\scriptsize 107}}$, K.M.
Sielewicz${}^{\textrm{\scriptsize 35}}$, T.
Siemiarczuk${}^{\textrm{\scriptsize 78}}$, D.
Silvermyr${}^{\textrm{\scriptsize 34}}$, C. Silvestre${}^{\textrm{\scriptsize
72}}$, G. Simatovic${}^{\textrm{\scriptsize 132}}$, G.
Simonetti${}^{\textrm{\scriptsize 35}}$, R. Singaraju${}^{\textrm{\scriptsize
136}}$, R. Singh${}^{\textrm{\scriptsize 80}}$, V.
Singhal${}^{\textrm{\scriptsize 136}}$, T. Sinha${}^{\textrm{\scriptsize
102}}$, B. Sitar${}^{\textrm{\scriptsize 38}}$, M.
Sitta${}^{\textrm{\scriptsize 32}}$, T.B. Skaali${}^{\textrm{\scriptsize
21}}$, M. Slupecki${}^{\textrm{\scriptsize 126}}$, N.
Smirnov${}^{\textrm{\scriptsize 140}}$, R.J.M.
Snellings${}^{\textrm{\scriptsize 54}}$, T.W. Snellman${}^{\textrm{\scriptsize
126}}$, J. Song${}^{\textrm{\scriptsize 98}}$, M. Song${}^{\textrm{\scriptsize
141}}$, Z. Song${}^{\textrm{\scriptsize 7}}$, F.
Soramel${}^{\textrm{\scriptsize 29}}$, S. Sorensen${}^{\textrm{\scriptsize
128}}$, F. Sozzi${}^{\textrm{\scriptsize 99}}$, E.
Spiriti${}^{\textrm{\scriptsize 73}}$, I. Sputowska${}^{\textrm{\scriptsize
119}}$, M. Spyropoulou-Stassinaki${}^{\textrm{\scriptsize 90}}$, J.
Stachel${}^{\textrm{\scriptsize 95}}$, I. Stan${}^{\textrm{\scriptsize 59}}$,
P. Stankus${}^{\textrm{\scriptsize 86}}$, E. Stenlund${}^{\textrm{\scriptsize
34}}$, G. Steyn${}^{\textrm{\scriptsize 66}}$, J.H.
Stiller${}^{\textrm{\scriptsize 95}}$, D. Stocco${}^{\textrm{\scriptsize
115}}$, P. Strmen${}^{\textrm{\scriptsize 38}}$, A.A.P.
Suaide${}^{\textrm{\scriptsize 122}}$, T. Sugitate${}^{\textrm{\scriptsize
47}}$, C. Suire${}^{\textrm{\scriptsize 52}}$, M.
Suleymanov${}^{\textrm{\scriptsize 16}}$, M. Suljic${}^{\textrm{\scriptsize
25}}$, R. Sultanov${}^{\textrm{\scriptsize 55}}$, M.
Šumbera${}^{\textrm{\scriptsize 85}}$, S. Sumowidagdo${}^{\textrm{\scriptsize
50}}$, A. Szabo${}^{\textrm{\scriptsize 38}}$, I.
Szarka${}^{\textrm{\scriptsize 38}}$, A.
Szczepankiewicz${}^{\textrm{\scriptsize 137}}$, M.
Szymanski${}^{\textrm{\scriptsize 137}}$, U. Tabassam${}^{\textrm{\scriptsize
16}}$, J. Takahashi${}^{\textrm{\scriptsize 123}}$, G.J.
Tambave${}^{\textrm{\scriptsize 22}}$, N. Tanaka${}^{\textrm{\scriptsize
131}}$, M. Tarhini${}^{\textrm{\scriptsize 52}}$, M.
Tariq${}^{\textrm{\scriptsize 18}}$, M.G. Tarzila${}^{\textrm{\scriptsize
79}}$, A. Tauro${}^{\textrm{\scriptsize 35}}$, G. Tejeda
Muñoz${}^{\textrm{\scriptsize 2}}$, A. Telesca${}^{\textrm{\scriptsize 35}}$,
K. Terasaki${}^{\textrm{\scriptsize 130}}$, C.
Terrevoli${}^{\textrm{\scriptsize 29}}$, B. Teyssier${}^{\textrm{\scriptsize
133}}$, J. Thäder${}^{\textrm{\scriptsize 75}}$, D.
Thakur${}^{\textrm{\scriptsize 49}}$, D. Thomas${}^{\textrm{\scriptsize
120}}$, R. Tieulent${}^{\textrm{\scriptsize 133}}$, A.
Tikhonov${}^{\textrm{\scriptsize 53}}$, A.R. Timmins${}^{\textrm{\scriptsize
125}}$, A. Toia${}^{\textrm{\scriptsize 61}}$, S.
Trogolo${}^{\textrm{\scriptsize 26}}$, G. Trombetta${}^{\textrm{\scriptsize
33}}$, V. Trubnikov${}^{\textrm{\scriptsize 3}}$, W.H.
Trzaska${}^{\textrm{\scriptsize 126}}$, T. Tsuji${}^{\textrm{\scriptsize
130}}$, A. Tumkin${}^{\textrm{\scriptsize 101}}$, R.
Turrisi${}^{\textrm{\scriptsize 109}}$, T.S. Tveter${}^{\textrm{\scriptsize
21}}$, K. Ullaland${}^{\textrm{\scriptsize 22}}$, A.
Uras${}^{\textrm{\scriptsize 133}}$, G.L. Usai${}^{\textrm{\scriptsize 24}}$,
A. Utrobicic${}^{\textrm{\scriptsize 132}}$, M. Vala${}^{\textrm{\scriptsize
56}}$, L. Valencia Palomo${}^{\textrm{\scriptsize 71}}$, J. Van Der
Maarel${}^{\textrm{\scriptsize 54}}$, J.W. Van Hoorne${}^{\textrm{\scriptsize
114}}$,${}^{\textrm{\scriptsize 35}}$, M. van Leeuwen${}^{\textrm{\scriptsize
54}}$, T. Vanat${}^{\textrm{\scriptsize 85}}$, P. Vande
Vyvre${}^{\textrm{\scriptsize 35}}$, D. Varga${}^{\textrm{\scriptsize 139}}$,
A. Vargas${}^{\textrm{\scriptsize 2}}$, M. Vargyas${}^{\textrm{\scriptsize
126}}$, R. Varma${}^{\textrm{\scriptsize 48}}$, M.
Vasileiou${}^{\textrm{\scriptsize 90}}$, A. Vasiliev${}^{\textrm{\scriptsize
81}}$, A. Vauthier${}^{\textrm{\scriptsize 72}}$, O. Vázquez
