Inverse Uncertainty Quantification — Markov Chain Monte Carlo
The goal of inverse uncertainty quantification analysis is to determine the probability distribution of unknown model parameters. These parameters can be unknown for example because they cannot directly be measured. These parameters are known as the calibration parameters and they are estimated based on experimental data that relates to the parameters through a computational model. Inverse uncertainty quantification propagates the experimental data information backward to gain knowledge of the calibration parameters. Compare this to Uncertainty Propagation that propagates a known parameters distribution forward to gain knowledge of the QoIs.
Consider a computational model represented by y = M(x, θ). Here, x is the vector of experimental parameters, θ is the vector of calibration parameters, and y is the vector of QoIs. The experimental parameters x are independent variables with a known prescribed PDF p(x). The calibration parameters θ are parameters to be calibrated from the inverse uncertainty quantification study, the existing knowledge on the calibration parameters are used to prescribe their prior distribution p(θ). Note that the calibration parameters are considered as inputs to the computer model but are unknown when conducting the experiments. Experiment parameters are known for both the computational model and physical experiment, and they are used to describe the conditions under which the experiments have been conducted.
To connect the computational model and the experimental observation yE, a discrepancy model is introduced as,
where δ ∼ Ν(0, ε) represents an additive error and is assumed to be normally distributed with mean equals to 0 and an unknown diagonal covariance matrix ε = σ2I, where σ is inferred with the calibration parameters. The definition assumes that the instrumentation, or the computer model, has no systematic bias.
Bayesian inference theory is used to determine the posterior distribution of the calibration parameters which is defined as p(θ| y, yE), where y and yE represent the computational model and experimental data, respectively. According to the Bayesian theory, we have
where p(θ) is the prior distribution of the calibration parameter, and p(y, yE|θ) is the likelihood function. Here the prior and posterior distribution reflects the degree of belief on possible values of θ before and after considering the experimental data. The likelihood function based on all the experimental data is defined as
where N is the number of experimental data, m is the number of QoIs. Here, the experimental data is considered as independent samples conditioned on θ.
The computational model is a surrogate model constructed with the known distribution of the experimental parameters and the prior distribution of the calibration parameters. In practice, the posterior distribution defined by Bayesian theory do not have a closed-form solution. We rely on MCMC method to solve the inverse problem. The basic idea of MCMC is to build a Markov chain with an invariant distribution that equals to the posterior distribution. MCMC method produces chains of sample points that follows the posterior distribution. In practice, users need to make decision about the termination of the Markov chain based on the number of samples and the step size between two consecutive samples. To diminish the influence of the starting values, we discard the first half of each chain and only keep the second half of the chain. The discarded samples are considered as burn-in samples or warm-up samples. More details related to MCMC can be found in Ref. 14.
You can perform a uncertainty propagation study after the inverse uncertainty quantification by using the calibrated parameters to do the forward Monte Carlo analysis and study the calibrated QoIs. Note that inverse uncertainty quantification can also be used to calibrate the predicted QoIs to real world observations. By changing the status from input parameter with a known distribution, to a calibration parameters, we can use experimental observations to fine-tune these inputs and reduce the difference between the computational model and observations.