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Reference and error bars in FCI

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Manuscript/rsdft-cipsi-qmc.bib
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@ -935,3 +935,19 @@
Url = {https://arxiv.org/abs/1812.06902v1}, Url = {https://arxiv.org/abs/1812.06902v1},
Year = {2018}, Year = {2018},
Bdsk-Url-1 = {https://arxiv.org/abs/1812.06902v1}} Bdsk-Url-1 = {https://arxiv.org/abs/1812.06902v1}}
@software{qp2_2020,
author = {Anthony Scemama and
Emmanuel Giner and
Anouar Benali and
Thomas Applencourt and
Kevin Gasperich},
title = {QuantumPackage/qp2: Version 2.1.2},
month = feb,
year = 2020,
publisher = {Zenodo},
version = {2.1.2},
doi = {10.5281/zenodo.3677565},
url = {https://doi.org/10.5281/zenodo.3677565}
}

145
Manuscript/rsdft-cipsi-qmc.tex
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@ -13,10 +13,19 @@
]{hyperref} ]{hyperref}
\urlstyle{same} \urlstyle{same}
\newcommand{\ie}{\textit{i.e.}}
\newcommand{\eg}{\textit{e.g.}}
\newcommand{\alert}[1]{\textcolor{red}{#1}} \newcommand{\alert}[1]{\textcolor{red}{#1}}
\newcommand{\eg}[1]{\textcolor{blue}{#1}}
\definecolor{darkgreen}{HTML}{009900} \definecolor{darkgreen}{HTML}{009900}
\usepackage[normalem]{ulem} \usepackage[normalem]{ulem}
\newcommand{\titou}[1]{\textcolor{red}{#1}}
\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
\newcommand{\mc}{\multicolumn}
\newcommand{\fnm}{\footnotemark}
\newcommand{\fnt}{\footnotetext}
\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
\newcommand{\EPT}{E_{\text{PT2}}} \newcommand{\EPT}{E_{\text{PT2}}}
\newcommand{\EDMC}{E_{\text{FN-DMC}}} \newcommand{\EDMC}{E_{\text{FN-DMC}}}
@ -93,20 +102,20 @@ sets because the constraints imposed by the FN approximation
are less severe than the constraints imposed by the finite-basis are less severe than the constraints imposed by the finite-basis
approximation. approximation.
\alert{However, it is usually harder to control the FN error in DMC, and this might affect energy differences such as atomization energies. \titou{However, it is usually harder to control the FN error in DMC, and this might affect energy differences such as atomization energies.
Moreover, improving systematically the nodal surface of the trial wave function can be a tricky job as there is no variational principle for the nodes.} Moreover, improving systematically the nodal surface of the trial wave function can be a tricky job as there is no variational principle for the nodes.}
The qualitative picture of the electronic structure of weakly The qualitative picture of the electronic structure of weakly
correlated systems, such as organic molecules near their equilibrium correlated systems, such as organic molecules near their equilibrium
geometry, is usually well represented with a single Slater geometry, is usually well represented with a single Slater
determinant. This feature is in part responsible for the success of determinant. This feature is in part responsible for the success of
density-functional theory (DFT) and coupled cluster. density-functional theory (DFT) and coupled cluster theory.
DMC with a single-determinant trial wave function can be used as a DMC with a single-determinant trial wave function can be used as a
single-reference post-Hatree-Fock method, with an accuracy comparable single-reference post-Hatree-Fock method, with an accuracy comparable
to coupled cluster.\cite{Dubecky_2014,Grossman_2002} to coupled cluster.\cite{Dubecky_2014,Grossman_2002}
The favorable scaling of QMC, its very low memory requirements and The favorable scaling of QMC, its very low memory requirements and
its adequacy with massively parallel architectures make it a its adequacy with massively parallel architectures make it a
serious alternative for high-accuracy simulations on large systems. serious alternative for high-accuracy simulations of large systems.
As it is not possible to minimize directly the FN-DMC energy with respect As it is not possible to minimize directly the FN-DMC energy with respect
to the variational parameters of the trial wave function, the to the variational parameters of the trial wave function, the
@ -115,7 +124,7 @@ finite-basis approximation.
