ABSTRACT
In this chapter, we first present the basic theory of hierarchical sparse Bayesian learning and then discuss its fundamental connections with the Bayesian Ockham razor, which trades off the fit to the data by the model against the amount of information about the model that is extracted from the data. Then we illustrate the sparse Bayesian learning theory with an application to structural health monitoring where we use the prior knowledge that structural stiffness losses due to excessive loading or environmental degradation typically occur in localized areas in the absence of structural collapse. By exploiting the spatial sparseness of the structural stiffness losses using our theory, we can reduce the ill-conditioning in the structural stiffness identification problem. A fast algorithm is presented that establishes the probability of localized stiffness reductions by using noisy incomplete modal data from before and after damage. The simulated damage assessment example illustrates the effectiveness and robustness of the proposed algorithms. Several nice features of our theory from both theoretical and computational perspectives are discussed. We note that the developed methods also have much broader applicability for inverse problems in science and technology where system matrices are to be inferred from noisy partial information about their eigenquantities.