Doce${}^{\textrm{\scriptsize 96}}$,${}^{\textrm{\scriptsize 36}}$, V.
Vechernin${}^{\textrm{\scriptsize 135}}$, A.M. Veen${}^{\textrm{\scriptsize
54}}$, A. Velure${}^{\textrm{\scriptsize 22}}$, E.
Vercellin${}^{\textrm{\scriptsize 26}}$, S. Vergara
Limón${}^{\textrm{\scriptsize 2}}$, R. Vernet${}^{\textrm{\scriptsize 8}}$, L.
Vickovic${}^{\textrm{\scriptsize 118}}$, J.
Viinikainen${}^{\textrm{\scriptsize 126}}$, Z.
Vilakazi${}^{\textrm{\scriptsize 129}}$, O. Villalobos
Baillie${}^{\textrm{\scriptsize 103}}$, A. Villatoro
Tello${}^{\textrm{\scriptsize 2}}$, A. Vinogradov${}^{\textrm{\scriptsize
81}}$, L. Vinogradov${}^{\textrm{\scriptsize 135}}$, T.
Virgili${}^{\textrm{\scriptsize 30}}$, V. Vislavicius${}^{\textrm{\scriptsize
34}}$, Y.P. Viyogi${}^{\textrm{\scriptsize 136}}$, A.
Vodopyanov${}^{\textrm{\scriptsize 67}}$, M.A. Völkl${}^{\textrm{\scriptsize
95}}$, K. Voloshin${}^{\textrm{\scriptsize 55}}$, S.A.
Voloshin${}^{\textrm{\scriptsize 138}}$, G. Volpe${}^{\textrm{\scriptsize
33}}$,${}^{\textrm{\scriptsize 139}}$, B. von Haller${}^{\textrm{\scriptsize
35}}$, I. Vorobyev${}^{\textrm{\scriptsize 36}}$,${}^{\textrm{\scriptsize
96}}$, D. Vranic${}^{\textrm{\scriptsize 35}}$,${}^{\textrm{\scriptsize 99}}$,
J. Vrláková${}^{\textrm{\scriptsize 40}}$, B. Vulpescu${}^{\textrm{\scriptsize
71}}$, B. Wagner${}^{\textrm{\scriptsize 22}}$, J.
Wagner${}^{\textrm{\scriptsize 99}}$, H. Wang${}^{\textrm{\scriptsize 54}}$,
M. Wang${}^{\textrm{\scriptsize 7}}$, D. Watanabe${}^{\textrm{\scriptsize
131}}$, Y. Watanabe${}^{\textrm{\scriptsize 130}}$, M.
Weber${}^{\textrm{\scriptsize 35}}$,${}^{\textrm{\scriptsize 114}}$, S.G.
Weber${}^{\textrm{\scriptsize 99}}$, D.F. Weiser${}^{\textrm{\scriptsize
95}}$, J.P. Wessels${}^{\textrm{\scriptsize 62}}$, U.
Westerhoff${}^{\textrm{\scriptsize 62}}$, A.M.
Whitehead${}^{\textrm{\scriptsize 91}}$, J. Wiechula${}^{\textrm{\scriptsize
61}}$,${}^{\textrm{\scriptsize 94}}$, J. Wikne${}^{\textrm{\scriptsize 21}}$,
G. Wilk${}^{\textrm{\scriptsize 78}}$, J. Wilkinson${}^{\textrm{\scriptsize
95}}$, G.A. Willems${}^{\textrm{\scriptsize 62}}$, M.C.S.
Williams${}^{\textrm{\scriptsize 106}}$, B. Windelband${}^{\textrm{\scriptsize
95}}$, M. Winn${}^{\textrm{\scriptsize 95}}$, S.
Yalcin${}^{\textrm{\scriptsize 70}}$, P. Yang${}^{\textrm{\scriptsize 7}}$, S.
Yano${}^{\textrm{\scriptsize 47}}$, Z. Yin${}^{\textrm{\scriptsize 7}}$, H.
Yokoyama${}^{\textrm{\scriptsize 131}}$,${}^{\textrm{\scriptsize 72}}$, I.-K.
Yoo${}^{\textrm{\scriptsize 98}}$, J.H. Yoon${}^{\textrm{\scriptsize 51}}$, V.
Yurchenko${}^{\textrm{\scriptsize 3}}$, A. Zaborowska${}^{\textrm{\scriptsize
137}}$, V. Zaccolo${}^{\textrm{\scriptsize 82}}$, A.
Zaman${}^{\textrm{\scriptsize 16}}$, C. Zampolli${}^{\textrm{\scriptsize
106}}$,${}^{\textrm{\scriptsize 35}}$, H.J.C. Zanoli${}^{\textrm{\scriptsize
122}}$, S. Zaporozhets${}^{\textrm{\scriptsize 67}}$, N.
Zardoshti${}^{\textrm{\scriptsize 103}}$, A.
Zarochentsev${}^{\textrm{\scriptsize 135}}$, P. Závada${}^{\textrm{\scriptsize
57}}$, N. Zaviyalov${}^{\textrm{\scriptsize 101}}$, H.
Zbroszczyk${}^{\textrm{\scriptsize 137}}$, I.S. Zgura${}^{\textrm{\scriptsize
59}}$, M. Zhalov${}^{\textrm{\scriptsize 87}}$, H.
Zhang${}^{\textrm{\scriptsize 22}}$,${}^{\textrm{\scriptsize 7}}$, X.
Zhang${}^{\textrm{\scriptsize 7}}$,${}^{\textrm{\scriptsize 75}}$, Y.
Zhang${}^{\textrm{\scriptsize 7}}$, C. Zhang${}^{\textrm{\scriptsize 54}}$, Z.
Zhang${}^{\textrm{\scriptsize 7}}$, C. Zhao${}^{\textrm{\scriptsize 21}}$, N.
Zhigareva${}^{\textrm{\scriptsize 55}}$, D. Zhou${}^{\textrm{\scriptsize 7}}$,
Y. Zhou${}^{\textrm{\scriptsize 82}}$, Z. Zhou${}^{\textrm{\scriptsize 22}}$,
H. Zhu${}^{\textrm{\scriptsize 7}}$,${}^{\textrm{\scriptsize 22}}$, J.
Zhu${}^{\textrm{\scriptsize 115}}$,${}^{\textrm{\scriptsize 7}}$, A.
Zichichi${}^{\textrm{\scriptsize 12}}$,${}^{\textrm{\scriptsize 27}}$, A.
Zimmermann${}^{\textrm{\scriptsize 95}}$, M.B.
Zimmermann${}^{\textrm{\scriptsize 62}}$,${}^{\textrm{\scriptsize 35}}$, G.
Zinovjev${}^{\textrm{\scriptsize 3}}$, M. Zyzak${}^{\textrm{\scriptsize 42}}$

