Algebraic Multigrid Solvers for Complex-Valued Matrices

Algebraic Multigrid Solvers for Complex-Valued Matrices

In the mathematical modeling of real-life applications, systems of equations with complex coefficients often arise. While many techniques of numerical linear algebra, e.g., Krylovsubspace methods, extend directly to the case of complex-valued matrices, some of the most effective preconditioning techniques and linear solvers are limited to the real-valued case. Here, we consider the extension of...

Algebraic Multigrid for Complex Symmetric Systems

The two dimensional quasistatic time-harmonic Maxwell formulations yield complex Helmholtz equations. Multigrid techniques are known to be efficient for solving the discretization of real valued diffusion equations. In this paper these multigrid techniques are extended to handle the complex equation. The implementation of geometric multigrid techniques can be cumbersome for practical engineerin...

Detection of Strong Coupling in Algebraic Multigrid Solvers.

Complex-Valued Neural Network for Hermitian Matrices

This paper proposes neural network for computing the eigenvectors of Hermitian matrices. For the eigenvalues of Hermitian matrices, we establish an explicit representation for the solution of the neural network based on Hermitian matrices and analyze its convergence property. We also consider to compute the eigenvectors of skew-symmetric matrices and skew-Hermitian matrices, corresponding to th...

Heuristics for the Automatic Construction of Coarse Grids in Multigrid Solvers for Finite Element Matrices

The nite element method (FEM) has proven to be a popular spatial discretization technique in the simulation of complex physical systems in many areas of science and engineering. Implicit solution schemes, used in the time discretization, require the application of the inverse of an operator which is usually linear or linearized; thus, after the FEM discretization, a linear set of discrete equat...