Optimizing functionals using Differential Evolution Cantún-Avila, K.B.; González-Sánchez, D.; Díaz-Infante, S. ...
Engineering applications of artificial intelligence,
January 2021, 2021-01-00, Volume:
97
Journal Article
Peer reviewed
Metaheuristic algorithms are typically used for optimizing a function f:A→R, where A is a subset of RN. Nevertheless, many real-life problems require A to be a set of functions which makes f a ...functional. In this paper, we present a methodology to address the optimization of functionals by using the evolutionary algorithm known as Differential Evolution. Unlike traditional techniques where continuity and differentiability assumptions are required to solve some associated differential equations—like calculus of variations, Pontryagin’s principle or dynamic programming, the optimization is carried out directly on the functional without the need of any of the assumptions mentioned before. Lagrangians involving derivatives are considered, these derivatives are computed implementing Automatic Differentiation with dual numbers. To the best of our knowledge, this is the first time that a metaheuristic optimization approach has been applied to directly optimize a broad variety of functionals. The effectiveness of our methodology is validated by solving two problems. The first problem is related to the implementation of quarantine and isolation in SARS epidemics and the second validation problem deals with the well-known brachistochrone curve problem. The results of both validation problems are in outstanding agreement with those obtained with the application of traditional techniques, specifically with the Forward–Backward-Sweep method in the first problem, and with the calculus of variations for the latter problem. We also found that interpolation may be employed to solve the large scale global optimization problems arisen in the optimization of functionals.
•A methodology to optimize functionals is presented.•Functionals depending on derivatives are considered.•Automatic differentiation is used to compute derivatives.•Interpolation is proposed to cope with the originated LSGO problem.
Charged particle multiplicity distributions in positron-proton deep inelastic scattering at a centre-of-mass energy
s
=
319
GeV are measured. The data are collected with the H1 detector at HERA ...corresponding to an integrated luminosity of 136 pb
-
1
. Charged particle multiplicities are measured as a function of photon virtuality
Q
2
, inelasticity
y
and pseudorapidity
η
in the laboratory and the hadronic centre-of-mass frames. Predictions from different Monte Carlo models are compared to the data. The first and second moments of the multiplicity distributions are determined and the KNO scaling behaviour is investigated. The multiplicity distributions as a function of
Q
2
and the Bjorken variable
x
bj
are converted to the hadron entropy
S
hadron
, and predictions from a quantum entanglement model are tested.
This contribution proposes a new formulation to efficiently compute directional derivatives of order one to fourth. The formulation is based on automatic differentiation implemented with dual ...numbers. Directional derivatives are particular cases of symmetric multilinear forms; therefore, using their symmetric properties and their coordinate representation, we implement functions to calculate mixed partial derivatives. Moreover, with directional derivatives, we deduce concise formulas for the velocity, acceleration, jerk, and jounce/snap vectors. The utility of our formulation is proved with three examples. The first example presents a comparison against the forward mode of finite differences to compute the fourth-order directional derivative of a scalar function. To this end, we have coded the finite differences method to calculate partial derivatives until the fourth order, to any order of approximation. The second example presents efficient computations of the velocity, acceleration, jerk, and jounce/snap. Finally, the third example is related to the computation of some partial derivatives. The implemented code of the proposed formulation and the finite differences method is proportioned as additional material to this article.
•A dual number formulation to efficiently compute directional derivatives for vector functions is presented.•Four Matlab classes to compute directional derivatives of orders one to fourth are implemented.•Formulas to compute directional derivatives along different vectors using dual numbers are deduced.•Formulas for the velocity and acceleration in terms of directional derivatives are presented.•The forward mode of finite differences to any order of approximation is implemented.
A precision measurement of jet cross sections in neutral current deep-inelastic scattering for photon virtualities
5.5
<
Q
2
<
80
GeV
2
and inelasticities
0.2
<
y
<
0.6
is presented, using data taken ...with the H1 detector at HERA, corresponding to an integrated luminosity of
290
pb
-
1
. Double-differential inclusive jet, dijet and trijet cross sections are measured simultaneously and are presented as a function of jet transverse momentum observables and as a function of
Q
2
. Jet cross sections normalised to the inclusive neutral current DIS cross section in the respective
Q
2
-interval are also determined. Previous results of inclusive jet cross sections in the range
150
<
Q
2
<
15
,
000
GeV
2
are extended to low transverse jet momenta
5
<
P
T
jet
<
7
GeV
. The data are compared to predictions from perturbative QCD in next-to-leading order in the strong coupling, in approximate next-to-next-to-leading order and in full next-to-next-to-leading order. Using also the recently published H1 jet data at high values of
Q
2
, the strong coupling constant
α
s
(
M
Z
)
is determined in next-to-leading order.