The conventional approach consists in multiplying the trial wave The conventional approach consists in multiplying the trial wave
function by a positive function, the \emph{Jastrow factor}, taking function by a positive function, the \emph{Jastrow factor}, taking
account of the electron-electron cusp and the short-range correlation account of the electron-electron cusp and the short-range correlation
effects. The wave function is then re-optimized within Variational effects. The wave function is then re-optimized within variational
Monte Carlo (VMC) in the presence of the Jastrow factor and the nodal Monte Carlo (VMC) in the presence of the Jastrow factor and the nodal
surface is expected to be improved. Using this technique, it has been surface is expected to be improved. Using this technique, it has been
shown that the chemical accuracy could be reached within shown that the chemical accuracy could be reached within
@ -125,13 +134,13 @@ Another approach consists in considering the FN-DMC method as a
\emph{post-FCI method}. The trial wave function is obtained by \emph{post-FCI method}. The trial wave function is obtained by
approaching the FCI with a selected configuration interaction (sCI) approaching the FCI with a selected configuration interaction (sCI)
method such as CIPSI for instance.\cite{Giner_2013,Caffarel_2016_2} method such as CIPSI for instance.\cite{Giner_2013,Caffarel_2016_2}
When the basis set is increased, the trial wave function gets closer \titou{When the basis set is increased, the trial wave function gets closer
to the exact wave function, so the nodal surface can be systematically to the exact wave function, so the nodal surface can be systematically
improved.\cite{Caffarel_2016} improved.\cite{Caffarel_2016} WRONG}
This technique has the advantage that using FCI nodes in a given basis This technique has the advantage that using FCI nodes in a given basis
set is well defined, so the calculations are reproducible in a set is well defined, so the calculations are reproducible in a
black-box way without needing any expertise in QMC. black-box way without needing any expertise in QMC.
But this technique can't be applied to large systems because of the But this technique cannot be applied to large systems because of the
exponential scaling of the size of the trial wave function. exponential scaling of the size of the trial wave function.
Extrapolation techniques have been used to estimate the FN-DMC energies Extrapolation techniques have been used to estimate the FN-DMC energies
obtained with FCI wave functions,\cite{Scemama_2018} and other authors obtained with FCI wave functions,\cite{Scemama_2018} and other authors
@ -339,9 +348,9 @@ post-HF method of interest.
All the calculations were made using BFD All the calculations were made using BFD
pseudopotentials\cite{Burkatzki_2008} with the associated double, pseudopotentials\cite{Burkatzki_2008} with the associated double-,
triple and quadruple zeta basis sets (BFD-V$n$Z). triple-, and quadruple-$\zeta$ basis sets (BFD-VXZ).
CCSD(T) and DFT calculations were made with CCSD(T) and KS-DFT calculations were made with
\emph{Gaussian09},\cite{g16} using an unrestricted Hartree-Fock \emph{Gaussian09},\cite{g16} using an unrestricted Hartree-Fock
determinant as a reference for open-shell systems. determinant as a reference for open-shell systems.
@ -361,7 +370,7 @@ in the determinant localization approximation (DLA),\cite{Zen_2019}
where only the determinantal component of the trial wave where only the determinantal component of the trial wave
function is present in the expression of the wave function on which function is present in the expression of the wave function on which
the pseudopotential is localized. Hence, in the DLA the fixed-node the pseudopotential is localized. Hence, in the DLA the fixed-node
energy is independent of the Jatrow factor, as in all-electron energy is independent of the Jastrow factor, as in all-electron
calculations. Simple Jastrow factors were used to reduce the calculations. Simple Jastrow factors were used to reduce the
fluctuations of the local energy. fluctuations of the local energy.