## Affiliation notes

0Deceased

idp1821440Also at: Georgia State University, Atlanta, Georgia, United States

idp3238368Also at: Also at Department of Applied Physics, Aligarh Muslim
University, Aligarh, India

idp3964480Also at: M.V. Lomonosov Moscow State University, D.V. Skobeltsyn
Institute of Nuclear, Physics, Moscow, Russia

## Collaboration Institutes

1A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute)
Foundation, Yerevan, Armenia
2Benemérita Universidad Autónoma de Puebla, Puebla, Mexico
3Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine
4Bose Institute, Department of Physics and Centre for Astroparticle Physics
and Space Science (CAPSS), Kolkata, India
5Budker Institute for Nuclear Physics, Novosibirsk, Russia
6California Polytechnic State University, San Luis Obispo, California, United
States
7Central China Normal University, Wuhan, China
8Centre de Calcul de l’IN2P3, Villeurbanne, Lyon, France
9Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana,
Cuba
10Centro de Investigaciones Energéticas Medioambientales y Tecnológicas
(CIEMAT), Madrid, Spain
11Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City and
Mérida, Mexico
12Centro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche “Enrico
Fermi’, Rome, Italy
13Chicago State University, Chicago, Illinois, United States
14China Institute of Atomic Energy, Beijing, China
15Commissariat à l’Energie Atomique, IRFU, Saclay, France
16COMSATS Institute of Information Technology (CIIT), Islamabad, Pakistan
17Departamento de Física de Partículas and IGFAE, Universidad de Santiago de
Compostela, Santiago de Compostela, Spain
18Department of Physics, Aligarh Muslim University, Aligarh, India
19Department of Physics, Ohio State University, Columbus, Ohio, United States
20Department of Physics, Sejong University, Seoul, South Korea
21Department of Physics, University of Oslo, Oslo, Norway
22Department of Physics and Technology, University of Bergen, Bergen, Norway
23Dipartimento di Fisica dell’Università ’La Sapienza’ and Sezione INFN, Rome,
Italy
24Dipartimento di Fisica dell’Università and Sezione INFN, Cagliari, Italy
25Dipartimento di Fisica dell’Università and Sezione INFN, Trieste, Italy
26Dipartimento di Fisica dell’Università and Sezione INFN, Turin, Italy
27Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Bologna, Italy
28Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Catania, Italy
29Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Padova, Italy
30Dipartimento di Fisica ‘E.R. Caianiello’ dell’Università and Gruppo
Collegato INFN, Salerno, Italy
31Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy
32Dipartimento di Scienze e Innovazione Tecnologica dell’Università del
Piemonte Orientale and INFN Sezione di Torino, Alessandria, Italy
33Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari, Italy
34Division of Experimental High Energy Physics, University of Lund, Lund,
Sweden
35European Organization for Nuclear Research (CERN), Geneva, Switzerland
36Excellence Cluster Universe, Technische Universität München, Munich, Germany
37Faculty of Engineering, Bergen University College, Bergen, Norway
38Faculty of Mathematics, Physics and Informatics, Comenius University,
Bratislava, Slovakia
39Faculty of Nuclear Sciences and Physical Engineering, Czech Technical
University in Prague, Prague, Czech Republic
40Faculty of Science, P.J. Šafárik University, Košice, Slovakia
41Faculty of Technology, Buskerud and Vestfold University College, Tonsberg,
Norway
42Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universität
Frankfurt, Frankfurt, Germany
43Gangneung-Wonju National University, Gangneung, South Korea
44Gauhati University, Department of Physics, Guwahati, India
45Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-
Wilhelms-Universität Bonn, Bonn, Germany
46Helsinki Institute of Physics (HIP), Helsinki, Finland
47Hiroshima University, Hiroshima, Japan
48Indian Institute of Technology Bombay (IIT), Mumbai, India
49Indian Institute of Technology Indore, Indore, India
50Indonesian Institute of Sciences, Jakarta, Indonesia
51Inha University, Incheon, South Korea
52Institut de Physique Nucléaire d’Orsay (IPNO), Université Paris-Sud, CNRS-
IN2P3, Orsay, France
53Institute for Nuclear Research, Academy of Sciences, Moscow, Russia
54Institute for Subatomic Physics of Utrecht University, Utrecht, Netherlands
55Institute for Theoretical and Experimental Physics, Moscow, Russia
56Institute of Experimental Physics, Slovak Academy of Sciences, Košice,
Slovakia
57Institute of Physics, Academy of Sciences of the Czech Republic, Prague,
Czech Republic
58Institute of Physics, Bhubaneswar, India
59Institute of Space Science (ISS), Bucharest, Romania
60Institut für Informatik, Johann Wolfgang Goethe-Universität Frankfurt,
Frankfurt, Germany
61Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt,
Frankfurt, Germany
62Institut für Kernphysik, Westfälische Wilhelms-Universität Münster, Münster,
Germany
63Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
Mexico City, Mexico
64Instituto de Física, Universidad Nacional Autónoma de México, Mexico City,
Mexico
65Institut Pluridisciplinaire Hubert Curien (IPHC), Université de Strasbourg,
CNRS-IN2P3, Strasbourg, France
66iThemba LABS, National Research Foundation, Somerset West, South Africa
67Joint Institute for Nuclear Research (JINR), Dubna, Russia
68Konkuk University, Seoul, South Korea
69Korea Institute of Science and Technology Information, Daejeon, South Korea
70KTO Karatay University, Konya, Turkey
