Adaptive System Identification for GMRES(m) Based on DMDc

The Restarted Generalized Minimal Residual Method (GMRES(m)) is a widely used iterative solver for large-scale linear systems, particularly when the coefficient matrix is sparse, non-singular, and non-symmetric. However, the method may suffer from slow convergence or even stagnation. To address this, the restart parameter mmm must be carefully tuned, yet no general rule exists for selecting its optimal value. Designing such a control strategy is further complicated by the need for internal insights into the algorithm’s behavior. In this work, we aim to capture the internal dynamics of GMRES(m) using a system identification framework based on Dynamic Mode Decomposition with Control (DMDc).