@ -372,36 +381,37 @@ fluctuations of the local energy.
error} error}
\label{sec:mu-dmc} \label{sec:mu-dmc}
\begin{table} \begin{table}
\caption{Fixed-node energies and number of determinants in the water \caption{Fixed-node energies (in hartree) and number of determinants in \ce{H2O} and \ce{F2} with various trial wave functions.}
molecule and the fluorine dimer with different trial wave functions.}
\label{tab:h2o-dmc} \label{tab:h2o-dmc}
\centering \centering
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{crlrl} \begin{tabular}{ccrlrl}
& \multicolumn{2}{c}{BFD-VDZ} & \multicolumn{2}{c}{BFD-VTZ} \\ & & \multicolumn{2}{c}{BFD-VDZ} & \multicolumn{2}{c}{BFD-VTZ} \\
$\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\EDMC$ \\ \cline{3-4} \cline{5-6}
System & $\mu$ & $\Ndet$ & $\EDMC$ & $\Ndet$ & $\EDMC$ \\
\hline \hline
& \multicolumn{4}{c}{H$_2$O} \\ \ce{H2O}
$0.00$ & $11$ & $-17.253\,59(6)$ & $23$ & $-17.256\,74(7)$ \\ & $0.00$ & $11$ & $-17.253\,59(6)$ & $23$ & $-17.256\,74(7)$ \\
$0.20$ & $23$ & $-17.253\,73(7)$ & $23$ & $-17.256\,73(8)$ \\ & $0.20$ & $23$ & $-17.253\,73(7)$ & $23$ & $-17.256\,73(8)$ \\
$0.30$ & $53$ & $-17.253\,4(2)$ & $219$ & $-17.253\,7(5)$ \\ & $0.30$ & $53$ & $-17.253\,4(2)$ & $219$ & $-17.253\,7(5)$ \\
$0.50$ & $1\,442$ & $-17.253\,9(2)$ & $16\,99$ & $-17.257\,7(2)$ \\ & $0.50$ & $1\,442$ & $-17.253\,9(2)$ & $16\,99$ & $-17.257\,7(2)$ \\
$0.75$ & $3\,213$ & $-17.255\,1(2)$ & $13\,362$ & $-17.258\,4(3)$ \\ & $0.75$ & $3\,213$ & $-17.255\,1(2)$ & $13\,362$ & $-17.258\,4(3)$ \\
$1.00$ & $6\,743$ & $-17.256\,6(2)$ & $256\,73$ & $-17.261\,0(2)$ \\ & $1.00$ & $6\,743$ & $-17.256\,6(2)$ & $256\,73$ & $-17.261\,0(2)$ \\
$1.75$ & $54\,540$ & $-17.259\,5(3)$ & $207\,475$ & $-17.263\,5(2)$ \\ & $1.75$ & $54\,540$ & $-17.259\,5(3)$ & $207\,475$ & $-17.263\,5(2)$ \\
$2.50$ & $51\,691$ & $-17.259\,4(3)$ & $858\,123$ & $-17.264\,3(3)$ \\ & $2.50$ & $51\,691$ & $-17.259\,4(3)$ & $858\,123$ & $-17.264\,3(3)$ \\
$3.80$ & $103\,059$ & $-17.258\,7(3)$ & $1\,621\,513$ & $-17.263\,7(3)$ \\ & $3.80$ & $103\,059$ & $-17.258\,7(3)$ & $1\,621\,513$ & $-17.263\,7(3)$ \\
$5.70$ & $102\,599$ & $-17.257\,7(3)$ & $1\,629\,655$ & $-17.263\,2(3)$ \\ & $5.70$ & $102\,599$ & $-17.257\,7(3)$ & $1\,629\,655$ & $-17.263\,2(3)$ \\
$8.50$ & $101\,803$ & $-17.257\,3(3)$ & $1\,643\,301$ & $-17.263\,3(4)$ \\ & $8.50$ & $101\,803$ & $-17.257\,3(3)$ & $1\,643\,301$ & $-17.263\,3(4)$ \\
$\infty$ & $200\,521$ & $-17.256\,8(6)$ & $1\,631\,982$ & $-17.263\,9(3)$ \\ & $\infty$ & $200\,521$ & $-17.256\,8(6)$ & $1\,631\,982$ & $-17.263\,9(3)$ \\
& \multicolumn{3}{c}{F$_2$} \\ \\
$0.00$ & $23$ & $-48.419\,5(4)$ \\ \ce{F2}
$0.25$ & $8$ & $-48.421\,9(4)$ \\ & $0.00$ & $23$ & $-48.419\,5(4)$ \\
$0.50$ & $1743$ & $-48.424\,8(8)$ \\ & $0.25$ & $8$ & $-48.421\,9(4)$ \\
$1.00$ & $11952$ & $-48.432\,4(3)$ \\ & $0.50$ & $1743$ & $-48.424\,8(8)$ \\
$2.00$ & $829438$ & $-48.441\,0(7)$ \\ & $1.00$ & $11952$ & $-48.432\,4(3)$ \\
$5.00$ & $5326459$ & $-48.445(2)$ \\ & $2.00$ & $829438$ & $-48.441\,0(7)$ \\
$\infty$ & $8302442$ & $-48.437(3)$ \\ & $5.00$ & $5326459$ & $-48.445(2)$ \\
& $\infty$ & $8302442$ & $-48.437(3)$ \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\end{table} \end{table}
@ -515,7 +525,7 @@ the density and of electron correlation, but also reduces the
imbalance in the quality of the description of the atoms and the imbalance in the quality of the description of the atoms and the
molecule, leading to more accurate atomization energies. molecule, leading to more accurate atomization energies.