71Laboratoire de Physique Corpusculaire (LPC), Clermont Université, Université
Blaise Pascal, CNRS–IN2P3, Clermont-Ferrand, France
72Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-
Alpes, CNRS-IN2P3, Grenoble, France
73Laboratori Nazionali di Frascati, INFN, Frascati, Italy
74Laboratori Nazionali di Legnaro, INFN, Legnaro, Italy
75Lawrence Berkeley National Laboratory, Berkeley, California, United States
76Moscow Engineering Physics Institute, Moscow, Russia
77Nagasaki Institute of Applied Science, Nagasaki, Japan
78National Centre for Nuclear Studies, Warsaw, Poland
79National Institute for Physics and Nuclear Engineering, Bucharest, Romania
80National Institute of Science Education and Research, Bhubaneswar, India
81National Research Centre Kurchatov Institute, Moscow, Russia
82Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark
83Nikhef, Nationaal instituut voor subatomaire fysica, Amsterdam, Netherlands
84Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom
85Nuclear Physics Institute, Academy of Sciences of the Czech Republic, Řež u
Prahy, Czech Republic
86Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States
87Petersburg Nuclear Physics Institute, Gatchina, Russia
88Physics Department, Creighton University, Omaha, Nebraska, United States
89Physics Department, Panjab University, Chandigarh, India
90Physics Department, University of Athens, Athens, Greece
91Physics Department, University of Cape Town, Cape Town, South Africa
92Physics Department, University of Jammu, Jammu, India
93Physics Department, University of Rajasthan, Jaipur, India
94Physikalisches Institut, Eberhard Karls Universität Tübingen, Tübingen,
Germany
95Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
96Physik Department, Technische Universität München, Munich, Germany
97Purdue University, West Lafayette, Indiana, United States
98Pusan National University, Pusan, South Korea
99Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum
für Schwerionenforschung, Darmstadt, Germany
100Rudjer Bošković Institute, Zagreb, Croatia
101Russian Federal Nuclear Center (VNIIEF), Sarov, Russia
102Saha Institute of Nuclear Physics, Kolkata, India
103School of Physics and Astronomy, University of Birmingham, Birmingham,
United Kingdom
104Sección Física, Departamento de Ciencias, Pontificia Universidad Católica
del Perú, Lima, Peru
105Sezione INFN, Bari, Italy
106Sezione INFN, Bologna, Italy
107Sezione INFN, Cagliari, Italy
108Sezione INFN, Catania, Italy
109Sezione INFN, Padova, Italy
110Sezione INFN, Rome, Italy
111Sezione INFN, Trieste, Italy
112Sezione INFN, Turin, Italy
113SSC IHEP of NRC Kurchatov institute, Protvino, Russia
114Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria
115SUBATECH, Ecole des Mines de Nantes, Université de Nantes, CNRS-IN2P3,
Nantes, France
116Suranaree University of Technology, Nakhon Ratchasima, Thailand
117Technical University of Košice, Košice, Slovakia
118Technical University of Split FESB, Split, Croatia
119The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of
Sciences, Cracow, Poland
120The University of Texas at Austin, Physics Department, Austin, Texas,
United States
121Universidad Autónoma de Sinaloa, Culiacán, Mexico
122Universidade de São Paulo (USP), São Paulo, Brazil
123Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil
124Universidade Federal do ABC, Santo Andre, Brazil
125University of Houston, Houston, Texas, United States
126University of Jyväskylä, Jyväskylä, Finland
127University of Liverpool, Liverpool, United Kingdom
128University of Tennessee, Knoxville, Tennessee, United States
129University of the Witwatersrand, Johannesburg, South Africa
130University of Tokyo, Tokyo, Japan
131University of Tsukuba, Tsukuba, Japan
132University of Zagreb, Zagreb, Croatia
133Université de Lyon, Université Lyon 1, CNRS/IN2P3, IPN-Lyon, Villeurbanne,
Lyon, France
134Università di Brescia, Brescia, Italy
135V. Fock Institute for Physics, St. Petersburg State University, St.
Petersburg, Russia
136Variable Energy Cyclotron Centre, Kolkata, India
137Warsaw University of Technology, Warsaw, Poland
138Wayne State University, Detroit, Michigan, United States
139Wigner Research Centre for Physics, Hungarian Academy of Sciences,
Budapest, Hungary
140Yale University, New Haven, Connecticut, United States
141Yonsei University, Seoul, South Korea
142Zentrum für Technologietransfer und Telekommunikation (ZTT), Fachhochschule
Worms, Worms, Germany


# 4-Bit High-Speed Binary Ling Adder

Projjal Gupta, Electronics and Communication Engineering SRM Institute of
Science and Technology, Kattankulathur Email<EMAIL_ADDRESS>
###### Abstract

Binary addition is one of the most primitive and most commonly used
applications in computer arithmetic. A large variety of algorithms and
implementations have been proposed for binary addition. Huey Ling proposed a
simpler form of CLA equations which rely on adjacent pair bits (ai, bi) and
(ai−1, bi−1). Along with bit generate and bit propagate, we introduce another
prefix bit, the half sum bit. Ling adder increases the speed of n-bit binary
addition, which is an upgrade from the existing Carry-Look-Ahead adder.
Several variants of the carry look-ahead equations, like Ling carries, have
been presented that simplify carry computation and can lead to faster
structures. Ling adders, make use of Ling carry and propagate bits, in order
to calculate the sum bit. As a result, dependency on the previous bit addition
is reduced; that is, ripple effect is lowered. This paper provides a
comparative study on the implementation of the above mentioned high-speed
adders.

