In recent years, contour-based eigensolvers have emerged as a standard approach for the solution of large and sparse eigenvalue problems. Building upon performance improvements through non-linear least square optimization so-called rational filters, we introduce systematic method to design these filters by minimizing worst-case convergence ratio eliminate parametric dependence on weight functio...

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...