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<meta http-equiv="Content-Type" content="text/html;charset=utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Quantum Monte Carlo</title>
@ -329,76 +329,75 @@ for the JavaScript code in this tag.
<h2>Table of Contents</h2>
<div id="text-table-of-contents">
<ul>
<li><a href="#org813b90b">1. Introduction</a>
<li><a href="#org3c1efae">1. Introduction</a>
<ul>
<li><a href="#org6dfa1bc">1.1. Energy and local energy</a></li>
<li><a href="#org85a0a0b">1.1. Energy and local energy</a></li>
</ul>
</li>
<li><a href="#orgf59e340">2. Numerical evaluation of the energy of the hydrogen atom</a>
<li><a href="#org6f77d3f">2. Numerical evaluation of the energy of the hydrogen atom</a>
<ul>
<li><a href="#orgd9bbb38">2.1. Local energy</a>
<li><a href="#org9e28ed2">2.1. Local energy</a>
<ul>
<li><a href="#org19971eb">2.1.1. Exercise 1</a></li>
<li><a href="#orgc86d08a">2.1.2. Exercise 2</a></li>
<li><a href="#orgf2b1ece">2.1.3. Exercise 3</a></li>
<li><a href="#org30c5c6a">2.1.4. Exercise 4</a></li>
<li><a href="#org5ba9684">2.1.5. Exercise 5</a></li>
<li><a href="#orgf6097b0">2.1.1. Exercise 1</a></li>
<li><a href="#org23dae5b">2.1.2. Exercise 2</a></li>
<li><a href="#orgfa0dd30">2.1.3. Exercise 3</a></li>
<li><a href="#orgf9422b3">2.1.4. Exercise 4</a></li>
<li><a href="#org9a98a25">2.1.5. Exercise 5</a></li>
</ul>
</li>
<li><a href="#org78ac743">2.2. Plot of the local energy along the \(x\) axis</a>
<li><a href="#org4dc200a">2.2. Plot of the local energy along the \(x\) axis</a>
<ul>
<li><a href="#org674310c">2.2.1. Exercise</a></li>
<li><a href="#org57af737">2.2.1. Exercise</a></li>
</ul>
</li>
<li><a href="#org79e713c">2.3. Numerical estimation of the energy</a>
<li><a href="#orgf856ad9">2.3. Numerical estimation of the energy</a>
<ul>
<li><a href="#orgb32b9f2">2.3.1. Exercise</a></li>
<li><a href="#org5b336d3">2.3.1. Exercise</a></li>
</ul>
</li>
<li><a href="#orgcd32e90">2.4. Variance of the local energy</a>
<li><a href="#orgcf8dd2d">2.4. Variance of the local energy</a>
<ul>
<li><a href="#org56c6c46">2.4.1. Exercise (optional)</a></li>
<li><a href="#orge76103e">2.4.2. Exercise</a></li>
<li><a href="#org1932bf3">2.4.1. Exercise (optional)</a></li>
<li><a href="#org7e92320">2.4.2. Exercise</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#orgd2b410e">3. Variational Monte Carlo</a>
<li><a href="#org344d166">3. Variational Monte Carlo</a>
<ul>
<li><a href="#org7e2d1ab">3.1. Computation of the statistical error</a>
<li><a href="#org7557f27">3.1. Computation of the statistical error</a>
<ul>
<li><a href="#org202952d">3.1.1. Exercise</a></li>
<li><a href="#org2e5da03">3.1.1. Exercise</a></li>
</ul>
</li>
<li><a href="#org8009dba">3.2. Uniform sampling in the box</a>
<li><a href="#org6d5f40f">3.2. Uniform sampling in the box</a>
<ul>
<li><a href="#org2827619">3.2.1. Exercise</a></li>
<li><a href="#org29364d9">3.2.1. Exercise</a></li>
</ul>
</li>
<li><a href="#org0de0313">3.3. Metropolis sampling with \(\Psi^2\)</a>
<li><a href="#orgd8b523a">3.3. Metropolis sampling with \(\Psi^2\)</a>
<ul>
<li><a href="#orgdf21cbc">3.3.1. Optimal step size</a></li>
<li><a href="#org7e22285">3.3.2. Exercise</a></li>
<li><a href="#org3f8f8a0">3.3.1. Optimal step size</a></li>
<li><a href="#org0bb36e4">3.3.2. Exercise</a></li>
</ul>
</li>
<li><a href="#orgcfba2cf">3.4. Generalized Metropolis algorithm</a>
<li><a href="#orgd2b8682">3.4. Generalized Metropolis algorithm</a>
<ul>
<li><a href="#org32d2aaa">3.4.1. Gaussian random number generator</a></li>
