Software (as well as experience on the designer's part) is invaluable in the computationally intensive process of creating optical systems that change the path of light rays in certain desirable ways. Optical designers begin interacting with the software by entering requirements. These may include optical qualities and physical size as well as building parameters that describe a possible starting configuration for the system. To fulfill design targets within tolerances determined by an application, these parameters are automatically changed during optimization, a critical feature of modern optical design software.1
The sort most frequently used in applications is so-called local optimization. Mathematically speaking, a ‘local minimum’ of an error function based on deviations from requirements (e.g., image defects) is then found. If the error function is visualized as the elevation in a hill landscape with several valleys, and a starting configuration is thought of as a specific location on a hill where a ball is placed, then intuitively one expects that two balls starting close to each other will roll downhill until they reach the same valley (i.e., the same local minimum where the slope is zero in all directions). Local optimization is thus usually assumed to reach the nearest local minimum to the starting configuration, and starting configurations close to each other are expected to converge, typically, to the same minimum.
Figure 1. Fractal basins of attraction for a 7-dimensional Double Gauss optimization problem.
When a familiar design is optimized for new specifications, designers usually prefer to obtain a system shape and correction properties that are similar to the ones that are already known to be satisfactory. A higher degree of predictability is also desirable when attempting to reach the design goal in several stages. In such cases, each stage should preserve what has been achieved previously. When the outcome is not the expected one, the cause is usually attributed to the inherent complexity of a design landscape with many local minima.
Our objective is to make optical designers aware of another possible source of unpredictability in the optimization process. Situations exist where starting points close to each other converge to distinct minima, often with very different imaging quality. Then, obtaining the best of them becomes a matter of luck.2 Figure 1 was produced using commercial optical design software for a typical optimization problem with seven variables, where the goal is to design a Double Gauss photographic objective. By locally optimizing equally spaced starting points on a 2D grid of 1001 × 1001 points, we find three distinct local minima. Each grid point has a color depending on which local minimum we obtain from the corresponding starting configuration. The two coordinates of the point are related to the initial values of two of the variables (the initial values for the other variables are the same for all points in the figure). For the examples shown in this paper, the colored regions are 2D cuts through the fractal basins of attraction of the existing local minima.2, 3
Figure 2. (a) Double Gauss system. (b) Another local minimum of the same optimization problem.
Figure 3. Four optimization paths in a 2D variable space. The upper and lower figures have been obtained using different commercial programs (and starting points), but the results are qualitatively similar. The minima A, B, C, and D are essentially the same for both programs. The starting points for optimization are chosen very close to each other at the position START, and the iterations are shown as small points.
In addition to the expected Double Gauss solution (the red region in Figure 1)—see Figure 2(a)—we also obtain the minimum shown in Figure 2(b) (the green region) and a third minimum (the blue region) closely resembling the one in Figure 2(b). The green and blue solutions have error functions about 30 times larger than the Double Gauss design. Figure 1 shows that the regions leading to the desirable and undesirable solutions are mixed together in a very complicated way.
To understand how these instabilities occur, we examine a simple 2D optimization problem with four local minima (blue points). Details are available elsewhere.2 A movie4 shows the history of a set of 10 starting points that are very close to each other. For each starting point, the iteration trajectories are first attracted, and then repelled, by regions close to the red and purple lines in the movie. The history of earlier trajectories is shown by green points.
The distance between the 10 trajectories increases rapidly with the number of iterations before the attracting-repelling regions are reached. The extreme sensitivity to initial conditions occurs because the trajectories are then already far apart from each other. Consequently, despite the fact that they have their origin in almost the same point, they converge toward different local minima after the trajectories are repelled.5 Similar behavior can be observed in Figure 3 for four trajectories, initially very close to each other, that arrive at four different local minima (A, B, C, and D).2 A large number of iteration points in the middle of the lower part of Figure 3 indicates the presence of a horizontal attracting-repelling region there.
When starting points that are very close to each other lead to different local minima after optimization, the design problem has an extreme sensitivity to change of initial conditions. Such instabilities in optimization, if present, decrease predictability by mixing regions of desirable and undesirable solutions together. A better understanding of these instabilities is necessary to assess whether, and to what degree, they affect design productivity.
As a next step, it is also worth investigating whether it is possible to modify practical optimization algorithms in a way that makes the conflict between speed and stability less acute.
Maarten van Turnhout, Florian Bociort
Delft University of Technology
Delft, The Netherlands