Comparative studies on damage identification with Tikhonov regularization and sparse regularization

Summary Structural damage identification is essentially an inverse problem. Ill‐posedness is a common obstacle encountered in solving such an inverse problem, especially in the context of a sensitivity‐based model updating for damage identification. Tikhonov regularization, also termed as ℓ2‐norm regularization, is a common approach to handle the ill‐posedness problem and yields an acceptable and smooth solution. Tikhonov regularization enjoys a more popular application as its explicit solution, computational efficiency, and convenience for implementation. However, as the ℓ2‐norm term promotes smoothness, the solution is sometimes over smoothed, especially in the case that the sensor number is limited. On the other side, the solution of the inverse problem bears sparse properties because typically, only a small number of components of the structure are damaged in comparison with the whole structure. In this regard, this paper proposes an alternative way, sparse regularization, or specifically ℓ1‐norm regularization, to handle the ill‐posedness problem in response sensitivity‐based damage identification. The motivation and implementation of sparse regularization are firstly introduced, and the differences with Tikhonov regularization are highlighted. Reweighting sparse regularization is adopted to enhance the sparsity in the solution. Simulation studies on a planar frame and a simply supported overhanging beam show that the sparse regularization exhibits certain superiority over Tikhonov regularization as less false‐positive errors exist in damage identification results. The experimental result of the overhanging beam further demonstrates the effectiveness and superiorities of the sparse regularization in response sensitivity‐based damage identification. Copyright © 2015 John Wiley & Sons, Ltd.