Approximation of the Scattering Amplitude using Nonsymmetric Saddle Point Matrices
For computational scientists solving scattering amplitude problems, this method offers a more consistent and faster iterative solver.
The paper proposes a conjugate gradient-like iteration for nonsymmetric saddle point matrices to approximate the scattering amplitude, achieving more consistent convergence than GLSQR or QMR and, with preconditioning, faster convergence than state-of-the-art methods.
In this paper we examine iterative methods for solving the forward ($A{\bf x}={\bf b}$) and adjoint ($A^{T}{\bf y}={\bf g}$) systems of linear equations used to approximate the scattering amplitude, defined by ${\bf g}^{T}{\bf x}={\bf y}^{T}{\bf b}$. Based on an idea first proposed by Gene Golub, we use a conjugate gradient-like iteration for a nonsymmetric saddle point matrix that is constructed so as to have a real positive spectrum. Numerical experiments show that this method is more consistent than known methods for computing the scattering amplitude such as GLSQR or QMR. We then demonstrate that when combined with known preconditioning techniques, the proposed method exhibits more rapid convergence than state-of-the-art iterative methods for nonsymmetric systems.