Second-order flows for approaching stationary points of a class of non-convex energies via convex-splitting schemes
<|reference_start|>Second-order flows for approaching stationary points of a class of non-convex energies via convex-splitting schemes: The use of accelerated gradient flows is an emerging field in optimization, scientific computing and beyond. This paper contributes to the theoretical underpinnings of a recently-introduced computational paradigm known as second-order flows, which demonstrate significant performance particularly for the minimization of non-convex energy functionals defined on Sobolev spaces, and are characterized by novel dissipative hyperbolic partial differential equations. Our approach hinges upon convex-splitting schemes, a tool which is not only pivotal for clarifying the well-posedness of second-order flows, but also yields a versatile array of robust numerical schemes through temporal and spatial discretization. We prove the convergence to stationary points of such schemes in the semi-discrete setting. Further, we establish their convergence to time-continuous solutions as the time-step tends to zero, and perform a comprehensive error analysis in the fully discrete case. Finally, these algorithms undergo thorough testing and validation in approaching stationary points of non-convex variational models in applied sciences, such as the Ginzburg-Landau energy in phase-field modeling and a specific case of the Landau-de Gennes energy of the Q-tensor model for liquid crystals.<|reference_end|>
@article{chen2024second-order,
    title={Second-order flows for approaching stationary points of a class of
  non-convex energies via convex-splitting schemes},
    author={Haifan Chen, Guozhi Dong, Jos'e A. Iglesias, Wei Liu, Ziqing Xie},
    journal={arXiv preprint arXiv:2402.12173},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.12173},
    primaryClass={math.NA cs.NA math.AP math.OC}
}