\subsection{Size-consistency} \subsection{Size consistency}
An extremely important feature required to get accurate An extremely important feature required to get accurate
atomization energies is size-consistency (or strict separability), atomization energies is size-consistency (or strict separability),
@ -589,12 +599,12 @@ Ref.~\onlinecite{Scemama_2015}).
%\begin{squeezetable} %\begin{squeezetable}
\begin{table} \begin{table}
\caption{FN-DMC Energies of the fluorine atom and the dissociated fluorine \caption{FN-DMC energies (in hartree) of the fluorine atom and the dissociated fluorine
dimer, and size-consistency error.} dimer, and size-consistency error. \titou{BASIS?}}
\label{tab:size-cons} \label{tab:size-cons}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{cccc} \begin{tabular}{cccc}
$\mu$ & F & Dissociated F$_2$ & Size-consistency error \\ $\mu$ & \ce{F} & Dissociated \ce{F2} & Size-consistency error \\
\hline \hline
0.00 & $-24.188\,7(3)$ & $-48.377\,7(3)$ & $-0.000\,3(4)$ \\ 0.00 & $-24.188\,7(3)$ & $-48.377\,7(3)$ & $-0.000\,3(4)$ \\
0.25 & $-24.188\,7(3)$ & $-48.377\,2(4)$ & $+0.000\,2(5)$ \\ 0.25 & $-24.188\,7(3)$ & $-48.377\,2(4)$ & $+0.000\,2(5)$ \\
@ -619,7 +629,7 @@ size-consistent FN-DMC energies for all values of $\mu$ (within
$2\times$ statistical error bars). $2\times$ statistical error bars).
\subsection{Spin-invariance} \subsection{Spin invariance}
Closed-shell molecules often dissociate into open-shell Closed-shell molecules often dissociate into open-shell
fragments. To get reliable atomization energies, it is important to fragments. To get reliable atomization energies, it is important to
@ -644,7 +654,7 @@ So in the context of RS-DFT, the determinantal expansions will be
impacted by this spurious effect, as opposed to FCI. impacted by this spurious effect, as opposed to FCI.
\begin{table} \begin{table}
\caption{FN-DMC Energies of the triplet carbon atom (BFD-VDZ) with \caption{FN-DMC energies (in hartree) of the triplet carbon atom (BFD-VDZ) with
different values of $m_s$.} different values of $m_s$.}
\label{tab:spin} \label{tab:spin}
\begin{ruledtabular} \begin{ruledtabular}
@ -687,38 +697,39 @@ noticeable with $\mu=5$~bohr$^{-1}$.
