Fast multi-level iteration schemes with compression technique for eigen-problems of compact integral operators

Abstract

In this paper, we present multi-level iteration schemes to solve the eigen-problems of compact integral operators based on the multiscale Galerkin methods. By using the compression technique, the wavelet coefficient matrix is truncated into sparse. This technique results in a fast algorithm. To further minimize computational complexity, we establish Jacobi, Gauss-Seidel and L-H iteration schemes for solving the final algebraic eigenvalue problems obtained from discretizing the integral equation using fast multiscale Galerkin methods. We show that the proposed schemes require only linear computational complexity and obtain the desired convergence order for the approximate eigenvalues and eigenvectors. Numerical experiments are conducted and analyzed, with comparisons to the standard routine and the power iteration algorithm included. Ultimately, our multi-level iteration schemes yield faster solutions with fewer iterations, which is independent of the dimension of approximate space.