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.

我们为将 Nesterov 的加速梯度下降法 (AGD) 应用于各种偏微分方程 (PDE) 的解的近似开发了理论基础。这是通过在应用其预处理版本 (PAGD) 来最小化局部 Lipschitz时证明存在不变集和指数收敛率来实现的平滑、强凸的目标函数。我们引入了一个内置预调节器的二阶常微分方程 (ODE)，并表明 PAGD 是该 ODE 的显式时间离散化，它需要自然时间步长限制以实现能量稳定性。在连续时间级别，我们使用简单的能量参数展示了 ODE 解到其稳态的指数收敛。在离散层面，假设上述步长限制，证明了不变集的存在，并通过模拟能量参数和连续层面的收敛性推导出 PAGD 方案的匹配指数收敛速度。PAGD 方法在数值 PDE 中的应用通过使用伪谱方法进行空间离散化的某些非线性椭圆 PDE 来证明，并进行了多次数值实验。结果证实了 PAGD 方法的全局几何和网格大小独立收敛，其加速速率比预处理梯度下降 (PGD) 方法有所提高。