Initial Investigations

We present here the initial investigations that were carried out before implementing the final trend optimization scheme for our self-assembly problem. It would not be fair to present the final method without a discussion of the previous work that lead to confidence in our results. This previous work lead to the development of the trend optimization method that is expected to be useful in many other applications with similarly difficult objective functions.

We seek the lowest point of the objective function in the regions of the relevant local minima of the trend. We first briefly discuss the method we used to construct the trend and envelopes.

Points from the domain are chosen using Monte Carlo sampling, and the trend, and upper and lower envelopes, are constructed from the objective function evaluations at these points, as shown in Figure 5.9. We first define a radius in parameter space, rnbhd, that is the distance, or neighborhood, around each point that will be taken into account when con- structing the envelope. These envelopes are interpolated based on local averaging. For each point, find the minimum and maximum value of the objective function in a neighborhood around that point,

f_nbhdmin(xp) = min{f(x) :|x−xp|< rnbhd}

f_nbhdmax(xp) = max{f(x) :|x−xp|< rnbhd}. (5.4)

Our upper envelope is a surface that goes through these f_nbhdmax(x) values and the lower envelope is a surface that goes through thesef_nbhdmin(x) values. An infinite number of surfaces can qualify as an upper envelope (likewise for lower); we require only that it should touch

f(x) at least at one point and that it should be smoother than f(x).

The trend is a smoothed version of the objective function; it shows the simple smooth shape underlying the objective function. A straightforward way to find a trend is by finding the average value of the objective function in a neighborhood around each point, f_nbhdavg (x), and constructing the surface through these values.

In our method we fit a linear regression such that the difference between f(x) and the trend atx is minimized in some norm. We tested a variety of basis functions for the trend.

Figure 5.9: Schematic of the trend and upper and lower envelopes for a noisy objective function with a simple trend. The objective function is shown in green, the trend is red, and the upper and lower envelopes are blue.

Once a satisfactory trend has been found, any desired method for finding the global minimum of it can be used. In the regions of the minima, local searches using more objective function evaluations can be conducted to find the multiple relevant minima. The evaluation of the trend is very fast, unlike the evaluation of the objective function, so it is feasible to evaluate the trend at a large number of points, and thus perform a robust global optimization of the trend.

The process of choosing points by Monte Carlo sampling and evaluating the trend and envelopes at these points can be repeated any number of times to get better and better surfaces. Successive selection of points should be chosen not only around the region of the global minimum of the trend but also in other regions of lowest point density.

Initial sensitivity studies were conducted on a one-dimensional slice of the objective function with the goal of tuning the trend optimization method to our particular lattice quality objective function. These investigations varied the number of points sampled at each step, as well as the number of steps in the procedure. They revealed appropriate basis functions and methods to use to fit the trend, tolerances on acceptable variability, and methods for optimizing the trend. We present our final method below.

Trend Optimization Algorithm used in Self-Assembly Problem

distribution.

2. Construct a trend by computing a least squares fit to these points.

3. Evaluate the quadratic at 10,000 Monte Carlo-selected points and determine the points at which the trend has local minima.

4. Refine the size of the domain by reducing the length along each dimension by a factor of 3 and re-centering the refined domain on the minimum of the quadratic.

5. Repeat the above steps, using all objective function evaluations performed thus far in Step 2.

6. Evaluate the objective function at another 20 points in this doubly refined area of the global minimum.

7. The optimal parameter values (minimum of the objective function) is declared to be the point in this refined area at which the objective function is lowest.

This procedure is fast because although it is expensive to evaluate the objective function, fitting the quadratic and evaluating it at 10,000 points requires very little time. Each set of 20 objective function evaluations may also be done in parallel thus reducing the wall-clock time for the optimization. We also note that all previous objective function evaluations are used to construct the trend and to find the global minimum in the final step. It is these features and the simple smooth nature of the trend that make this algorithm so efficient for our particular implementation.

We note that up to 14 successive sampling steps were considered. The factor by which the search area could be refined at each step was also explored, using factors from 0.1 to 0.5. These studies indicated that 2 sampling steps were sufficient, each consisting of 20 sampled points, and that the search area could be refined by a factor of 1/3. It is this realisation that lead to this method being 100 times faster than a simulated annealing procedure.

These values were demonstrated to be sufficient by examining the upper and lower envelopes. Flat upper and lower envelopes after two sampling steps indicate that all points in this region exhibit the same variability in repeated evaluations of the objective function, and thus, are equally good. Hence, potentials using parameters from this region are equally robust in the sense of producing the same quality lattices with the same potential. The

robustness of the potential with respect to changes in its parameters is proportional to the distance to the center of the flat region of the envelopes (Section 5.1.4.4).

Although designed with the potential for the self-assembly of a honeycomb lattice in mind, the above procedure has been successful in optimizing the parameters for other self- assembly potentials as well.