*Handouts: printouts of multimin_sa_compare.cpp and
multimin_sim.cpp, excerpt on simulated annealing *

In this session, we'll consider multidimensional minimization with standard downhill routines for local minima and then an application of Monte Carlo techniques to find the global minimum: simulated annealing.

In Session 10, you tried the gsl routines for multidimensional minimization. Here we finish up that exploration, then consider a real physics problem: minimizing the potential energy of molecular configurations built from Na and Cl ions to find stable configurations. This will work without complications for small configurations, but bigger clusters will be tricky.

- [Move on to the next part if you completed this in Session 10.] Returning to "multimin_test.cpp", what algorithm is used initially to do the minimization? Modify the code to try some other algorithms (some major changes would be necessary to use the simplex algorithm; why?). Which is "best" for this problem?
- Consider how you would modify the code to study the problem
described in the Pang handout: determining the stable structures of
molecules of Na and Cl ions.
- Read the Pang handout. What is the physics of the interaction energy (i.e., what does each term represent?)? What is the minimization problem to be solved?
- What information to you need to find r
_{ij}for every possible pair of ions? What information do you want to have when the program is finished? (I.e., what should the output be?) - Does the vector you pass to the minimizer include the x,y,z
coordinates of every ion? Write down what is passed for the cases
of NaCl, Na
_{2}Cl, and Na_{2}Cl_{2}. - What will the functions "my_f" and "my_df" look like? Remember that they do their calculation based on the vector passed to them, which will not include every x,y,z coordinate (why not?).
- Sketch out a plan to solve the problem using the gsl minimization routines. If at all possible, devise your plan in collaboration with a classmate.

- Now consider the code multimin_nacl.cpp, which I wrote
to address this problem.
(Warning: This is not a great code.)
Compile and link with make_multimin_nacl.
- Compare your plan to my solution. Explain why your plan is better. :)
- In what ways can you check if the problem is coded correctly?
List some of the tests you would make. [Hint 1: Think of some
things you can change and still expect to get the same answer from
the minimization.] [Hint 2: Can you check a simple case a
different way?]
What symmetry can you use to (partially) check Na
_{2}Cl? - Are the answers for two (1 Na and 1 Cl) and three atoms
(2 Na's and 1 Cl) physically
reasonable? Here are some questions that address this point:
Should the final energy be positive or negative?
What is a reasonable number of eV? What should the distance be
roughly in angstroms? (E.g., what are typical atomic spacings?)
Should the atoms be closer or further apart in NaCl compared to
Na
_{2}Cl? - Now try to find the structure of Na
_{2}Cl_{2}. Keep in mind that the program identifies*local*minima. There may be more stable configurations. How would you tell which is more stable? - Now consider Na
_{3}Cl_{2}^{+}and Na_{3}Cl_{3}. Can you reproduce the claimed minimum energy structures found in Fig. 4.2? (Note: Make sure the solution has converged to a minimum; you may need to increase the number of iterations allowed, adjust the initial step size, or adjust the tolerance.) - To check for better local minima, try different starting points (for the x-vector) and try different algorithms (it makes a difference!!!). Finding a global minimum (and being sure you've found it) is a tough problem! (We'll return to this problem later when discussing simulated annealing.)

Standard optimization methods are very good in
finding local minima near where the minimization was started, but not
good at finding the *global* minimum. Finding the global minimum
of a function (such as the energy) is often (but not always) the goal.
One strategy using conventional minimizers is to start the minimization
at many different places in the parameter space (perhaps chosen at
random) and keep the best minimum found.

Here we'll adapt our Monte Carlo algorithm for generating a canonical
Boltzmann distribution of configurations at a temperature T to mimic how
physical systems find their ground states (i.e., the energy minimum at
T=0). At high temperature (compared to characteristic energy
spacings), the equilibrium distribution will include many states. If
the system is cooled *slowly*, then it will have the opportunity
to settle into the lowest energy state as T goes to zero. This is
called *annealing*. If the system is cooled quickly, it can get
stuck in a state that is not the minimum ("quenching"); this is
analogous to the routines we looked at in Session 21, which rapidly go
"downhill" to find local minima.

The strategy here is to simulate the annealing process by treating the function to be minimized as an energy (it might actually be an energy!), introducing an artificial temperature T, generating a sequence of states in a canonical distribution via the Metropolis algorithm. Then we lower the temperature according to a "schedule" and let the system settle into (hopefully!) a configuration that minimizes the energy.

We first consider the simple example of a damped sine wave in one
dimension, which has
many local minima.

`f(x) = e ^{-(x-1)2}sin(8x)`

(This example is from the GSL documentation.) Suppose we start with an initial guess of 15.5; how do "downhill" minimization methods compare to simulated annealing?

- Plot the function in Mathematica so we know what to expect ahead of time (remember that in the general case of interest we won't be able to do this!). Using FindMinimum starting near 15.5 to find the local minimum. Then find the global minimum.
- The code multimin_sa_compare.cpp (compile and link with make_multimin_sa_compare) applies the Gsl routines we used in a previous session to find the minimum and then applies a simple simulated annealing algorithm. Run the program and look at the code to figure out where the simulated annealing control parameters (starting and finishing temperatures, maximum step size, cooling rate, # of iterations at each temperature) are set. How well does each method work?
- Now adjust the simulated annealing control parameters to give it a chance! The step size should be large enough (or small enough) so that successes are neither assured nor happen too infrequently (50% would be ideal!). The initial temperature should be considerably larger than the largest typical delta_E normally encountered. You shouldn't cool to rapidly. Make these adjustments and rerun the code. Can you get it to work?
- Just to check that the Metropolis algorithm is doing something, once the code is working, change the line with "random < exp()" to "random > exp()" and verify that it no longer works.

Now we'll apply essentially the same approach to the problem of the shape of molecules built from Na and Cl atoms.

- The code "multimin_sim.cpp" is basically the "multimin_nacl.cpp" code from an earlier session with a simulated annealing step inserted between two "downhill" minimizations. The starting point is still the arbitrary set of coordinates from the original code. Look at the code and find where the simulated annealing control parameters are set. Note that the starting values mean that the simulated annealing part is skipped. What Metropolis algorithm is used in this code compared to the "multimin_sa_compare.cpp" code?
- Compile, link, and run the code (use "make_multimin_sim") and record the best answers for three Na's and two and three Cl's.
- Now play with the control parameters and try to do better. Good luck!

furnstahl.1@osu.edu