2020/06/11 by Jea-Hyun Park, Park, Jea-Hyun, Abner J. Salgado +3 · 1 citation
Engineering · Mathematics · Computer Science · #Advanced Numerical Methods in Computational Mathematics #Numerical methods for differential equations #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.2006.06732
We develop a theoretical foundation for the application of Nesterov's\naccelerated gradient descent method (AGD) to the approximation of solutions of\na wide class of partial differential equations (PDEs). This is achieved by\nproving the existence of an invariant set and exponential convergence rates\nwhen its preconditioned version (PAGD) is applied to minimize locally Lipschitz\nsmooth, strongly convex objective functionals. We introduce a second-order\nordinary differential equation (ODE) with a preconditioner built-in and show\nthat PAGD is an explicit time-discretization of this ODE, which requires a\nnatural time step restriction for energy stability. At the continuous time\nlevel, we show an exponential convergence of the ODE solution to its steady\nstate using a simple energy argument. At the discrete level, assuming the\naforementioned step size restriction, the existence of an invariant set is\nproved and a matching exponential rate of convergence of the PAGD scheme is\nderived by mimicking the energy argument and the convergence at the continuous\nlevel. Applications of the PAGD method to numerical PDEs are demonstrated with\ncertain nonlinear elliptic PDEs using pseudo-spectral methods for spatial\ndiscretization, and several numerical experiments are conducted. The results\nconfirm the global geometric and mesh size-independent convergence of the PAGD\nmethod, with an accelerated rate that is improved over the preconditioned\ngradient descent (PGD) method.\n