While the theoretical formulation of uncertainty quantification is straightforward, i.e., calculating the probability distribution of the quantity of our interest following some basic rules in probability theory, the actual computation is challenging in practice. Firstly, many realistic problems involve strongly nonlinear models, which lead to complex non-Gaussian posteriors. In such a situation, linearization-based methods (e.g., Kalman filter/smoother) may fail to capture the posterior, whereas fully nonlinear approaches (e.g., Monte Carlo/ sampling method) may be prohibitively expensive because the convergence of such approaches relies on large number of model evaluations. Another computationally challenging situation is when the model contains many uncertain parameters, since high dimensional parameter space is notoriously sparse and difficult to explore, which is the phenomenon known as the “curse of dimensionality”.

In our previous studies, new approaches based on statistical/machine learning techniques were developed to address the above-mentioned computational challenges. For example, we investigated adaptive construction and refinement of Gaussian process-based surrogates to approximate nonlinear models, which greatly facilitate the sampling process while keeping the computational cost (measured by the number of model evaluations) at a minimum level. Another example is the use of clustering algorithm and Gaussian mixture model to build proper proposal distributions that approximate complex posterior non-Gaussian distributions, and thus improve the efficiency of a sampling algorithm. To address the high-dimensionality issue, we noted that many high-dimensional models in fact conceal low-dimensional structures, i.e., the simulation of the quantity of our interest depends only on a small number of linear combinations of the large number of uncertain parameters. Based on this observation, we studied the inverse regression-based dimension reduction method, which seeks low-dimensional representations for originally high-dimensional models by exploiting such structures, and thus mitigates the curse of dimensionality. Numerical experiments showed that the new approaches could lead to significant saving of computational cost (often by orders of magnitude), and thus greatly enhance the practicality of probabilistic inverse modeling for complex systems.