In addition to comparing model fitting to the CLS method, we are also interested in evaluating the robustness of the model fitting approach to variations in the chosen model, which may not always be known a priori. Figure shows plots of C(𝐩𝐩) and |∇C| � versus fitting iteration for three scenarios in which the fitting model 𝐌𝐌(𝐩𝐩) has one less (left), the same number (middle), and one more (right) Kubo component(s) than are actually present in the simulated data 𝐃𝐃. We start with analysis of the middle panels (C) and (D), which correspond to the fitting results shown in Figure . Plots of C(𝐩𝐩) in panel (C) show that C(𝐩𝐩) is useful for broadly comparing the quality of fit for a given iteration of 𝐩𝐩 but is not particularly useful for understanding convergence because progress near the global minimum is difficult to evaluate on a relative scale. On the other hand, |∇C| � shows simple and reproducible behavior that clearly indicates convergence. The variation of |∇C| � with iteration shows three phases. First, when 𝐩𝐩 is far away from a minimum of the cost function, the gradient is ~10 -1 . Next, 𝐩𝐩 reaches the neighborhood of a minimum where the second order approximation of C(𝐩𝐩) in Eq. 3 is highly accurate, and hence, |∇C| � rapidly descends many orders of magnitude in as little as 2-3 iterations. Finally, 𝐩𝐩 reaches the minimum of C(𝐩𝐩) but |∇C| � approaches an asymptote. This asymptote should be essentially invariant at the global minimum because |∇C| � is, effectively, the relative roundoff error in C(𝐩𝐩), and hence, is determined by accumulated roundoff errors in Eq. 8 on top of machine precision (~2 × 10 -16 ). In a few cases, |∇C| � descends halfway and stalls. The algorithm responds to stalling by restarting from a new randomly generated 𝐩𝐩 within the usersupplied boundaries corresponding to the sudden increases seen in both plots (C) and (D). The dashed line in (C) is a user defined limit associated with the stopping criteria (section C of Materials and Methods). Video S3 shows fitting trajectories for every trial. Panels (E) and (F) show plots of C(𝐩𝐩) and |∇C| � for 20 trials of fitting a two-Kubo model to data simulated by a one-Kubo lineshape. The many jumps in both plots are evidence of frequent stalling and random restarts, which is opposite of the behavior seen in panels (A) and (B). We ran this experiment for only 20 trials due to the large number of iterations, the resulting crowding in the plots, and the larger memory requirement for the video.