Showing papers by "Long Chen published in 2019"

A Divergence Free Weak Virtual Element Method for the Stokes Problem on Polytopal Meshes

Abstract: Some virtual element methods on polytopal meshes for the Stokes problem are proposed and analyzed. The pressure is approximated by discontinuous polynomials, while the velocity is discretized by H(div) virtual elements enriched with some tangential polynomials on the element boundaries. A weak symmetric gradient of the velocity is computed using the corresponding degree of freedoms. The main feature of the method is that it exactly preserves the divergence free constraint, and therefore the error estimates for the velocity does not explicitly depend on the pressure.

Nonconforming Virtual Element Method for $2m$th Order Partial Differential Equations in $\mathbb {R}^n$

Abstract: A unified construction of the $H^m$-nonconforming virtual elements of any order $k$ is developed on any shape of polytope in $\mathbb R^n$ with constraints $m\leq n$ and $k\geq m$. As a vital tool in the construction, a generalized Green's identity for $H^m$ inner product is derived. The $H^m$-nonconforming virtual element methods are then used to approximate solutions of the $m$-harmonic equation. After establishing a bound on the jump related to the weak continuity, the optimal error estimate of the canonical interpolation, and the norm equivalence of the stabilization term, the optimal error estimates are derived for the $H^m$-nonconforming virtual element methods.

From differential equation solvers to accelerated first-order methods for convex optimization

Abstract: Convergence analyses of accelerated first-order methods for convex optimization problems are presented from the point of view of ordinary differential equation (ODE) solvers. A new dynamical system, which is called Nesterov accelerated gradient (NAG) flow, has been derived from the connection between acceleration mechanism and $A$-stability for ODE solvers. The exponential decay to the equilibrium of the solution trajectory is firstly established in the continuous level. Numerical discretizations are then considered and the convergence analyses are established via a tailored Lyapunov function. The proposed ODE solvers approach can not only cover existing methods, such as Nesterov's accelerated gradient method, but also produce new algorithms for solving composite convex optimization problems that possesses accelerated convergence rates.

Anisotropic Error Estimates of the Linear Nonconforming Virtual Element Methods

Abstract: A refined a priori error analysis of the lowest-order (linear) nonconforming virtual element method (VEM) for approximating a model Poisson problem is developed in both 2D and 3D. A set of new geom...

First order optimization methods based on Hessian-driven Nesterov accelerated gradient flow

Abstract: A novel dynamical inertial Newton system, which is called Hessian-driven Nesterov accelerated gradient (H-NAG) flow is proposed. Convergence of the continuous trajectory are established via tailored Lyapunov function, and new first-order accelerated optimization methods are proposed from ODE solvers. It is shown that (semi-)implicit schemes can always achieve linear rate and explicit schemes have the optimal(accelerated) rates for convex and strongly convex objectives. In particular, Nesterov's optimal method is recovered from an explicit scheme for our H-NAG flow. Furthermore, accelerated splitting algorithms for composite optimization problems are also developed.