Core Concepts

The GADI (Generalized Alternating Direction Implicit) iterative method is an efficient approach for solving large sparse complex symmetric linear systems, with applications in solving Lyapunov and Riccati equations with complex coefficients.

Abstract

The paper introduces the GADI iterative method for solving large sparse complex symmetric linear systems of the form Ax = b, where A = W + iT, with W being a symmetric positive definite matrix and T being a symmetric positive semi-definite matrix.
Key highlights:
The GADI method is presented and its convergence properties are analyzed, showing that it unconditionally converges.
Numerical experiments demonstrate that the GADI method outperforms existing methods like MHSS, PMHSS, CRI, and TSCSP in terms of iteration count and computational time.
The GADI method is applied to solve Lyapunov equations with complex coefficients, where it is shown to be more efficient than the HSS method.
For solving Riccati equations with complex coefficients, the paper combines the Newton method and the GADI method, providing a Newton-GADI algorithm that effectively solves the problem.
Detailed complexity analysis and numerical results are provided to validate the efficiency of the proposed methods.

Stats

Ax = (W + iT)x = b
A ∈ C^(n×n), x, b ∈ C^n
W, T ∈ R^(n×n) are real symmetric matrices, with at least one being positive definite

Quotes

"We have introduced the generalized alternating direction implicit iteration (GADI) method for solving large sparse complex symmetric linear systems and proved its convergence properties."
"Furthermore, as an application of the GADI method in solving complex symmetric linear systems, we utilized the flattening operator and Kronecker product properties to solve Lyapunov and Riccati equations with complex coefficients using the GADI method."

Key Insights Distilled From

by Juan Zhang,W... at **arxiv.org** 04-19-2024

Deeper Inquiries

The GADI method can be extended to solve other types of complex linear systems by considering different matrix structures and properties. One approach is to adapt the GADI algorithm to handle non-symmetric matrices by modifying the iteration scheme to accommodate the specific characteristics of the matrix. This may involve adjusting the splitting parameters, convergence criteria, or the way the linear subsystems are solved within each iteration.
Additionally, the GADI method can be applied to systems with different properties such as non-Hermitian matrices, indefinite matrices, or systems with specific sparsity patterns. By tailoring the GADI algorithm to suit the properties of the linear system, it can be effectively extended to a broader range of complex linear systems beyond the symmetric case.

One potential limitation of the GADI method is the computational cost associated with solving the linear subsystems at each iteration. As the method requires solving two linear systems with symmetric positive definite matrices, the computational complexity can be significant for large-scale problems. To address this limitation, efficient solvers and preconditioning techniques can be employed to improve the convergence rate and reduce the computational burden.
Another drawback of the GADI method is the sensitivity to the choice of parameters such as the splitting parameters α and ω. Suboptimal parameter selection can lead to slower convergence or even divergence of the iterative process. To mitigate this issue, parameter tuning strategies, adaptive parameter selection schemes, or automated parameter optimization algorithms can be implemented to enhance the robustness and efficiency of the GADI method.

Yes, the GADI method can be combined with various preconditioning and acceleration techniques to enhance its performance for large-scale problems. One common approach is to incorporate efficient preconditioners that can reduce the condition number of the coefficient matrix and improve the convergence rate of the iterative solver. Techniques such as incomplete Cholesky factorization, algebraic multigrid, or domain decomposition methods can be utilized as preconditioners in conjunction with the GADI method.
Furthermore, acceleration techniques like Krylov subspace methods (e.g., GMRES, BiCGSTAB) can be integrated into the GADI algorithm to accelerate the convergence of the iterative process. By leveraging advanced iterative solvers and preconditioning strategies, the GADI method can achieve faster convergence and better scalability for solving large-scale complex linear systems.

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