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In this paper we examine iterative methods for solving the forward ($A{\bfx}={\bf b}$) and adjoint ($A^{T}{\bf y}={\bf g}$) systems of linear equationsused to approximate the scattering amplitude, defined by ${\bf g}^{T}{\bfx}={\bf y}^{T}{\bf b}$. Based on an idea first proposed by Gene Golub, we use aconjugate gradient-like iteration for a nonsymmetric saddle point matrix thatis constructed so as to have a real positive spectrum. Numerical experimentsshow that this method is more consistent than known methods for computing thescattering amplitude such as GLSQR or QMR. We then demonstrate that whencombined with known preconditioning techniques, the proposed method exhibitsmore rapid convergence than state-of-the-art iterative methods for nonsymmetricsystems.

Approximation of the Scattering Amplitude using Nonsymmetric Saddle Point Matrices