  We develop a theoretical foundation for the application of Nesterov's
accelerated gradient descent method (AGD) to the approximation of solutions of
a wide class of partial differential equations (PDEs). This is achieved by
proving the existence of an invariant set and exponential convergence rates
when its preconditioned version (PAGD) is applied to minimize locally Lipschitz
smooth, strongly convex objective functionals. We introduce a second-order
ordinary differential equation (ODE) with a preconditioner built-in and show
that PAGD is an explicit time-discretization of this ODE, which requires a
natural time step restriction for energy stability. At the continuous time
level, we show an exponential convergence of the ODE solution to its steady
state using a simple energy argument. At the discrete level, assuming the
aforementioned step size restriction, the existence of an invariant set is
proved and a matching exponential rate of convergence of the PAGD scheme is
derived by mimicking the energy argument and the convergence at the continuous
level. Applications of the PAGD method to numerical PDEs are demonstrated with
certain nonlinear elliptic PDEs using pseudo-spectral methods for spatial
discretization, and several numerical experiments are conducted. The results
confirm the global geometric and mesh size-independent convergence of the PAGD
method, with an accelerated rate that is improved over the preconditioned
gradient descent (PGD) method.