###### Index Terms:

Ling Adder, High Speed Binary Adder, Binary Addition.

##  I Introduction

The family of Ling adders is a particularly fast adder and is designed using
H. Ling’s equations and generally implemented in BiCMOS. It is an upgrade to
the already existing Carry-Look-Ahead Adders and is mathematically faster, as
it requires lesser steps for the computation of a sum. The circuit of a Ling
adder is particularly more complex, and is less favourable for use in VLSI
systems due to its complexity and it requires far more extra components than
traditional systems. The circuit is divided into 4 parts, which can be denoted
by H. Ling’s equations.

##  II Analysis of Ling’s Equation

###  II-A Initial Generation of Bits

Ling Adders require to form the bit generate and bit propogate that are used
in the regular Carry look ahead adders. It is denoted by the 3 symbols
$g_{i}$, $p_{i}$ and $d_{i}$.
The generate and propagate bits follow CLA so they can be denoted as,

$g_{i}=a_{i}\cdot b_{i}$ $p_{i}=a_{i}+b_{i}$

However, Ling adder requires an extra half bit term which later on simplifies
the circuit design, while increasing the overall efficiency of the adder. This
half bit generate is denoted by $d_{i}$ and can be mathematically shown by,

$d_{i}=a_{i}\oplus b_{i}$

The above mentioned Generate Bit $g_{i}$ and Propagate Bit $p_{i}$ are used
further to derive the Ling Generates, which are terms that will go on to
simplify the final equation. This is particularly important because these
generates will form the base of the Ling adder circuit design. These are
denoted by $G_{i}^{*}$ and $P_{i}^{*}$

###  II-B The CLA Basis of Ling’s Equations

The CLA depends upon the carry out term of the previous for the new carry
terms.

$c_{i+i}=g_{i}+p_{i}\cdot c_{i}$

which is similar to the Simple Ripple Adder which uses the carry output of the
preceding data bits for forward addition.

Similarly based on the above concept, Ling created a new theoretical carry
generate which he denoted by H. This is used later on in the Adder to generate
the sum $S_{i}$. The term H is given by,

$H_{i}=c_{i}+c_{i-1}$

where,

$c_{i}=H_{i}\cdot p_{i}$

Introduction of ling carry $H_{i}$ is one of the major reasons why Ling Adder
is a fast yet complex adder. Use of Ling carry equations decreases the number
of boolean terms during its operations, but increases the design complexity.

###  II-C Ling Generate and Propagate

Ling proposed the use of Ling Propagate and Ling Generate to simplify the
operations of the Ling adder. It is very important, as this is the first step
where we can see how the terms are generated by using the $i^{th}$ and
$(i-1)^{th}$ terms. These terms can be derived by

$G_{i}^{*}=g_{i}+g_{i-1}$

and

$P_{i}^{*}=p_{i}\cdot p_{i-1}$

Ling generate and propagate terms are used to calculate the Ling carry term
$H$. Later on in the Adder design, the sum terms are directly influenced by
the all the Ling terms.

###  II-D Ling Sum Term

The final sum term for the $i^{th}$ pair terms of a and b are devised by
following Ling sum equations, which take in lesser number of inputs, and hence
decrease the lag in the system.

If we assume that all input gates have only two inputs, we can see that
calculation of CLA carry $C$ requires 5 logic levels, whereas that for ling
carry $H$ requires only four. Although the computation of carry is simplified,
calculation of the sum bits using Ling carries is much more complicated. The
sum bit, when calculated by using traditional carry, is given to be,

$s_{i}=d_{i}\oplus c_{i-1}$

We note that we require to use both the carry output and half-bit term from
the first operation block.

$c_{i}=H_{i}\cdot p_{i}$

Using the above term in the Ling Sum Equation,

$s_{i}=d_{i}\oplus p_{i-1}\cdot H_{i-1}$

on break down,

$s_{i}=H^{\prime}_{i-1}\oplus d_{i}+H_{i-1}(d_{i}\oplus p_{i-1})$

Hence, the output value for $a_{i}+b_{i}$ is given by $s_{i}$ and $c_{i}$.

##  III Ling Carry Equation

###  III-A General expansion and Substitution

Earlier in the CLA basis subsection of Ling Equation analysis, we came across
2 equations,

$H_{i}=c_{i}+c_{i-1}$

and,

$c_{i}=H_{i}\cdot p_{i}$

Since we know that,

$c_{i+1}=g_{i}+p_{i}\cdot c_{i}$

Thus we can write the Carry Output as,

$H_{i}=g_{i}+g_{i-1}+p_{i-1}\cdot g_{i-1}+p_{i-1}\cdot p_{i-2}\cdot
g_{i-2}+.....$ $+p_{i-1}\cdot p_{i-2}\cdot p_{i-1}\cdot.....\cdot p_{1}\cdot
p_{0}\cdot g_{0}$

But we know that,

$G_{i}^{*}=g_{i}+g_{i-1}$ $P_{i}^{*}=p_{i}\cdot p_{i-1}$

Thus we can simplify the $H$ equations to Ling generate-propagate terms.

###  III-B Application in 4-Bit System

In a 4-bit adder design, we require the terms $H_{3}$, $H_{2}$, $H_{1}$ and
$H_{0}$.

From the Ling generate-propagate equations and the expanded Ling Carry
equation in the previous subsection, we can write the 4 terms as,

$H_{3}=G_{3}+P_{2}\cdot G_{1}$ $H_{2}=G_{2}+P_{1}\cdot G_{0}$ $H_{1}=G_{1}$
$H_{0}=G_{0}$

It is noted that the complexity of the system will increase with increase of
Input terms.

##  IV Logic design of 4-Bit Ling Adder

From all the above sections and designs, we can design the 4 bit Ling Adder.

