Using deep neural networks to solve PDEs has attracted a lot of attentions recently. However, why the learning method works is falling far behind its empirical success. In this paper, we provide rigorous numerical analysis on Ritz (DRM) \cite{wan11} for second order elliptic equations with Neumann boundary conditions. We establish first nonasymptotic convergence rate in $H^1$ norm DRM using $\m...

It is shown that certain classes of high order accurate Galerkin approximations for homogeneous second-order hyperbolic equations, known to possess optimal 2 °° order rate of convergence in L , also possess optimal order rate of convergence in L . This is attainable with particular smoothness assumptions on the initial data. We establish sufficient conditions for optimal L -convergence of the a...

We consider Broyden's 1965 method for solving nonlinear equations. If the mapping is linear, then a simple modification of this method guarantees global and Qsuperlinear convergence. For nonlinear mappings it is shown that the hybrid strategy for nonlinear equations due to Powell leads to R-superlinear convergence provided the search directions form a uniformly linearly independent sequence. We...

Plenty of researches have been performed to probe the diverse properties of ZnO nanowires, but only a few have focused on the physical properties of a single nanowire since to analyze the optical confinement and their correlation lengths along a single nanowire is difficult. In this study, a single ZnO nanowire was synthesized using a Ti-assisted chemical vapor deposition (CVD) method to avoid ...