Unbiased estimation of the gradient of the log-likelihood in inverse problems

We consider the problem of estimating a parameter $$\theta \in \Theta \subseteq {\mathbb {R}}^{d_{\theta }}$$ θ ∈ Θ ⊆ R d θ associated with a Bayesian inverse problem. Typically one must resort to a numerical approximation of gradient of the log-likelihood and also adopt a discretization of the problem in space and/or time. We develop a new methodology to unbiasedly estimate the gradient of the log-likelihood with respect to the unknown parameter, i.e. the expectation of the estimate has no discretization bias. Such a property is not only useful for estimation in terms of the original stochastic model of interest, but can be used in stochastic gradient algorithms which benefit from unbiased estimates. Under appropriate assumptions, we prove that our estimator is not only unbiased but of finite variance. In addition, when implemented on a single processor, we show that the cost to achieve a given level of error is comparable to multilevel Monte Carlo methods, both practically and theoretically. However, the new algorithm is highly amenable to parallel computation.

Nonlinear Ultrasonics for Early Damage Detection

Structural Health Monitoring (SHM) is an emerging discipline that aims at improving the management of the life cycle of industrial components. The scope of this chapter is to present the integration of nonlinear ultrasonics with the Bayesian inverse problem as an appropriate tool to estimate the updated health state of a component taking into account the associated uncertainties. This updated information can be further used by prognostics algorithms to estimate the future damage stages. Nonlinear ultrasonics allows an early detection of damage moving forward the achievement of reliable predictions, while the inverse problem emerges as a rigorous method to extract the slight signature of early damage inside the experimental signals using theoretical models. The Bayesian version of the inverse problem allows measuring the underlying uncertainties, improving the prediction process. This chapter presents the fundamentals of nonlinear ultrasonics, their practical application for SHM, and the Bayesian inverse problem as a method to unveil damage and manage uncertainty.

Nonlinear Ultrasonics for Early Damage Detection

Structural Health Monitoring (SHM) is an emerging discipline that aims at improving the management of the life cycle of industrial components. The scope of this chapter is to present the integration of nonlinear ultrasonics with the Bayesian inverse problem as an appropriate tool to estimate the updated health state of a component taking into account the associated uncertainties. This updated information can be further used by prognostics algorithms to estimate the future damage stages. Nonlinear ultrasonics allows an early detection of damage moving forward the achievement of reliable predictions, while the inverse problem emerges as a rigorous method to extract the slight signature of early damage inside the experimental signals using theoretical models. The Bayesian version of the inverse problem allows measuring the underlying uncertainties, improving the prediction process. This chapter presents the fundamentals of nonlinear ultrasonics, their practical application for SHM, and the Bayesian inverse problem as a method to unveil damage and manage uncertainty.