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JOSS/analyticcontinuation/paper.md

@@ -35,14 +35,14 @@ In this work, we present the analytic continuation component of the GreenX libra
 
 # Statement of need
 
-Analytic continuation (AC) is used in different scientific fields where complex analysis is relevant, like mathematical function theory, engineering and theoretical physics/chemistry in e.g.  quantum mechanics [@golze2019gw], quantum field theory [@nekrasov2024analytic], numerical methods for solving differential equations [@lope2002analytic] and real-time propagation methods [@li2020real]. In the following, we discuss the four examples depicted in Figure 1. The first example, shown in the top left of Figure 1, involves the application of AC to model functions, which may include Gamma functions [@luke1975error], Zeta functions [@iriguchi2007estimation], and others. 
+Analytic continuation (AC) is used in different scientific fields where complex analysis is relevant, like mathematical function theory, engineering and theoretical physics/chemistry in e.g.  quantum mechanics [@golze2019gw], quantum field theory [@nekrasov2024analytic], numerical methods for solving differential equations [@lope2002analytic] and real-time propagation methods [@li2020real]. In the following, we discuss the four examples depicted in Figure 1. The first example, shown in the top left of Figure 1, involves the application of AC to model functions, which may include Gamma functions [@luke1975error], Zeta functions [@iriguchi2007estimation], and others. 
 
-In quantum field theory, AC can be applied to the frequently arising, complex-valued Green's functions, like the Green's function of the Hubbard model [@schott2016analytic]. However, Green's functions also appear in ab-initio many-body perturbation theory methods like the $GW$ approximation. The $GW$ method [@hedin1965new] is considered the method of choice for predicting band structures of solids as well as electron removal and addition energies of molecules, as measured in direct and indirect photoemission experiments [@golze2019gw]. The complex-valued self energy is a central quantity in the $GW$ method, computed as the convolution of the Green's function $G$ and the screened interaction $W$. AC is a frequently used tool for continuing the self energy from the imaginary to the real frequency axis in conventional scaling $GW$ implementations [@gonze2009abinit; @ren2012resolution; @van2015gw; @wilhelm2017periodic] and low-scaling implementations [@liu2016cubic; @wilhelm2018toward;  @forster2020low;@wilhelm2021low; @forster2021low; @forster2021gw100; @forster2023two; @graml2024low]. More recently, AC has also been applied to the screened Coulomb interaction [@springer1998first; @friedrich2019tetrahedron; @duchemin2020robust; @voora2020molecular; @duchemin2021cubic; @samal2022modeling; @cdwac; @kehry2023robust] to e.g. reduce the computational scaling associated with core-level excitations [@cdwac]. The AC of a self energy $\Sigma$ and a screened Coulomb interaction $W$ are depicted as the second and third example in the bottom panel of Figure 1.
+In quantum field theory, AC can be applied to the frequently arising, complex-valued Green's functions, like the Green's function of the Hubbard model [@schott2016analytic]. However, Green's functions also appear in ab-initio many-body perturbation theory methods like the $GW$ approximation. The $GW$ method [@hedin1965new] is considered the method of choice for predicting band structures of solids as well as electron removal and addition energies of molecules, as measured in direct and indirect photoemission experiments [@golze2019gw]. The complex-valued self energy is a central quantity in the $GW$ method, computed as the convolution of the Green's function $G$ and the screened Coulomb interaction $W$. AC is a frequently used tool for continuing the self energy from the imaginary to the real frequency axis in conventional scaling $GW$ implementations [@gonze2009abinit; @ren2012resolution; @van2015gw; @wilhelm2017periodic] and low-scaling implementations [@liu2016cubic; @wilhelm2018toward;  @forster2020low;@wilhelm2021low; @forster2021low; @forster2021gw100; @forster2023two; @graml2024low]. More recently, AC has also been applied to the screened Coulomb interaction [@springer1998first; @friedrich2019tetrahedron; @duchemin2020robust; @voora2020molecular; @duchemin2021cubic; @samal2022modeling; @cdwac; @kehry2023robust] to e.g. reduce the computational scaling associated with core-level excitations [@cdwac]. The AC of a self energy $\Sigma$ and a screened Coulomb interaction $W$ are depicted as the second and third example in the bottom panel of Figure 1.
 
-![Application of the GX-AnalyticContinuation component to a model function with two poles (top left), an RT-TDDFT UV-vis Absorption spectrum (top right), the $GW$ self energy (bottom left) and the $GW$ screened coulomb interaction (bottom right). More information about the functions that are presented here can be found on the [website of the GX-AC component](https://nomad-coe.github.io/greenX/gx_ac.html).](ac_overview.pdf){label="overview"}
+![Application of the GX-AnalyticContinuation component to a model function with two poles (top left), an RT-TDDFT UV-vis absorption spectrum (top right), the $GW$ self energy (bottom left) and the $GW$ screened Coulomb interaction (bottom right). More information about the functions that are presented here can be found on the [website of the GX-AC component](https://nomad-coe.github.io/greenX/gx_ac.html).](ac_overview.pdf){label="overview"}
 
 Our fourth and final example is the usage of AC in real-time propagation algorithms, such as real-time time-dependent density functional theory (RT-TDDFT) [@li2020real]. RT-TDDFT yields, for example, access to the absorption spectra of molecules and solids via the complex-valued dynamic polarizability tensor. The resolution of the RT-TDDFT absorption spectrum depends on the simulation length. It has been shown that applying Padé approximants to the dynamic polarizability tensor is an effective strategy for achieving higher spectral resolution with much shorter simulation times [@bruner2016accelerated; @mattiat2018efficient].
-An illustrative UV-vis absorption spectrum, with and without the use of AC, is shown in the top right of Figure 1.
+An illustrative UV-vis absorption spectrum, with and without the use of AC, is shown in the top right of Figure 1.
 
 AC of analytic (holomorphic) functions is typically performed by  approximating the function with a rational function in one domain of the complex plane, typically along the imaginary axis. According to the identity theorem, the resulting rational function can then be evaluated over a broader domain of the complex plane, for example, along the real axis. 
 Padé approximants are an established choice for rational functions. Their flexibility enables the approximation of functions with complicated pole structures [@golze2019gw]. Padé approximants can be expressed by the ratio of two polynomials with arbitrary order, or alternatively by a continued fraction [@ThielePade_Baker].