<li><a href="#org0f3c095">3.4.2. Exercise 1</a></li>
<li><a href="#org2493e9d">3.4.3. Exercise 2</a></li>
<li><a href="#orgd0861e4">3.4.1. Gaussian random number generator</a></li>
<li><a href="#org0cf72a5">3.4.2. Exercise 1</a></li>
<li><a href="#orgb298efc">3.4.3. Exercise 2</a></li>
</ul>
</li>
</ul>
</li>
<li><a href="#org15f595a">4. Project</a></li>
<li><a href="#orge21458a">5. <span class="todo TODO">TODO</span> Last things to do</a></li>
<li><a href="#org56dd2e2">6. Schedule</a></li>
<li><a href="#orgd5eecd6">4. Project</a></li>
<li><a href="#orgdf684f7">5. Acknowledgments</a></li>
</ul>
</div>
</div>
<div id="outline-container-org813b90b" class="outline-2">
<h2 id="org813b90b"><span class="section-number-2">1</span> Introduction</h2>
<div id="outline-container-org3c1efae" class="outline-2">
<h2 id="org3c1efae"><span class="section-number-2">1</span> Introduction</h2>
<div class="outline-text-2" id="text-1">
<p>
This website contains the QMC tutorial of the 2021 LTTC winter school
@ -438,8 +437,8 @@ coordinates, etc).
</p>
</div>
<div id="outline-container-org6dfa1bc" class="outline-3">
<h3 id="org6dfa1bc"><span class="section-number-3">1.1</span> Energy and local energy</h3>
<div id="outline-container-org85a0a0b" class="outline-3">
<h3 id="org85a0a0b"><span class="section-number-3">1.1</span> Energy and local energy</h3>
<div class="outline-text-3" id="text-1-1">
<p>
For a given system with Hamiltonian \(\hat{H}\) and wave function \(\Psi\), we define the local energy as
@ -522,8 +521,8 @@ energy computed over these configurations:
</div>
</div>
<div id="outline-container-orgf59e340" class="outline-2">
<h2 id="orgf59e340"><span class="section-number-2">2</span> Numerical evaluation of the energy of the hydrogen atom</h2>
<div id="outline-container-org6f77d3f" class="outline-2">
<h2 id="org6f77d3f"><span class="section-number-2">2</span> Numerical evaluation of the energy of the hydrogen atom</h2>
<div class="outline-text-2" id="text-2">
<p>
In this section, we consider the hydrogen atom with the following
@ -552,8 +551,8 @@ To do that, we will compute the local energy and check whether it is constant.
</p>
</div>
<div id="outline-container-orgd9bbb38" class="outline-3">
<h3 id="orgd9bbb38"><span class="section-number-3">2.1</span> Local energy</h3>
<div id="outline-container-org9e28ed2" class="outline-3">
<h3 id="org9e28ed2"><span class="section-number-3">2.1</span> Local energy</h3>
<div class="outline-text-3" id="text-2-1">
<p>
You will now program all quantities needed to compute the local energy of the H atom for the given wave function.
@ -580,8 +579,8 @@ to catch the error.
</div>
</div>
<div id="outline-container-org19971eb" class="outline-4">
<h4 id="org19971eb"><span class="section-number-4">2.1.1</span> Exercise 1</h4>
<div id="outline-container-orgf6097b0" class="outline-4">
<h4 id="orgf6097b0"><span class="section-number-4">2.1.1</span> Exercise 1</h4>
<div class="outline-text-4" id="text-2-1-1">
<div class="exercise">
<p>
@ -627,8 +626,8 @@ and returns the potential.
</div>
</div>
<div id="outline-container-orgc86d08a" class="outline-4">
<h4 id="orgc86d08a"><span class="section-number-4">2.1.2</span> Exercise 2</h4>
<div id="outline-container-org23dae5b" class="outline-4">
<h4 id="org23dae5b"><span class="section-number-4">2.1.2</span> Exercise 2</h4>
<div class="outline-text-4" id="text-2-1-2">
<div class="exercise">
<p>
@ -664,8 +663,8 @@ input arguments, and returns a scalar.
