Certification and the Potential Energy Landscape

TL;DR

This paper discusses how Smale's α-theory can be used to certify that numerical solutions to potential energy landscapes are true stationary points, ensuring reliable convergence and avoiding misleading approximations.

Contribution

It introduces a method using Smale's α-theory to mathematically certify stationary points in potential energy landscapes, guaranteeing their validity regardless of numerical precision.

Findings

Certification guarantees quadratic convergence to stationary points.

Smale's α-theory provides a rigorous mathematical proof of solution validity.

The method distinguishes true solutions from non-solutions in numerical approximations.

Abstract

Typically, there is no guarantee that a numerical approximation obtained using standard nonlinear equation solvers is indeed an actual solution, meaning that it lies in the quadratic convergence basin. Instead, it may lie only in the linear convergence basin, or even in a chaotic region, and hence not converge to the corresponding stationary point when further optimization is attempted. In some cases, these non-solutions could be misleading. Proving that a numerical approximation will quadratically converge to a stationary point is termed \textit{certification}. In this report, we provide details of how Smale's $\alpha$-theory can be used to certify numerically obtained stationary points of a potential energy landscape, providing a \textit{mathematical proof} that the numerical approximation does indeed correspond to an actual stationary point, independent of the precision employed.