A new sensitivity-based damage detection method is proposed to identify and estimate the location and severity of structural damage using incomplete noisy modal data. For these purposes, an improved sensitivity function of modal strain energy (MSE) based on Lagrange optimization problem is derived to adapt the initial sensitivity formulation of MSE to damage detection problem with the aid of new mathematical approaches. In the presence of incomplete noisy modal data, the sensitivity matrix is sparse, rectangular, and ill-conditioned, which leads to an ill-posed damage equation. To overcome this issue, a new regularization method named as Regularized Least Squares Minimal Residual (RLSMR) is proposed to solve the ill-posed damage equation. This method relies on Krylov subspace and exploits bidiagonalization and iterative algorithms to solve linear mathematical systems. For the majority of Krylov subspace methods, conventional direct methods for the determination of an optimal regularization parameter may not be proper. To cope with this limitation, a hybrid technique is introduced that depends on the residual of RLSMR method, the number of iterations, and the bidiagonalization algorithm. The accuracy and performance of the improved and proposed methods are numerically examined by a planner truss by incorporating incomplete noisy modal parameters and finite element modeling errors. A comparative study on the initial and improved sensitivity functions is conduced to investigate damage detectability of these sensitivity formulations. Furthermore, the accuracy and robustness of RLSMR method in detecting damage are compared with the well-known Tikhonov regularization method. Results show that the improved sensitivity of MSE is an efficient tool for using in the damage detection problem due to a high sensitivity to damage and reliable damage detectability in comparison with the initial sensitivity function. Additionally, it is observed that the RLSMR method with the aid of the hybrid technique successfully solves the ill-posed damage equation and provides better damage detection results compared with the Tikhonov regularization technique

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