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updated project and codes

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mhjensen
2017-10-04 14:40:31 +02:00
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@@ -45,20 +45,24 @@ div { text-align: justify; text-justify: inter-word; }
'___sec0'),
('Introduction', 3, None, '___sec1'),
('Part a): Producing the data', 3, None, '___sec2'),
('Part b): Fitting the data using regression analysis and other methods',
('Part b): Estimating the standard deviation',
3,
None,
'___sec3'),
('Part c): Introducing Bayesian statistics',
('Part c): Fitting the data using regression analysis and other methods',
3,
None,
'___sec4'),
('Part d): Studying the Ising model or the VMC results with Neural networks',
('Part d): Introducing Bayesian statistics',
3,
None,
'___sec5'),
('Background literature', 2, None, '___sec6'),
('Introduction to numerical projects', 2, None, '___sec7')]}
('Part e): Studying the Ising model or the VMC results with Neural networks',
3,
None,
'___sec6'),
('Background literature', 2, None, '___sec7'),
('Introduction to numerical projects', 2, None, '___sec8')]}
end of tocinfo -->
<body>
@@ -73,8 +77,8 @@ MathJax.Hub.Config({
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@@ -107,7 +111,7 @@ MathJax.Hub.Config({
<h3 id="___sec1">Introduction </h3>
<p>
The aim of this project is to use an already developed Monte Carlo program (either the ising Model or a variational Monte Carlo code) to produce, in case of the Ising model, the energy as function of temperature. For the variational Monte carlo calculation of interacting electrons in an oscilaltor trap, the data are represented by the ground state energies as functions of the variational parameters.
The aim of this project is to use an already developed Monte Carlo program (either the ising Model or a variational Monte Carlo code) to produce, in case of the <a href="https://github.com/CompPhysics/MachineLearning/tree/master/doc/Programs/IsingModel" target="_blank">Ising model</a>, the energy as function of temperature. For the <a href="https://github.com/CompPhysics/MachineLearning/tree/master/doc/Programs/VMC2Electrons/" target="_blank">variational Monte Carlo calculation</a> of interacting electrons in an oscilaltor trap, the data are represented by the ground state energies as functions of the variational parameters.
<p>
In its simplest form
@@ -122,34 +126,54 @@ constant expressing the strength of the interaction between
neighboring spins. The symbol \( < kl> \) indicates that we sum over
nearest neighbors only. We will assume that we have a ferromagnetic
ordering, viz \( J> 0 \). We will use periodic boundary conditions and
the Metropolis algorithm only. Alternatively we can use the supplied variational Monte Carlo program which solves Schroedinger's equation for two interacting electrons in a harmonic oscillator trap. Both codes can be found at the webpage of the course under programs.
the Metropolis algorithm only. Alternatively we can use the supplied variational Monte Carlo program which solves Schroedinger's equation for two interacting electrons in a harmonic oscillator trap. Both codes can be found at the webpage of the <a href="https://github.com/CompPhysics/MachineLearning/tree/master/doc/Programs/" target="_blank">course under programs</a>.
<h3 id="___sec2">Part a): Producing the data </h3>
<p>
\( \langle E\rangle \) and \( \langle \vert M\vert \rangle \), the specific heat
\( C_V \) and the susceptibility \( \chi \) as functions of \( T \) for \( L=40 \),
\( L=60 \), \( L=100 \) and \( L=140 \) for \( T\in [2.0,2.3] \) with a step in
temperature \( \Delta T=0.05 \) or smaller. You may find it convenient narrow the domain for \( T \).
If we opt for the Ising model code, we need to generate for every span over the lattice and output of the total energy and the magnetic moment (if we wish to study this quantity as well), that is compute and write to file as function of temperature
\( \langle E\rangle \) and \( \langle \vert M\vert \rangle \).
We will use a fixed lattice siste of \( L\times L = 40 \times 40 \). Make sure the calculations have been equilibrated and compute these expectation values for
for \( T\in [1.0,3.0] \) with a step in
temperature \( \Delta T=0.1 \) or smaller. This are the data you will use to estimate the standard deviation in the next part of the project.
You should keep a file for the different temperatures as you will need to compute the standard deviation for each temperature.
<p>
Plot \( \langle E\rangle \),
\( \langle \vert M\vert\rangle \), \( C_V \) and \( \chi \) as functions of \( T \).
Alternatively you can run the variational Monte Carlo program for two interacting electrons confined to move in a harmonic oscillator trap. Here you will produce the variational expectaction value of the energy as a function of the variational parameters \( \alpha \) and \( \beta \). You could single out only a set of such parameters after you have found the minimum energy.
<h3 id="___sec3">Part b): Fitting the data using regression analysis and other methods </h3>
<p>
These two cases form then our training data which we will use for our estimates of the standard deviation.
<h3 id="___sec3">Part b): Estimating the standard deviation </h3>
This part deals with widely used resampling methods to find the standard deviation on our data sets. The methods we can use are the Blocking method, the Bootstrap method and the Jackknife method. The latter two are examples of so-called resampling methods. <a href="https://github.com/CompPhysics/MachineLearning/tree/master/doc/Programs/Sampling" target="_blank">The functions provided under the program folder</a> encode all three methods. The typical situation is to generate data using a C++ or Fortan code, as in part a) above. These data are then used by a post-analysis program in order to perform a statistical analysis.
<p>
In case of the Ising model, the final product will be a table of the expectation values of the energy (or the magnetic moment) as function of temperature and with a proper standard deviation. These data will then enter our analysis in parts c-e). We will call these data the training data.
<p>
If you opt for the variational Monte Carlo program, you need to make a table of the ground state energy as function of various variational parameters (you can limit yourself to \( \alpha \) only) with proper standard deviation estimations.
<p>
Your task here is thus to use the data from part a) and generate the standard deviation using one (or more) of the methods discussed above. In general, the Bootstrap method is the most widely used one.
<p>
These data will then be used in our next step, where we will try to obtain a functional approximation to our data sets.
<h3 id="___sec4">Part c): Fitting the data using regression analysis and other methods </h3>
<p>
More text to come
<h3 id="___sec4">Part c): Introducing Bayesian statistics </h3>
<h3 id="___sec5">Part d): Introducing Bayesian statistics </h3>
More text to come
<h3 id="___sec5">Part d): Studying the Ising model or the VMC results with Neural networks </h3>
<h3 id="___sec6">Part e): Studying the Ising model or the VMC results with Neural networks </h3>
More text to come
<h2 id="___sec6">Background literature </h2>
<h2 id="___sec7">Background literature </h2>
<p>
If you wish to read more about the Ising model and statistical physics here are three suggestions.
@@ -160,7 +184,7 @@ If you wish to read more about the Ising model and statistical physics here are
<li> <a href="https://global.oup.com/academic/product/monte-carlo-methods-in-statistical-physics-9780198517979?cc=no&lang=en&" target="_blank">M. E. J. Newman and T. Barkema</a>, <em>Monte Carlo Methods in Statistical Physics</em>, Oxford, see chapters 3 and 4.</li>
</ul>
<h2 id="___sec7">Introduction to numerical projects </h2>
<h2 id="___sec8">Introduction to numerical projects </h2>
<p>
Here follows a brief recipe and recommendation on how to write a report for each