This report introduces our multi-level correction algorithms for solving semi-linear elliptic equations, eigenvalue problems, and inequality constraint optimization problems. This report will first start from the understanding of the most basic Aubin-Nitsche technique in the simplest finite element theory, introduce a new low-dimensional subspace we defined and the Aubin-Nitsche estimation on it, and then use this technique Iterative algorithm applied to construct nonlinear equations. We will analyze the convergence speed and computational efficiency of this iterative algorithm, and use tensor technology to improve the efficiency of solving polynomial nonlinear equations to an asymptotically optimal degree regardless of the number of nonlinear iterations. Finally, we will introduce the application of this idea in eigenvalue problems, optimization problems with inequality constraints, and the latest developments.