</div>
</div>
<div id="outline-container-orgf2b1ece" class="outline-4">
<h4 id="orgf2b1ece"><span class="section-number-4">2.1.3</span> Exercise 3</h4>
<div id="outline-container-orgfa0dd30" class="outline-4">
<h4 id="orgfa0dd30"><span class="section-number-4">2.1.3</span> Exercise 3</h4>
<div class="outline-text-4" id="text-2-1-3">
<div class="exercise">
<p>
@ -747,8 +746,8 @@ Therefore, the local kinetic energy is
</div>
</div>
<div id="outline-container-org30c5c6a" class="outline-4">
<h4 id="org30c5c6a"><span class="section-number-4">2.1.4</span> Exercise 4</h4>
<div id="outline-container-orgf9422b3" class="outline-4">
<h4 id="orgf9422b3"><span class="section-number-4">2.1.4</span> Exercise 4</h4>
<div class="outline-text-4" id="text-2-1-4">
<div class="exercise">
<p>
@ -808,8 +807,8 @@ are calling is yours.
</div>
</div>
<div id="outline-container-org5ba9684" class="outline-4">
<h4 id="org5ba9684"><span class="section-number-4">2.1.5</span> Exercise 5</h4>
<div id="outline-container-org9a98a25" class="outline-4">
<h4 id="org9a98a25"><span class="section-number-4">2.1.5</span> Exercise 5</h4>
<div class="outline-text-4" id="text-2-1-5">
<div class="exercise">
<p>
@ -821,8 +820,8 @@ Find the theoretical value of \(a\) for which \(\Psi\) is an eigenfunction of \(
</div>
</div>
<div id="outline-container-org78ac743" class="outline-3">
<h3 id="org78ac743"><span class="section-number-3">2.2</span> Plot of the local energy along the \(x\) axis</h3>
<div id="outline-container-org4dc200a" class="outline-3">
<h3 id="org4dc200a"><span class="section-number-3">2.2</span> Plot of the local energy along the \(x\) axis</h3>
<div class="outline-text-3" id="text-2-2">
<p>
The program you will write in this section will be written in
@ -853,8 +852,8 @@ In Fortran, you will need to compile all the source files together:
</div>
</div>
<div id="outline-container-org674310c" class="outline-4">
<h4 id="org674310c"><span class="section-number-4">2.2.1</span> Exercise</h4>
<div id="outline-container-org57af737" class="outline-4">
<h4 id="org57af737"><span class="section-number-4">2.2.1</span> Exercise</h4>
<div class="outline-text-4" id="text-2-2-1">
<div class="exercise">
<p>
@ -950,8 +949,8 @@ plot './data' index 0 using 1:2 with lines title 'a=0.1', \
</div>
</div>
<div id="outline-container-org79e713c" class="outline-3">
<h3 id="org79e713c"><span class="section-number-3">2.3</span> Numerical estimation of the energy</h3>
<div id="outline-container-orgf856ad9" class="outline-3">
<h3 id="orgf856ad9"><span class="section-number-3">2.3</span> Numerical estimation of the energy</h3>
<div class="outline-text-3" id="text-2-3">
<p>
If the space is discretized in small volume elements \(\mathbf{r}_i\)
@ -981,8 +980,8 @@ The energy is biased because:
</div>
<div id="outline-container-orgb32b9f2" class="outline-4">
<h4 id="orgb32b9f2"><span class="section-number-4">2.3.1</span> Exercise</h4>
<div id="outline-container-org5b336d3" class="outline-4">
<h4 id="org5b336d3"><span class="section-number-4">2.3.1</span> Exercise</h4>
<div class="outline-text-4" id="text-2-3-1">
<div class="exercise">
<p>
@ -1055,8 +1054,8 @@ To compile the Fortran and run it:
</div>
</div>
<div id="outline-container-orgcd32e90" class="outline-3">
<h3 id="orgcd32e90"><span class="section-number-3">2.4</span> Variance of the local energy</h3>
<div id="outline-container-orgcf8dd2d" class="outline-3">
<h3 id="orgcf8dd2d"><span class="section-number-3">2.4</span> Variance of the local energy</h3>
<div class="outline-text-3" id="text-2-4">
<p>
The variance of the local energy is a functional of \(\Psi\)
@ -1083,8 +1082,8 @@ energy can be used as a measure of the quality of a wave function.