\begin{squeezetable} \begin{squeezetable}
\begin{table*} \begin{table*}
\caption{Mean absolute error (MAE), mean signed errors (MSE) and \caption{Mean absolute errors (MAE), mean signed errors (MSE) and
standard deviations (RMSD) obtained with the different methods and standard deviations (RMSD) obtained with various methods and
basis sets.} basis sets.}
\label{tab:mad} \label{tab:mad}
\begin{ruledtabular} \begin{ruledtabular}
\begin{tabular}{ll rrr rrr rrr} \begin{tabular}{ll ddd ddd ddd}
Method & \(\mu\) & \phantom{} & VDZ-BFD & \phantom{} & \phantom{} & VTZ-BFD & \phantom{} & \phantom{} & VQZ-BFD & \phantom{} \\ & & \mc{3}{c}{VDZ-BFD} & \mc{3}{c}{VTZ-BFD} & \mc{3}{c}{VQZ-BFD} \\
\phantom{} & \phantom{} & MAE & MSE & RMSD & MAE & MSE & RMSD & MAE & MSE & RMSD \\ \cline{3-5} \cline{6-8} \cline{9-11}
Method & $\mu$ & \tabc{MAE} & \tabc{MSE} & \tabc{RMSD} & \tabc{MAE} & \tabc{MSE} & \tabc{RMSD} & \tabc{MAE} & \tabc{MSE} & \tabc{RMSD} \\
\hline \hline
PBE & 0 & 5.02 & -3.70 & 6.04 & 4.57 & 1.00 & 5.32 & 5.31 & 0.79 & 6.27 \\ PBE & 0 & 5.02 & -3.70 & 6.04 & 4.57 & 1.00 & 5.32 & 5.31 & 0.79 & 6.27 \\
BLYP & 0 & 9.53 & -9.21 & 7.91 & 5.58 & -4.44 & 5.80 & 5.86 & -4.47 & 6.43 \\ BLYP & 0 & 9.53 & -9.21 & 7.91 & 5.58 & -4.44 & 5.80 & 5.86 & -4.47 & 6.43 \\
PBE0 & 0 & 11.20 & -10.98 & 8.68 & 6.40 & -5.78 & 5.49 & 6.28 & -5.65 & 5.08 \\ PBE0 & 0 & 11.20 & -10.98 & 8.68 & 6.40 & -5.78 & 5.49 & 6.28 & -5.65 & 5.08 \\
B3LYP & 0 & 11.27 & -10.98 & 9.59 & 7.27 & -5.77 & 6.63 & 6.75 & -5.53 & 6.09 \\ B3LYP & 0 & 11.27 & -10.98 & 9.59 & 7.27 & -5.77 & 6.63 & 6.75 & -5.53 & 6.09 \\
\hline \\
CCSD(T) & \(\infty\) & 24.10 & -23.96 & 13.03 & 9.11 & -9.10 & 5.55 & 4.52 & -4.38 & 3.60 \\ CCSD(T) & \(\infty\) & 24.10 & -23.96 & 13.03 & 9.11 & -9.10 & 5.55 & 4.52 & -4.38 & 3.60 \\
\hline \\
RS-DFT-CIPSI & 0 & 4.53 & -1.66 & 5.91 & 6.31 & 0.91 & 7.93 & 6.35 & 3.88 & 7.20 \\ RS-DFT-CIPSI & 0 & 4.53 & -1.66 & 5.91 & 6.31 & 0.91 & 7.93 & 6.35 & 3.88 & 7.20 \\
\phantom{} & 1/4 & 5.55 & -4.66 & 5.52 & 4.58 & 1.06 & 5.72 & 5.48 & 1.52 & 6.93 \\ & 1/4 & 5.55 & -4.66 & 5.52 & 4.58 & 1.06 & 5.72 & 5.48 & 1.52 & 6.93 \\
\phantom{} & 1/2 & 13.42 & -13.27 & 7.36 & 6.77 & -6.71 & 4.56 & 6.35 & -5.89 & 5.18 \\ & 1/2 & 13.42 & -13.27 & 7.36 & 6.77 & -6.71 & 4.56 & 6.35 & -5.89 & 5.18 \\
\phantom{} & 1 & 17.07 & -16.92 & 9.83 & 9.06 & -9.06 & 5.88 & --- & --- & --- \\ & 1 & 17.07 & -16.92 & 9.83 & 9.06 & -9.06 & 5.88 & & & \\
\phantom{} & 2 & 19.20 & -19.05 & 10.91 & --- & --- & --- & --- & --- & --- \\ & 2 & 19.20 & -19.05 & 10.91 & & & & & & \\
