780.20: 1094 Session 13

Handouts: Excerpts on autocorrelation functions and variational Monte Carlo, printouts of codes

In this session, we'll continue our survey of Monte Carlo computational methods with a look at autocorrelation and variational Monte Carlo applied to simple problems. The discussion is easily generalized to the more complex problems for which these approaches are well suited.

Session 12 (cont.)

Continue with "The Two-D Ising Model", but move on to Session 13 "Autocorrelation" after an hour. If you have time later (possibly next week), go back and finish it.

Autocorrelation

When evaluating an average over Monte Carlo configurations, we want to skip the first n₁ steps and then use data taken every n0 Monte Carlo steps. How do we determine how large to take n0 and n1? We'll use a simple integration problem to explore these issues, by integrating x²e^-x2 from 0 to 10 and dividing by the integral of e^-x2 (so this is like < x² > with a probability distribution proportional to e^-x2).

1. Calculate the (normalized) integral in Mathematica for reference.
2. Consider first random sampling as a review/recap of our discussions.
   • Create an autocorrelation_test project and then run the autocorrelation_test.cpp code a number of times (at least 10) and see how the random sampling estimates vary. Is the calculation of errors in the code (e.g. error_random) consistent with the discussion in the background notes from Session 11e?
     
     
     
     
   • Are the estimated errors consistent with the actual deviations of the estimates from the exact result? (E.g., if the error is one standard deviation, about what fraction of the results should lie between the exact result +/- the error? What fraction do you observe?)
     
     
     
     
   • Vary the number N of configurations used to make the estimates to see how the error scales. (The number of configurations is just the number of x's sampled in this case.) With what power of N does it scale? Is this result consistent with expectations from the notes?
     
     
     
     
3. Next consider the Metropolis algorithm. There are three parameters for you to adjust (besides the total number of iterations): "max_step", "initial_skip" and "skip".
   • What do each of these affect? (Why do we skip steps at the beginning? Why do we skip configurations? How do we know how much to change x? See the Pang excerpt for answers, but discuss it first.)
     
     
     
     
   • Adjust "max_step" so the acceptance rate is around 50% (or less). [If stepping a vector X, we test every component to form a Monte Carlo step and look for choosing a step size so that the acceptance rate is around 50%.] What "max_step" did you use?
   • Devise a way to decide on the "initial_skip" that includes a plot. Set "initial_skip" in the code to the value you deduce. (Note that your result will depend on your choice of "max_step".) Explain your value.
   
   
   
   
   
4. To figure out a good value for "skip", we'll calculate the "autocorrelation function". This is defined in the Pang handout as equation (9.21). For us, "A" is "x²". Your task is to generate the analog of Fig. 9.1 for the current problem.
   • What is the value of the autocorrelation function C(l) at l=0?
     
     
   • What is the condition that makes C(l) tend to zero (based on the defining equation)?
     
     
     
     
   • You have everything you need in the code already for (9.21) except for the average in (9.22). Modify the code to calculate that and then print out and plot C(l). Attach the plot. [Hint: An easy way (although not the most efficient way!) is to introduce an array "result_save[i]" in the Monte Carlo step loop and save all of the "result" evaluations. Then after this loop you can step through values of l (say from 0 to 100) and calculate (9.22) for each l (you should only use iterations - l configurations to do this calculation; why?). Then you can calculate (9.21) and output it.]
5. Now that the Metropolis results are reliable, repeat step 2. above, but now for the Metropolis sampling.

Variational Monte Carlo

We'll do a simple example of variational Monte Carlo to illustrate the basic idea. Generalizing to more complex systems is straightforward (but takes a lot longer to run!).

1. Take a look at the "variational_SHO.cpp" code and how it implements variational Monte Carlo for a one-dimensional harmonic oscillator using the VariationalMC class. Make a variational_SHO project and add variational_SHO.cpp, random_seed.cpp, VaritionalMC.h, and VariationalMC.cpp. Run it. You'll be asked to supply a range and step size for the variational parameter "a". This will require some experimentation to make sure the minimum with respect to "a" is in the interval you select.
   
   
2. The variational_SHO.dat file is suitable for plotting in gnuplot. The plotfile "variational_SHO.plt" is provided to illustrate how to plot it with error bars. Try it out (and attach a plot). Does the graph make sense? How might you modify the code to find the minimum automatically (rather than graphically)?
   
   
   
   
3. Does this code implement the features explored in the "Autocorrelation" section? If not, how would you improve this code?
   
   
   
   
4. Try modifying the trial function to another reasonable form with one variational parameter "a" (use your imagination!). You'll have to modify the "Psi" and "E_local" functions. Find the minimum graphically and compare to the exact solution.