Certification and the Potential Energy Landscape

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.

7 pages, 4 figures. arXiv admin note: text overlap with arXiv:1302.6265

References in corpus (12)

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• Characterizing the network topology of the energy landscapes of atomic clusters
• Tumbling through a landscape: Evidence of instabilities in high-dimensional moduli spaces
• Finding all flux vacua in an explicit example
• Enumerating Gribov copies on the lattice
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• Certifying the Potential Energy Landscape
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Cited by in corpus (4)

• A Collection of Challenging Optimization Problems in Science, Engineering and Economics
• Potential Energy Landscape of the Two-Dimensional XY Model: Higher-Index Stationary Points
• Energy Landscape of the Finite-Size Mean-field 2-Spin Spherical Model and Topology Trivialization
• Statistics of Stationary Points of Random Finite Polynomial Potentials