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.

翻译：加速梯度流的使用是优化、科学计算及其他领域中的一个新兴研究方向。本文为一类最近引入的名为二阶流的计算范式提供理论基础，该方法在最小化定义于Sobolev空间上的非凸能量泛函时展现出显著性能，其特点由新颖的耗散型双曲偏微分方程描述。我们的方法依赖于凸分裂格式，这一工具不仅对阐明二阶流的适定性至关重要，而且通过时间和空间离散化衍生出一系列稳健的数值格式。我们证明了在半离散设定下此类格式收敛到驻点。进一步，我们建立了当时间步长趋近于零时，这些格式收敛到时间连续解，并在全离散情况下进行了全面的误差分析。最后，这些算法在逼近应用科学中非凸变分模型的驻点（例如相场建模中的Ginzburg-Landau能量和液晶Q张量模型中Landau-de Gennes能量的特定情形）上接受了彻底测试与验证。