Second-order flows for approaching stationary points of a class of non-convex energies via convex-splitting schemes

Abstract: This paper contributes to the exploration of a recently introduced computational paradigm known as second-order flows, which are characterized by novel dissipative hyperbolic partial differential equations extending accelerated gradient flows to energy functionals defined on Sobolev spaces, and exhibiting significant performance particularly for the minimization of non-convex energies. 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 timestep tends to zero. Finally, these algorithms undergo thorough testing and validation in approaching stationary points of representative non-convex variational models in scientific computing.