Iterative Solution of Forward and Adjoint Systems Using a Nonsymmetric Saddle Point Matrix With Preconditioning

ABSTRACT We present a framework for the simultaneous solution of forward () and adjoint () linear systems by reformulating the coupled problem as a single augmented nonsymmetric saddle point system. For this formulation, we derive conditions on a weight matrix under which the augmented matrix possesses a real positive spectrum, thereby permitting the use of a conjugate gradient‐like iteration in a transformed inner‐product setting. We also develop a Schur complement‐based preconditioning strategy implemented through incomplete QR factorization of , avoiding explicit formation of the normal equations. The numerical study comprises fourteen sparse test problems from the SuiteSparse Matrix Collection, drawn from a broad range of application domains, structural classes, and conditioning regimes. The principal experiment is a matched preconditioner comparison in which NspCG, GLSQR, and MINRES share the same incomplete QR factor, so that iteration‐phase runtime differences can be assessed without confounding by different factorization costs. In that setting, NspCG attains the smallest mean iteration count and the shortest mean iteration‐phase time, with a clear advantage over MINRES and an essentially tied per‐iteration cost with GLSQR. A brief secondary comparison with QMR and GLSQR under incomplete LU is reported for completeness, but is interpreted as a comparison of complete solver pipelines rather than of Krylov recurrences in isolation.