As per the design, The flowing outputs are passed through basic OR and AND
gates to satisfy $H$ equations.The carry is safely calculated as $c_{4}$

$c_{4}=H_{4}\cdot p_{4}$

Similarly, each block present in the Logic Diagram represents an operation
step described in each subsection of the logic analysis.
The use of free gates in the circuit represent the operations used to
calculate $H_{i}$. The initial $g_{-1}$ and $p_{-1}$ dont exist during the
case $i=0$ and hence they are taken as logical $false$ or $zero$ value by
grounding them.
Effectively, the overall circuit follows the final equation

$s=a+b$

and generates a carry term in-case the overall sum exceeds the 4-bit output
range.

##  V PCB and CAD Design

The system can be designed in real time by the use of actual logic gate ICs
belonging to the 74xx family. These ICs usually consist of 16 (DIL16) or 14
(DIL14) pins, and require low power. From the above Logisim design of the Ling
Adder, we can start designing the same circuit on any EDA or CAD software. Due
to its high complexity, the circuit has to be designed on the both sides of a
pcb and uses multiple vias for the on-board connections.

##  VI Conclusion

Hence, a basic 4-Bit Ling adder circuit was designed according to Huey Ling’s
equations. In 4-Bit arithmetic system, the CLA requires 5 terms, whereas the
Ling adder requires a maximum input of 4 terms, thereby decreasing the time
required for computation. When this adder is cascaded for higher number of Bit
input terms, the CLA will come across an increase in operation times. But in a
ling adder, this time increase would be much lesser than the other binary
adders.

## Acknowledgment

The authors would like to thank Mr. AVM Manikandan (Asst. Professor, ECE
Dept.) for his teachings, his support and guidance, under which the project
was successfully completed.

## References

  * [1] H. Ling, ”High Speed Binary Parallel Adder”, IEEE Transactions on Electronic Computers, EC-15, p. 799-809, October, 1966
  * [2] G. Dimitrakopoulos ,D. Nikolos, ”High-speed parallel-prefix VLSI Ling adders”, IEEE Transactions on Computers, January, 2005
  * [3] Deepa Yagain, Vijaya Krishna A, and Akansha Baliga, “Design of High-Speed Adders for Efficient Digital Design Blocks,” ISRN Electronics, vol. 2012, Article ID 253742, 9 pages, 2012. doi:10.5402/2012/253742
  * [4] N. T. Quach, M. J. Flynn, ”High-Speed Addition in CMOS”, IEEE Transactions on Computers, Vol.41, No.12, December, 1992.


119–126

# About the atomic and molecular databases in the planetary community – A
contribution in the Laboratory Astrophysics Data WG IAU 2022 GA session

M. Rengel1 1Max-Planck-Institut für Sonnensystem Forschung, Justus-von-Liebig-
Weg 3, 37077 Göttingen, Germany,
email<EMAIL_ADDRESS>
(2022)

###### Abstract

This paper corresponds to an invited oral contribution to the session 5A
organised by the IAU inter-commission B2-B5 working group (WG) “Laboratory
Astrophysics Data Compilation, Validation and Standardization: from the
Laboratory to FAIR usage in the Astronomical Community” at the IAU 2022
General Assembly (GA) [Rengel (2022)]. This WG provides a platform where to
discuss the Findability, Accessibility,Interoperability, Reuse (FAIR) usage of
laboratory Atomic and Molecular (A&M) data in astronomy and astrophysics.

A&M data play a key role in the understanding of the physics and chemistry of
processes in several research topics, including planetary science and
interdisciplinary research in particular the atmospheres of planets and
planetary explorations, etc. Databases, compilation of spectroscopic
parameters, and facility tools are used by computer codes to interpret
spectroscopic observations and simulate them. In this talk I presented
existing A&M databases of interest to the planetary community focusing on
access, organisation, infrastructures, limitations and issues, etc.

###### keywords:

planets, atmospheres, exoplanets, atomic data, molecular data, laboratory
astrophysics, experiment, databases, data network, data analysis

††volume: 371††journal: IAU 371 Symposium††editors: D. Soderblom & G. Nave,
eds.

##  1 Introduction

The talk represented a tour on A&M databases used in the planetary and
exoplanetary communities not from the perspective of a developer, but of an
user. A&M databases compile and provide detailed spectral information for A&M
to feed codes that predict and simulate radiation in gaseous media. Between
the applications, we find atmospheric physics: (exo) planetary atmospheres,
comets and small bodies. These databases are a critical input for the codes
which predict and interpret spectra of planetary atmospheres (hydrostatic
equilibrium atmospheres and expanding comas), and space and ground-based
telescopes facilities depend on the quality and extent of reference A&M
parameters.

Line lists typically contain hundreds to billions of individual transitions.
A&M databases contains detailed A&M spectroscopic parameters of A&M like
pressure-broadening (shapes), collision-induced absorption (CIA), transition
intensity or cross sections, line shape parameters, rotation-vibration
transition position (wavelength, frequency), parameters to describe how these
vary with temperature and pressure, aerosol indices of refraction,
microphysical and optical properties of atmospheric aerosols, for example.

##  2 Some existing A&M databases - Data and file formats

Several groups worldwide generate and compile A&M data through measurement
and/or calculation (e.g. HITRAN111https://hitran.org[[Gordon et al. (2022)]],
GEISA222https://geisa.aeris-data.fr/[[Jacquinet-Husson et al.(2016)]], JPL
Molecular Spectroscopy333https://spec.jpl.nasa.gov,
CDMS444https://cdms.astro.uni-koeln.de[[Endres et al.(2016)]], VAMDC
555https://vamdc.org[[Dubernet et al. (2010), Dubernet et al. (2016), Albert
et al. (2020)]], ExoMol666https://www.exomol.com[[Tennyson et al. (2020)]],
HITEMP777https://hitran.org/hitemp[[Rothman et al.(2010)]], VALD
http//vald.astro.uu.se, MoLLIST
http://bernath.uwaterloo.ca/molecularlists.php, Ames Molecular Spectroscopic
Data for Astrophysical and Atmospheric Studies (http://huang.seti.org,
TheoReTs https://theorets.univ-reims.fr/, etc.). Several secondary databases
and information services are fed with data from such sources in a fragmented
manner. A variety of data formats (cross sections, K-tables, line-by-line,
super-lines) and file formats (e.g. .hdf5, .pickle, .mp4, .txt, .npy) are
generated. There are online tools such as HAPI to extend functionalitties
https://hitran.org/hapi/[[Kochanov et al.(2016)]], and exo-k library to handle
radiative opacities https://pypi.org/project/exo-k/[[Leconte(2021)]], that
enable conversion between different formats. As part of the spectroscopic
input to atmospheric codes, the HITRAN molecular spectroscopic database is
already internationally recognised as standard in the planetary community, and
the ExoMol database, valid over extended temperature ranges, is widely used in
the exoplanetary community.