\phantom{} & 5 & 22.93 & -22.79 & 13.24 & --- & --- & --- & --- & --- & --- \\ & 5 & 22.93 & -22.79 & 13.24 & & & & & & \\
\phantom{} & \(\infty\) & 23.63(4) & -23.49(4) & 12.81(4) & 8.43(39) & -8.43(39) & 4.87(7) & 4.51(78) & -4.18(78) & 4.19(20) \\ & \(\infty\) & 23.63(4) & -23.49(4) & 12.81(4) & 8.43(39) & -8.43(39) & 4.87(7) & 4.51(78) & -4.18(78) & 4.19(20) \\
\hline \\
DMC@RS-DFT-CIPSI & 0 & 4.61(\phantom{0.}34) & -3.62(\phantom{0.}34) & 5.30(\phantom{0.}09) & 3.52(19) & -1.03(19) & 4.39(04) & 3.16(26) & -0.12(26) & 4.12(03) \\ DMC@ & 0 & 4.61(34) & -3.62(34) & 5.30(09) & 3.52(19) & -1.03(19) & 4.39(04) & 3.16(26) & -0.12(26) & 4.12(03) \\
\phantom{} & 1/4 & 4.04(\phantom{0.}37) & -3.13(\phantom{0.}37) & 4.88(\phantom{0.}10) & 3.39(77) & -0.59(77) & 4.44(34) & 2.90(25) & 0.25(25) & 3.745(5) \\ RS-DFT-CIPSI & 1/4 & 4.04(37) & -3.13(37) & 4.88(10) & 3.39(77) & -0.59(77) & 4.44(34) & 2.90(25) & 0.25(25) & 3.745(5) \\
\phantom{} & 1/2 & 3.74(\phantom{0.}35) & -3.53(\phantom{0.}35) & 4.03(\phantom{0.}23) & 2.46(18) & -1.72(18) & 3.02(06) & 2.06(35) & -0.44(35) & 2.74(13) \\ & 1/2 & 3.74(35) & -3.53(35) & 4.03(23) & 2.46(18) & -1.72(18) & 3.02(06) & 2.06(35) & -0.44(35) & 2.74(13) \\
\phantom{} & 1 & 5.42(\phantom{0.}29) & -5.14(\phantom{0.}29) & 4.55(\phantom{0.}03) & 4.38(94) & -4.24(94) & 5.11(31) & --- & --- & --- \\ & 1 & 5.42(29) & -5.14(29) & 4.55(03) & 4.38(94) & -4.24(94) & 5.11(31) & & & \\
\phantom{} & 2 & 5.98(\phantom{0.}83) & -5.91(\phantom{0.}83) & 4.79(\phantom{0.}71) & --- & --- & --- & --- & --- & --- \\ & 2 & 5.98(83) & -5.91(83) & 4.79(71) & & & & & & \\
\phantom{} & 5 & 6.18(\phantom{0.}84) & -6.13(\phantom{0.}84) & 4.87(\phantom{0.}55) & --- & --- & --- & --- & --- & --- \\ & 5 & 6.18(84) & -6.13(84) & 4.87(55) & & & & & & \\
\phantom{} & \(\infty\) & 7.38(1.08) & -7.38(1.08) & 5.67(\phantom{0.}68) & --- & --- & --- & --- & --- & --- \\ & \(\infty\) & 7.38(1.08) & -7.38(1.08) & 5.67(68) & & & & & & \\
\phantom{} & Opt. & 5.85(1.75) & -5.63(1.75) & 4.79(1.11) & --- & --- & --- & --- & --- & --- \\ & Opt. & 5.85(1.75) & -5.63(1.75) & 4.79(1.11) & & & & & & \\
\end{tabular} \end{tabular}
\end{ruledtabular} \end{ruledtabular}
\end{table*} \end{table*}
@ -874,7 +885,7 @@ large systems.
%%--------------------------------------- %%
\begin{acknowledgments} \begin{acknowledgments}
This work was performed using HPC resources from GENCI-TGCC (Grand This work was performed using HPC resources from GENCI-TGCC (Grand
Challenge 2019-gch0418) and from CALMIP (Toulouse) under allocation Challenge 2019-gch0418) and from CALMIP (Toulouse) under allocation