</p>
</div>
<div id="outline-container-org56c6c46" class="outline-4">
<h4 id="org56c6c46"><span class="section-number-4">2.4.1</span> Exercise (optional)</h4>
<div id="outline-container-org1932bf3" class="outline-4">
<h4 id="org1932bf3"><span class="section-number-4">2.4.1</span> Exercise (optional)</h4>
<div class="outline-text-4" id="text-2-4-1">
<div class="exercise">
<p>
@ -1095,8 +1094,8 @@ Prove that :
</div>
</div>
</div>
<div id="outline-container-orge76103e" class="outline-4">
<h4 id="orge76103e"><span class="section-number-4">2.4.2</span> Exercise</h4>
<div id="outline-container-org7e92320" class="outline-4">
<h4 id="org7e92320"><span class="section-number-4">2.4.2</span> Exercise</h4>
<div class="outline-text-4" id="text-2-4-2">
<div class="exercise">
<p>
@ -1175,8 +1174,8 @@ To compile and run:
</div>
</div>
<div id="outline-container-orgd2b410e" class="outline-2">
<h2 id="orgd2b410e"><span class="section-number-2">3</span> Variational Monte Carlo</h2>
<div id="outline-container-org344d166" class="outline-2">
<h2 id="org344d166"><span class="section-number-2">3</span> Variational Monte Carlo</h2>
<div class="outline-text-2" id="text-3">
<p>
Numerical integration with deterministic methods is very efficient
@ -1192,8 +1191,8 @@ interval.
</p>
</div>
<div id="outline-container-org7e2d1ab" class="outline-3">
<h3 id="org7e2d1ab"><span class="section-number-3">3.1</span> Computation of the statistical error</h3>
<div id="outline-container-org7557f27" class="outline-3">
<h3 id="org7557f27"><span class="section-number-3">3.1</span> Computation of the statistical error</h3>
<div class="outline-text-3" id="text-3-1">
<p>
To compute the statistical error, you need to perform \(M\)
@ -1233,8 +1232,8 @@ And the confidence interval is given by
</p>
</div>
<div id="outline-container-org202952d" class="outline-4">
<h4 id="org202952d"><span class="section-number-4">3.1.1</span> Exercise</h4>
<div id="outline-container-org2e5da03" class="outline-4">
<h4 id="org2e5da03"><span class="section-number-4">3.1.1</span> Exercise</h4>
<div class="outline-text-4" id="text-3-1-1">
<div class="exercise">
<p>
@ -1276,8 +1275,8 @@ input array.
</div>
</div>
<div id="outline-container-org8009dba" class="outline-3">
<h3 id="org8009dba"><span class="section-number-3">3.2</span> Uniform sampling in the box</h3>
<div id="outline-container-org6d5f40f" class="outline-3">
<h3 id="org6d5f40f"><span class="section-number-3">3.2</span> Uniform sampling in the box</h3>
<div class="outline-text-3" id="text-3-2">
<p>
We will now perform our first Monte Carlo calculation to compute the
@ -1338,8 +1337,8 @@ compute the statistical error.
</p>
</div>
<div id="outline-container-org2827619" class="outline-4">
<h4 id="org2827619"><span class="section-number-4">3.2.1</span> Exercise</h4>
<div id="outline-container-org29364d9" class="outline-4">
<h4 id="org29364d9"><span class="section-number-4">3.2.1</span> Exercise</h4>
<div class="outline-text-4" id="text-3-2-1">
<div class="exercise">
<p>
@ -1443,8 +1442,8 @@ well as the index of the current step.
</div>
</div>
<div id="outline-container-org0de0313" class="outline-3">
<h3 id="org0de0313"><span class="section-number-3">3.3</span> Metropolis sampling with \(\Psi^2\)</h3>
<div id="outline-container-orgd8b523a" class="outline-3">
<h3 id="orgd8b523a"><span class="section-number-3">3.3</span> Metropolis sampling with \(\Psi^2\)</h3>
<div class="outline-text-3" id="text-3-3">
<p>
We will now use the square of the wave function to sample random
@ -1563,8 +1562,8 @@ All samples should be kept, from both accepted <i>and</i> rejected moves.