##  3 Some (exo)planet atmospheric radiative transfer and inversion codes

One commonly used way to interpret the measured spectra is calculating a
synthetic spectrum for comparison with that measured by solving the radiative
transfer (forward model) -i.e., computation of the outgoing radiation from the
planetary surface for a given set of free parameters-, and inferring
parameters like temperature and abundance. This last step is called inversion
and consists in comparing the measured and best modelled spectra adjusting the
atmospheric parameters in such a way as to minimise any discrepancy. A number
of radiative transfer codes or forward models and inversion algorithms
(retrieval technique) are already generally available and used by the
planetary and exoplanetary characterisation communities (Table 1). When
calculating a synthetic atmospheric spectrum, lines are read from A&M
databases. The retrieval or parameter fitting techniques commonly used are
Optimal Estimation algorithm, nested sampling, Markov chain Monte Carlo
method, and Grid search. Some examples of applications of such algorithms are
given in [[Rengel et al.(2008), Hartogh et al.(2010), Rengel et al.(2014),
Shulyak et al.(2019), Shulyak et al.(2020), Rengel et al.(2022), Villanueva et
al.(2022)]]

Table 1: Some radiative transfer and inversion codes used in the (exo) planetary community Code name | Link | Reference
---|---|---
ARCiS | http://www.exoclouds.com | [Min et al.(2020)]
TauREx | https://taurex3-public.readthedocs.io/en/latest/ | [Al-Refaie et al.(2021)]
NEMESIS | https://users.ox.ac.uk/ atmp0035/nemesis.html | [Irwin et al.(2008)]
petitRADTRANS | https://petitradtrans.readthedocs.io/en/latest/ | [Mollière et al.(2019)]
PSG | https://psg.gsfc.nasa.gov/ | [Villanueva et al.(2018)]
CHIMERA | https://github.com/mrline/CHIMERA | [Line et al.(2013)]
PLATON | https://github.com/ideasrule/platon | [Zhang et al.(2020)]
ATMO | https://www.erc-atmo.eu | [Tremblin et al.(2015)]
MOLIERE | [Urban et al.(2012)] | [Urban(2004)]
ARTS | https://radiativetransfer.org/ | [Buehler et al.(2018)]
Home made (MPS) | - | [Jarchow, C. (1998)]
Helios-r2 | https://github.com/exoclime/Helios-r2 | [Kitzmann et al.(2020)]
SCARLET | - | [Benneke(2015)]
BART | https://github.com/exosports/BART | [Blecic(2016)]
INARA | https://gitlab.com/frontierdevelopmentlab/astrobiology/inara | [Soboczenski et al.(2018)]

##  4 Discussion: Needs and wish-list

In spite of the tremendous advances and current efforts in the generation of
databases, there are still improvements in progress. A growing demand for
spectroscopic data for (exo) planetary studies and other atmospheres is being
driven by scientists who are interested in modelling as well as observing
diverse bodies. Line lists are generated from experiments and/or ab initio
calculations and may be incomplete or contain errors. Databases differ in
completeness, and some ones do not accurately characterise high-frequency
spectral regions. Sometimes there are no datasets for a specific problem at
hand. Atmospheric codes used by the planetary and exoplanetary
characterisation communities, that are designed to solve the radiative
transfer equation by looking at the propagation of radiation through a medium
and simulate observations and infer parameters, have their own methods for the
computation of opacities and there are no community standards. Furthermore,
there are also no community standards in the selection of atmospheric codes in
mission planning. There are needs to increase accessibility of opacities
(computation, access, visualisation, manipulation), laboratory measurements of
molecular cross-sections, and pressure broadening description for some
species, among many other aspects.

Furthermore, going into details, the community identifies needs in the
following aspects: isotopologues: CIA (more lists of N2O, CH4, SO), expansion
of CIA of secondary species for wider temperature and wavelength ranges,
additional experiments for CO intensities, improvement of the CH4 quality of
the line shape parameters, continuum absorption by water vapour, partition
functions at higher temperatures for all species (around 5000 K), more
friendly ways implementing new data in the RT codes, more aerosol refractive
indexes above 500-600 K, more saturation vapour pressure data, kinetic data at
high temperature, Rayleigh scattering for non-H2, collisional xsecs (in
particular: mid-sized organics, H2O+, diatom-H2, CH3OH, HCN, SO2, CH2) and
high-resolution xsecs (R=1E6+) atmospheres, organic sulphide gases for the
line lists (biomarkers).

###### Acknowledgements.

I thank Sergey Yurchenko and Iouli Gordon for the input and discussions. I
thank also the members of the SOC of the session, and the members of the
inter-commission B2-B5 working group in special to Marie-Lise Dubernet. I
acknowledge the support by the DFG priority program SPP 1992 “Exploring the
Diversity of Extrasolar Planets” (DFG PR 36 24602/41). For the purpose of Open
Access, a CC-BY-SA 4.0 public copyright licence has been applied by the author
to the present document and will be applied to all subsequent versions up to
the Author Accepted Manuscript arising from this submission.

