A Bayesian and Laplace prior-based adaptive sparse regularization approach for structural damage diagnosis with uncertainty quantification

Abstract

The uncertainties arising from measurement noise and model error in operational structures often render damage detection problems ill-posed in practical applications. Limited sensor availability and the complexity of real-world damage scenarios further complicate damage identification. To address these challenges, this study proposes a novel Bayesian and Laplace prior-based adaptive sparse regularization (BLASR) method with two variants that integrate distinct strategies for selecting regularization factors within a Bayesian framework or a refinement scheme. These variants are designed to accommodate complex damage configurations and varying levels of uncertainty. Concurrently, a closed-form uncertainty quantification (UQ) approach is derived based on Bayesian inference, providing a quantitative measure of confidence in the estimated damage location and severity. However, the inclusion of a Laplace prior can complicate UQ by introducing nondifferentiability into the posterior distribution. Therefore, another key contribution of this work is to reformulate the nondifferentiable posterior into an equivalent differentiable form, enabling direct computation of the posterior covariance matrix for the damage parameters via the second-order derivatives of the negative log-posterior. This facilitates efficient UQ analysis through analytic confidence intervals (CIs). A comprehensive and extensive numerical experiment on a full-scale 3D frame structure subjected to realistic tridirectional seismic excitations is conducted under complex damage scenarios and multiple sources of uncertainty. The results demonstrate that the proposed method achieves superior accuracy and stability compared with two counterpart approaches. The innovation of the proposed method lies in two novel adaptive Bayesian sparse regularization strategies, which effectively reduce the impact of uncertainties on damage detection performance and consequently improve the accuracy of damage identification. Furthermore, the proposed method facilitates the derivation of closed-form CIs for the identified damage parameters, thereby enabling a transparent evaluation of residual uncertainty and enhancing both the interpretability and reliability of the results.