Preconditioned Accelerated Gradient Descent Methods for Locally Lipschitz Smooth Objectives with Applications to the Solution of Nonlinear PDEs

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