Optimization-based model calibration of marginal and joint output distributions utilizing analytical gradients

Abstract

Optimization-based model calibration (OBMC) determines statistical parameters of unknown input variables by optimizing statistical similarity between simulations and observations. Gradient-based optimization algorithms have been utilized for OBMC since they require less computational cost compared with non-gradient optimization algorithms. However, OBMC using numerical gradients such as finite differences has disadvantages in the aspect of computational cost and numerical instability. For these reasons, a previous study derived analytical gradients under the assumption that performance functions are a linear single response and unknown inputs follow a normal distribution. However, to apply the model calibration to general engineering problems, analytical gradients need to be derived without any assumption. Therefore, it is proposed in this study to derive analytical gradients of marginal and joint likelihood functions to calibrate both linear and nonlinear responses. In addition, the analytical gradients are derived for five distribution types of unknown input random variables to be estimated: normal, lognormal, Gumbel, extreme, and uniform distributions. For derivation of the analytical gradients, the chain rule is utilized to combine the derivatives of the inverse function of cumulative distribution function (CDF), performance function, and joint likelihood function. Numerical examples and one engineering example using finite element analysis (FEA) are employed to verify the accuracy and efficiency of the proposed analytical gradient, and it is demonstrated that accurately calibrated output responses can be obtained by the proposed method by finding a better optimum in OBMC.