The Breit frame provides a natural frame to analyze lepton–proton scattering events. In this reference frame, the parton model hard interactions between a quark and an exchanged boson defines the ...coordinate system such that the struck quark is back-scattered along the virtual photon momentum direction. In Quantum Chromodynamics (QCD), higher order perturbative or non-perturbative effects can change this picture drastically. As Bjorken-
x
decreases below one half, a rather peculiar event signature is predicted with increasing probability, where no radiation is present in one of the two Breit-frame hemispheres and all emissions are to be found in the other hemisphere. At higher orders in
α
s
or in the presence of soft QCD effects, predictions of the rate of these events are far from trivial, and that motivates measurements with real data. We report on the first observation of the empty current hemisphere events in electron–proton collisions at the HERA collider using data recorded with the H1 detector at a center-of-mass energy of 319 GeV. The fraction of inclusive neutral-current DIS events with an empty hemisphere is found to be
0.0112
±
3.9
%
stat
±
4.5
%
syst
±
1.6
%
mod
in the selected kinematic region of
150
<
Q
2
<
1500
GeV
2
and inelasticity
0.14
<
y
<
0.7
. The data sample corresponds to an integrated luminosity of 351.1 pb
-
1
, sufficient to enable differential cross section measurements of these events. The results show an enhanced discriminating power at lower Bjorken-
x
among different Monte Carlo event generator predictions.
The H1 Collaboration reports the first measurement of the 1-jettiness event shape observable
τ
1
b
in neutral-current deep-inelastic electron-proton scattering (DIS). The observable
τ
1
b
is ...equivalent to a thrust observable defined in the Breit frame. The data sample was collected at the HERA
ep
collider in the years 2003–2007 with center-of-mass energy of
s
=
319
GeV
, corresponding to an integrated luminosity of 351.1
pb
-
1
. Triple differential cross sections are provided as a function of
τ
1
b
, event virtuality
Q
2
, and inelasticity
y
, in the kinematic region
Q
2
>
150
GeV
2
. Single differential cross sections are provided as a function of
τ
1
b
in a limited kinematic range. Double differential cross sections are measured, in contrast, integrated over
τ
1
b
and represent the inclusive neutral-current DIS cross section measured as a function of
Q
2
and
y
. The data are compared to a variety of predictions and include long-standing and more recent Monte Carlo event generators, predictions in fixed-order perturbative QCD where calculations up to
O
(
α
s
3
)
are available for
τ
1
b
or inclusive DIS, and resummed predictions at next-to-leading logarithmic accuracy matched to fixed order predictions at
O
(
α
s
2
)
. These comparisons reveal sensitivity of the 1-jettiness observable to QCD parton shower and resummation effects, as well as the modeling of hadronization and fragmentation. Within their range of validity, the fixed-order predictions provide a good description of the data. Monte Carlo event generators are predictive over the full measured range and hence their underlying models and parameters can be constrained by comparing to the presented data.
A precision measurement of jet cross sections in neutral current deep-inelastic scattering for photon virtualities Formula omitted and inelasticities Formula omitted is presented, using data taken ...with the H1 detector at HERA, corresponding to an integrated luminosity of Formula omitted. Double-differential inclusive jet, dijet and trijet cross sections are measured simultaneously and are presented as a function of jet transverse momentum observables and as a function of Formula omitted. Jet cross sections normalised to the inclusive neutral current DIS cross section in the respective Formula omitted-interval are also determined. Previous results of inclusive jet cross sections in the range Formula omitted are extended to low transverse jet momenta Formula omitted. The data are compared to predictions from perturbative QCD in next-to-leading order in the strong coupling, in approximate next-to-next-to-leading order and in full next-to-next-to-leading order. Using also the recently published H1 jet data at high values of Formula omitted, the strong coupling constant Formula omitted is determined in next-to-leading order.
The measurement of the jet cross sections by the H1 collaboration had been compared to various predictions including the next-to-next-to-leading order (NNLO) QCD calculations which are corrected in ...this erratum for an implementation error in one of the components of the NNLO calculations. The jet data and the other predictions remain unchanged. Eight figures, one table and conclusions are adapted accordingly, exhibiting even better agreement between the corrected NNLO predictions and the jet data.
The determination of the strong coupling constant
α
s
(
m
Z
)
from H1 inclusive and dijet cross section data
1
exploits perturbative QCD predictions in next-to-next-to-leading order (NNLO)
2
–
4
. ...An implementation error in the NNLO predictions was found
4
which changes the numerical values of the predictions and the resulting values of the fits. Using the corrected NNLO predictions together with inclusive jet and dijet data, the strong coupling constant is determined to be
α
s
(
m
Z
)
=
0.1166
(
19
)
exp
(
24
)
th
. Complementarily,
α
s
(
m
Z
)
is determined together with parton distribution functions of the proton (PDFs) from jet and inclusive DIS data measured by the H1 experiment. The value
α
s
(
m
Z
)
=
0.1147
(
25
)
tot
obtained is consistent with the determination from jet data alone. Corrected figures and numerical results are provided and the discussion is adapted accordingly.
A first measurement is reported of the longitudinal proton structure function FL(X, Q(2)) at the ep collider HERA. It is based on inclusive deep inelastic e(+)p scattering cross section measurements ...with a positron beam energy of 27.5 GeV and proton beam energies of 920, 575 and 460 GeV. Employing the energy dependence of the cross section, FL is measured in a range of squared four-momentum transfers 12 <= Q2 <=, 90 GeV2 and low Bjorken x 0.00024 <= x <= 0.0036. The F-L values agree with higher order QCD calculations based on parton densities obtained using cross section data previously measured at HERA. (C) 2008 Elsevier B.V. All rights reserved.
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