MATH-UA 252/MA-UY 3204 - Fall 2022 - Homework #3

The  Thomson  problem.    Consider a system of N  particles at positions x1 , . . . , xN   (where xi   ∈ R3 ), where particle i has charge qi . The electrostatic potential energy stored in this system is:

UE (N) =    ke  ,

where rij   =  |xi  − xj |.  If we normalize the units so ke  = 1 and assume that qi  = 1 for all i, then UE (N) simplifies to:

UE (N) =     .

Now, assume that we have N electrons which are constrained to lie  on the surface  of the unit sphere in R3 . The Thomson problem is the following:

For N > 0, find the configuration of electrons on the unit sphere such that UE (N) is minimized.

That is, solve:

minimize       UE (N)

subject to       |xi | = 1,         i = 1, . . . , N.

For this problem, you should write a version of PGD to solve the Thomson problem. Comments:

• You will need to use line search to get your iteration to converge.   It’s recommended to just use backtracking line search.

• You will need a reasonable way of distributing points on the sphere. This is simple:

1. For each i, choose xi   =  (xi , yi , zi )  randomly such that xi , yi , zi   ∼ N(0, 1)  independently and identically. On Tuesday (9/27) we will discuss line search in more detail.

2. Set xi  := xi /|xi |.

You can do this in numpy as follows:

X = np.random.randn(N, 3)

X /= np.sqrt(np.sum(X**2, axis=1)).reshape(-1, 1)

Visualization.    To visualize your solution, you have two options:

1. You can use matplotlib’s mplot3d to make a 3D plot of the (xi , yi , zi ) coordinates (see this link for some ideas).

2. You can alternatively use PyVista to do the same.

Problem 1.    Solve the Thomson problem for several choices of N where 3 ≤ N ≤ 14, and verify that your results match the picture here:

https://tracer.lcc.uma.es/problems/thomson/thomson.html

Do this by visualizing the equilibrium distribution of nodes using the approach described in “Visualization” above, and also by comparing the resulting potential with what is listed in the table on the website linked  above.

Problem  2.    Solve the Thomson problem for a few choices of N  greater than N  =  14.  The Thomson problem is nonconve北, and thus has numerous local minima. If you re-run your optimization algorithm for different initializations, you may get different minimizing values.  Try re-running your solver for different starting configurations by selecting a new initial iterate randomly. If you like, you can use np.random.seed to make your results reproducible.  Visualize each result in 3D, as discussed above.  Are the final values of the electrostatic potential comparable, or are they very different?

Problem 3.    The URL above also gives the following fit to the value of the potential for different N :

Uapprox(N) =  (1 − aN−1/2  + bN−3/2),