Certification and the Potential Energy Landscape

摘要

Typically, there is no guarantee that a numerical approximation obtainedusing standard nonlinear equation solvers is indeed an actual solution, meaningthat it lies in the quadratic convergence basin. Instead, it may lie only inthe linear convergence basin, or even in a chaotic region, and hence notconverge to the corresponding stationary point when further optimization isattempted. In some cases, these non-solutions could be misleading. Proving thata numerical approximation will quadratically converge to a stationary point istermed \textit{certification}. In this report, we provide details of howSmale's $\alpha$-theory can be used to certify numerically obtained stationarypoints of a potential energy landscape, providing a \textit{mathematical proof}that the numerical approximation does indeed correspond to an actual stationarypoint, independent of the precision employed.