</div>
<div id="outline-container-orgdf21cbc" class="outline-4">
<h4 id="orgdf21cbc"><span class="section-number-4">3.3.1</span> Optimal step size</h4>
<div id="outline-container-org3f8f8a0" class="outline-4">
<h4 id="org3f8f8a0"><span class="section-number-4">3.3.1</span> Optimal step size</h4>
<div class="outline-text-4" id="text-3-3-1">
<p>
If the box is infinitely small, the ratio will be very close
@ -1599,8 +1598,8 @@ the same variable later on to store a time step.
</div>
<div id="outline-container-org7e22285" class="outline-4">
<h4 id="org7e22285"><span class="section-number-4">3.3.2</span> Exercise</h4>
<div id="outline-container-org0bb36e4" class="outline-4">
<h4 id="org0bb36e4"><span class="section-number-4">3.3.2</span> Exercise</h4>
<div class="outline-text-4" id="text-3-3-2">
<div class="exercise">
<p>
@ -1711,8 +1710,8 @@ Can you observe a reduction in the statistical error?
</div>
</div>
<div id="outline-container-orgcfba2cf" class="outline-3">
<h3 id="orgcfba2cf"><span class="section-number-3">3.4</span> Generalized Metropolis algorithm</h3>
<div id="outline-container-orgd2b8682" class="outline-3">
<h3 id="orgd2b8682"><span class="section-number-3">3.4</span> Generalized Metropolis algorithm</h3>
<div class="outline-text-3" id="text-3-4">
<p>
One can use more efficient numerical schemes to move the electrons by choosing a smarter expression for the transition probability.
@ -1833,8 +1832,8 @@ The algorithm of the previous exercise is only slighlty modified as:
</ol>
</div>
<div id="outline-container-org32d2aaa" class="outline-4">
<h4 id="org32d2aaa"><span class="section-number-4">3.4.1</span> Gaussian random number generator</h4>
<div id="outline-container-orgd0861e4" class="outline-4">
<h4 id="orgd0861e4"><span class="section-number-4">3.4.1</span> Gaussian random number generator</h4>
<div class="outline-text-4" id="text-3-4-1">
<p>
To obtain Gaussian-distributed random numbers, you can apply the
@ -1898,8 +1897,8 @@ In Python, you can use the <a href="https://numpy.org/doc/stable/reference/rando
</div>
<div id="outline-container-org0f3c095" class="outline-4">
<h4 id="org0f3c095"><span class="section-number-4">3.4.2</span> Exercise 1</h4>
<div id="outline-container-org0cf72a5" class="outline-4">
<h4 id="org0cf72a5"><span class="section-number-4">3.4.2</span> Exercise 1</h4>
<div class="outline-text-4" id="text-3-4-2">
<div class="exercise">
<p>
@ -1942,8 +1941,8 @@ Write a function to compute the drift vector \(\frac{\nabla \Psi(\mathbf{r})}{\P
</div>
</div>
<div id="outline-container-org2493e9d" class="outline-4">
<h4 id="org2493e9d"><span class="section-number-4">3.4.3</span> Exercise 2</h4>
<div id="outline-container-orgb298efc" class="outline-4">
<h4 id="orgb298efc"><span class="section-number-4">3.4.3</span> Exercise 2</h4>
<div class="outline-text-4" id="text-3-4-3">
<div class="exercise">
<p>
@ -2042,8 +2041,8 @@ Modify the previous program to introduce the drift-diffusion scheme.
</div>
</div>
<div id="outline-container-org15f595a" class="outline-2">
<h2 id="org15f595a"><span class="section-number-2">4</span> Project</h2>
<div id="outline-container-orgd5eecd6" class="outline-2">
<h2 id="orgd5eecd6"><span class="section-number-2">4</span> Project</h2>
<div class="outline-text-2" id="text-4">
<p>
Change your PDMC code for one of the following:
@ -2060,88 +2059,28 @@ And compute the ground state energy.
