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A Second-Order Method for Locating Critical Points of Prescribed Index on Energy Landscapes — arXiv:2609.11038

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.11038
License (as archived): CC0-1.0
Archived: 2026-10-16T06:10:02.229Z
SHA-256 of the archived text: c6c06313e04ab0652937dbb511ebfa2300f4519a96c0321982b833a77564e322
Stable URL of this page: https://metraipolis.com/documento/dc_bb9607466f28

How to cite

A Second-Order Method for Locating Critical Points of Prescribed Index on Energy Landscapes — arXiv:2609.11038 — arXiv (en), https://arxiv.org/abs/2609.11038 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_bb9607466f28 (Metraipolis archive dc_bb9607466f28, sha256 c6c06313e04ab0652937dbb511ebfa2300f4519a96c0321982b833a77564e322)

BibTeX

@misc{dcbb9607466f28,
  title = {A Second-Order Method for Locating Critical Points of Prescribed Index on Energy Landscapes — arXiv:2609.11038},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.11038},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_bb9607466f28; sha256 c6c06313e04ab0652937dbb511ebfa2300f4519a96c0321982b833a77564e322},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.11038. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Exploring high dimensional energy landscape is a challenging problem in many applications where both ground states and transition states offer important information about the underlying physical systems. To facilitate the computational exploration, we propose a cubic-regularized second-order method for locating critical points of prescribed index, including both energy minima (with an index $0$) and transition states (saddle points with positive index). At each step of the iteration, the proposed method involves several ingredients, such as updating the eigen-directions of the Hessian according to the prescribed index, reflecting the gradient and the Hessian along selected unstable eigen-directions, if any, and constructing the next step by solving a cubic-regularized subproblem. Under suitable assumptions, we show that, when the eigenspace and the cubic subproblem are solved exactly, the iteration converges locally and quadratically to a critical point of the prescribed index. We further establish convergence results in the inexact setting, where both the eigenspace computation and the solution of the cubic subproblem are performed approximately. We show that the resulting iteration retains local linear convergence, with the convergence rate depending on the accuracy of the inexact computations. Moreover, we prove that the fixed points of the algorithm are precisely critical points of the prescribed index. In addition, we introduce an adaptive strategy for updating the cubic regularization parameter to improve robustness and computational efficiency of the proposed algorithm. Numerical experiments confirm the predicted convergence behavior, demonstrate the effectiveness of the adaptive strategy, and illustrate the capability of the proposed method for locating critical points and constructing solution landscapes.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-11 in math.NA).
Categorie arXiv: Numerical Analysis (math.NA).
Pagina del lavoro: https://arxiv.org/abs/2609.11038 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-16T05:55:33.321Z dalla pagina https://arxiv.org/catchup/math.NA/2026-09-11?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).