Simultaneous preconditioning and symmetrization of non-symmetric linear systems
This work addresses the challenge of solving large non-symmetric linear systems by enabling the use of efficient symmetric solvers, but the results are preliminary and lack concrete numerical comparisons.
The paper proposes a new method that simultaneously preconditions and symmetrizes non-symmetric linear systems, enabling the use of symmetric solvers. The approach is shown to be efficient for ill-conditioned, highly non-symmetric systems.
Motivated by the theory of self-duality which provides a variational formulation and resolution for non self-adjoint partial differential equations \cite{G1, G2}, we propose new templates for solving large non-symmetric linear systems. The method consists of combining a new scheme that simultaneously preconditions and symmetrizes the problem, with various well known iterative methods for solving linear and symmetric problems. The approach seems to be efficient when dealing with certain ill-conditioned, and highly non-symmetric systems.