</div>
<div id="outline-container-orge21458a" class="outline-2">
<h2 id="orge21458a"><span class="section-number-2">5</span> <span class="todo TODO">TODO</span> Last things to do&#xa0;&#xa0;&#xa0;<span class="tag"><span class="noexport">noexport</span></span></h2>
<div id="outline-container-orgdf684f7" class="outline-2">
<h2 id="orgdf684f7"><span class="section-number-2">5</span> Acknowledgments</h2>
<div class="outline-text-2" id="text-5">
<ul class="org-ul">
<li class="off"><code>[&#xa0;]</code> Prepare 4 questions for the exam: multiple-choice questions
with 4 possible answers. Questions should be independent because
they will be asked in a random order.</li>
</ul>
</div>
<div class="figure">
<p><img src="https://trex-coe.eu/sites/default/files/inline-images/euflag.jpg" alt="euflag.jpg" />
</p>
</div>
<div id="outline-container-org56dd2e2" class="outline-2">
<h2 id="org56dd2e2"><span class="section-number-2">6</span> Schedule</h2>
<div class="outline-text-2" id="text-6">
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
<colgroup>
<col class="org-left" />
<col class="org-right" />
</colgroup>
<thead>
<tr>
<th scope="col" class="org-left"><span class="timestamp-wrapper"><span class="timestamp">&lt;2021-02-04 Thu 09:00&gt;&#x2013;&lt;2021-02-04 Thu 10:30&gt;</span></span></th>
<th scope="col" class="org-right">Lecture</th>
</tr>
</thead>
<tbody>
<tr>
<td class="org-left"><span class="timestamp-wrapper"><span class="timestamp">&lt;2021-02-04 Thu 10:45&gt;&#x2013;&lt;2021-02-04 Thu 11:10&gt;</span></span></td>
<td class="org-right">2.1</td>
</tr>
<tr>
<td class="org-left"><span class="timestamp-wrapper"><span class="timestamp">&lt;2021-02-04 Thu 11:10&gt;&#x2013;&lt;2021-02-04 Thu 11:30&gt;</span></span></td>
<td class="org-right">2.2</td>
</tr>
<tr>
<td class="org-left"><span class="timestamp-wrapper"><span class="timestamp">&lt;2021-02-04 Thu 11:30&gt;&#x2013;&lt;2021-02-04 Thu 12:15&gt;</span></span></td>
<td class="org-right">2.3</td>
</tr>
<tr>
<td class="org-left"><span class="timestamp-wrapper"><span class="timestamp">&lt;2021-02-04 Thu 12:15&gt;&#x2013;&lt;2021-02-04 Thu 12:30&gt;</span></span></td>
<td class="org-right">2.4</td>
</tr>
</tbody>
<tbody>
<tr>
<td class="org-left"><span class="timestamp-wrapper"><span class="timestamp">&lt;2021-02-04 Thu 14:00&gt;&#x2013;&lt;2021-02-04 Thu 14:10&gt;</span></span></td>
<td class="org-right">3.1</td>
</tr>
<tr>
<td class="org-left"><span class="timestamp-wrapper"><span class="timestamp">&lt;2021-02-04 Thu 14:10&gt;&#x2013;&lt;2021-02-04 Thu 14:30&gt;</span></span></td>
<td class="org-right">3.2</td>
</tr>
<tr>
<td class="org-left"><span class="timestamp-wrapper"><span class="timestamp">&lt;2021-02-04 Thu 14:30&gt;&#x2013;&lt;2021-02-04 Thu 15:30&gt;</span></span></td>
<td class="org-right">3.3</td>
</tr>
<tr>
<td class="org-left"><span class="timestamp-wrapper"><span class="timestamp">&lt;2021-02-04 Thu 15:30&gt;&#x2013;&lt;2021-02-04 Thu 16:30&gt;</span></span></td>
<td class="org-right">3.4</td>
</tr>
<tr>
<td class="org-left"><span class="timestamp-wrapper"><span class="timestamp">&lt;2021-02-04 Thu 16:30&gt;&#x2013;&lt;2021-02-04 Thu 18:30&gt;</span></span></td>
<td class="org-right">4.5</td>
</tr>
</tbody>
</table>
<p>
<a href="https://trex-coe.eu">TREX</a> : Targeting Real Chemical Accuracy at the Exascale project
has received funding from the European Unions Horizon 2020 - Research and
Innovation program - under grant agreement no. 952165. The content of this
document does not represent the opinion of the European Union, and the European
Union is not responsible for any use that might be made of such content.
</p>
</div>
</div>
</div>
<div id="postamble" class="status">
<p class="author">Author: Anthony Scemama, Claudia Filippi</p>
<p class="date">Created: 2021-02-03 Wed 16:26</p>
<p class="date">Created: 2021-02-03 Wed 16:59</p>
<p class="validation"><a href="http://validator.w3.org/check?uri=referer">Validate</a></p>
</div>
</body>