Solution of Preconditioned Nonsymmetric Saddle Point Systems through Modified Conjugate Gradient Iteration

In this dissertation, we present an iterative method (Preconditioned Nonsymmetric Saddle Point Conjugate Gradient) for simultaneously solving forward ($A{\bf x}={\bf b}$) and adjoint ($A^T{\bf y}={\bf g}$) linear systems. Our approach involves constructing an augmented nonsymmetric saddle point matrix that has a real positive spectrum and developing a conjugate gradient-like iteration for this matrix. We investigate the use of Schur Complement preconditioners with block-diagonal factorization computed by an incomplete QR factorization of $A$ to speed up the convergence of our method and compare the results to the preconditioned generalized least squares residual (GLSQR) and quasi-minimal residual (QMR) methods. We develop quadrature rules based on our NspCG method for estimating the generic bilinear form ${\bf u}^Tf(A){\bf v}$ necessary for approximating the scattering amplitude ${\bf g}^TA^{-1}{\bf b}$.