A method for the accurate determination of basins of attraction of jammed packings

The efficient computation of the volume of basins of attraction in the energy landscape of jammed solids has recently been enabled by Monte Carlo integration schemes akin to the Frenkel-Ladd method for calculating the free energy of atomic solids. The rate-limiting step for this approach is the energy minimization performed at every Monte Carlo step to determine whether a point lies within a given basin of attraction. Here, we show that directly solving the steepest descent trajectory using a variable-order variable-step implicit scheme based on the Backward Differentiation Formulas, implemented by the CVODE solver, identifies basins with near-perfect accuracy with comparable performance to other efficient optimizers (e.g., FIRE) that, however, do not preserve the basin geometry. We use this new scheme to produce accurate projections of basins of attraction of Hertzian particles in 2 dimensions. We also propose an efficient new method that uses the CVODE solver in non-convex regions and Newton's method otherwise, allowing us to improve performance further while significantly reducing the loss in accuracy characteristic of (quasi-) Newton methods.