Approximate Bayesian Computation considering Summary Statistics for Parameter Identification in Computational Mechanics

Abstract

Computational mechanics offers the possibility to construct numerical models that simulate the behavior of real-world engineering systems. These numerical models usually depend on a series of parameters such as material properties, whose precise characteristics may be unknown due to lack of knowledge. Under these circumstances, it is of interest to infer such characteristics using, e.g. Bayesian approaches since they allow to account for potential prior knowledge and direct risk assessment through posterior distributions. For that purpose, responses measured on the real system are contrasted with those coming from the numerical model, allowing the identification of the unknown parameters.

Bayesian approaches for identification are widely accepted by the engineering community. A key issue for its practical application is the evaluation of the likelihood function, which expresses the probability of observing the measured responses given the parameters that are being identified. However, for many practical cases, such likelihoods may be unavailable analytically or numerically instable to evaluate. To overcome these issues, this contribution explores the application of so-called likelihood-free methods in the context of mechanical engineering based on Approximate Bayesian Computation (ABC). ABC has been shown to be a feasible way to address the challenge described above. In particular, ABC does not require a likelihood or its evaluation but only requires the ability to simulate from the assumed system. The key is to use summary statistics (such as means, standard deviations etc.) to compare the measured and simulated responses. In addition, the use of these summary statistics introduces the possibility to introduce approximation concepts to speed up the numerical computation of the summary statistics. We compare the approach to other state of the art methods to illustrate its effectiveness and efficiency.