## References

  * [Albert et al. (2020)] Albert, D., Anthony, B.L. , Ba, Y.A. & al. 2020, Atoms, 8, 76 10.3390/atoms8040076
  * [Al-Refaie et al.(2021)] Al-Refaie, A. F., Changeat, Q., Waldmann, I. P., et al. 2021, Astrophysical Journal, 917, 37. doi:10.3847/1538-4357/ac0252
  * [Benneke(2015)] Benneke, B. 2015, arXiv:1504.07655
  * [Blecic(2016)] Blecic, J. 2016, arXiv:1604.02692
  * [Buehler et al.(2018)] Buehler, S. A., Mendrok, J., Eriksson, P., et al. 2018, Geoscientific Model Development, 11, 1537. doi:10.5194/gmd-11-1537-2018
  * [Dubernet et al. (2010)] Dubernet, M.-L., Boudon, V., Cullhane, J. L., & al. 2010, JQRST, 111, 2151-2159; 10.1016/j.jqsrt.2010.05.004
  * [Dubernet et al. (2016)] Dubernet, M.-L., Antony, B. K., Ba, Y.-A., & al. 2016, J. of Physics B, 49, 074003 10.1088/0953-4075/49/7/074003
  * [Endres et al.(2016)] Endres, C. P., Schlemmer, S., Schilke, P., et al. 2016, Journal of Molecular Spectroscopy, 327, 95. doi:10.1016/j.jms.2016.03.005
  * [Gordon et al. (2022)] Gordon, I. E., Rothman, L. S., Hargreaves, R. J., et al. 2022, JQSRT, 277, 107949 10.1016/j.jqsrt.2021.107949
  * [Hartogh et al.(2010)] Hartogh, P., Błecka, M. I., Jarchow, C., et al. 2010, A&A, 521, L48. doi:10.1051/0004-6361/201015159
  * [Irwin et al.(2008)] Irwin, P. G. J., Teanby, N. A., de Kok, R., et al. 2008, J. Quant. Spectrosc. and Rad. Trans, 109, 1136. doi:10.1016/j.jqsrt.2007.11.006
  * [Jacquinet-Husson et al.(2016)] Jacquinet-Husson, N., Armante, R., Scott, N. A., et al. 2016, Journal of Molecular Spectroscopy, 327, 31. doi:10.1016/j.jms.2016.06.007
  * [Jarchow, C. (1998)] Jarchow, C., 1998. Bestimmung atmosphärischer Wasserdampf- und Ozone profile mittels bodengebundener Millimeterwellen- Fernerkundung. Ph.D. thesis.
  * [Kitzmann et al.(2020)] Kitzmann, D., Heng, K., Oreshenko, M., et al. 2020, The Astrophysical Journal, 890, 174. doi:10.3847/1538-4357/ab6d71
  * [Kochanov et al.(2016)] Kochanov, R. V., Gordon, I. E., Rothman, L. S., et al. 2016, JQSRT, 177, 15. doi:10.1016/j.jqsrt.2016.03.005
  * [Leconte(2021)] Leconte, J. 2021, A&A, 645, A20. doi:10.1051/0004-6361/202039040
  * [Line et al.(2013)] Line, M. R., Wolf, A., Zhang, X., et al. 2013, The Astrophysical Journal
  * [Min et al.(2020)] Min, M., Ormel, C. W., Chubb, K., et al. 2020, A&A, 642, A28. doi:10.1051/0004-6361/201937377
  * [Mollière et al.(2019)] Mollière, P., Wardenier, J. P., van Boekel, R., et al. 2019, A&A, 627, A67. doi:10.1051/0004-6361/201935470
  * [Rengel et al.(2008)] Rengel, M., Hartogh, P., & Jarchow, C. 2008, Planet. Space Sci, 56, 1368. doi:10.1016/j.pss.2008.07.004
  * [Rengel et al.(2014)] Rengel, M., Sagawa, H., Hartogh, P., et al. 2014, A&A, 561, A4. doi:10.1051/0004-6361/201321945
  * [Rengel et al.(2022)] Rengel, M., Shulyak, D., Hartogh, P., et al. 2022, A&A, 658, A88. doi:10.1051/0004-6361/202141422
  * [Rengel (2022)] Rengel Miriam. (2022, September 1). About the atomic and molecular databases in the planetary community. Zenodo. https://doi.org/10.5281/zenodo.7040446
  * [Rothman et al.(2010)] Rothman, L. S., Gordon, I. E., Barber, R. J., et al. 2010, JQSRT, 111, 2139. doi:10.1016/j.jqsrt.2010.05.001
  * [Shulyak et al.(2019)] Shulyak, D., Rengel, M., Reiners, A., et al. 2019, A&A, 629, A109. doi:10.1051/0004-6361/201935691
  * [Shulyak et al.(2020)] Shulyak, D., Lara, L. M., Rengel, M., et al. 2020, A&A, 639, A48. doi:10.1051/0004-6361/201937210
  * [Soboczenski et al.(2018)] Soboczenski, F., Himes, M. D., O’Beirne, M. D., et al. 2018, arXiv:1811.03390
  * [Tennyson et al. (2020)] Tennyson, J., Yurchenko, S. N., Al-Refaie, A. F., et al. 2020, Journal of Quantitative Spectroscopy and Radiative Transfer, 255, 107228 10.1016/j.jqsrt.2020.107228
  * [Tremblin et al.(2015)] Tremblin, P., Amundsen, D. S., Mourier, P., et al. 2015, The Astrophysical Journal, 804, L17. doi:10.1088/2041-8205/804/1/L17
  * [Urban(2004)] Urban, J. 2004, Journal of Quantitative Spectroscopy and Radiative Transfer, 83, 529. doi:10.1016/S0022-4073(03)00104-3
  * [Urban et al.(2012)] Urban, J., Baron, P., & Lautie, N. 2012, Astrophysics Source Code Library. ascl:1212.004
  * [Villanueva et al.(2018)] Villanueva, G. L., Smith, M. D., Protopapa, S., et al. 2018, Journal of Quantitative Spectroscopy and Radiative Transfer, 217, 86. doi:10.1016/j.jqsrt.2018.05.023
  * [Villanueva et al.(2022)] Villanueva, G. L., Liuzzi, G., Faggi, S., et al. 2022, Fundamentals of the Planetary Spectrum Generator. 2022 edition of the handbook by G.L. Villanueva et al. ISBN 978-0-578-36143-7, 2022
  * [Zhang et al.(2020)] Zhang, M., Chachan, Y., Kempton, E. M.-R., et al. 2020, The Astrophysical Journal, 899, 27. doi:10.3847/1538-4357/aba1e6