### Achievement

Uncertainty quantification (UQ) includes the characterization, integration, and propagation of uncertainties that result from stochastic variations and a lack of knowledge or data in the natural world. Monte Carlo (MC) method is a sampling-based approach that has widely used for quantification and propagation of uncertainties. However, the standard MC method is often time-consuming if the simulation-based model is computationally intensive. This article gives an overview of modern MC methods to address the existing challenges of the standard MC in the context of UQ. Specifically, multilevel Monte Carlo (MLMC) extending the concept of control variates achieves a significant reduction of the computational cost by performing most evaluations with low accuracy and corresponding low cost, and relatively few evaluations at high accuracy and correspondingly high cost. Multifidelity Monte Carlo (MFMC) accelerates the convergence of standard Monte Carlo by generalizing the control variates with different models having varying fidelities and varying computational costs. Multimodel Monte Carlo method (MMMC), having a different set of MLMC and MFMC, aims to address the issue of uncertainty quantification and propagation when data for characterizing probability distributions are limited. Multimodel inference combined with importance sampling is proposed for quantifying and efficiently propagating the uncertainties resulting from small datasets. All of these three modern MC methods achieve a significant improvement of computational efficiency for probabilistic UQ, particularly uncertainty propagation. An algorithm summary and the corresponding code implementation are provided for each of the modern Monte Carlo methods. The extension and application of these methods are discussed in detail.

**Publications:**

Jiaxin Zhang. "Modern Monte Carlo methods for efficient uncertainty quantification and propagation: A survey". * WIREs Computational Statistics*, 2020. arXiv:2011.00680

**Multimodel Monte Carlo Method (MMMC) Related Publications:**

Jiaxin Zhang, and Michael Shields. "On the quantification and efficient propagation of imprecise probabilities resulting from small datasets." ** Mechanical Systems and Signal Processing** 98 (2018): 465-483.

Jiaxin Zhang, and Michael Shields. "The effect of prior probabilities on quantification and propagation of imprecise probabilities resulting from small datasets." ** Computer Methods in Applied Mechanics and Engineering** 334 (2018): 483-506.

Jiaxin Zhang, and Michael Shields. "Efficient monte carlo resampling for probability measure changes from bayesian updating." ** Probabilistic Engineering Mechanics** 55 (2019): 54-66.

Jiaxin Zhang, Michael Shields, and Stephanie TerMaath. "Probabilistic modeling and prediction of out-of-plane unidirectional composite lamina properties." * Mechanics of Advanced Materials and Structures* (2020): 1-17.

Jiaxin Zhang, Stephanie TerMaath, and Michael Shields. "Imprecise global sensitivity analysis using bayesian multimodel inference and importance sampling." * Mechanical Systems and Signal Processing* 148 (2020): 107162.

Jiaxin Zhang and Michael Shields. "On the quantification and efficient propagation of imprecise probabilities with copula dependence."* International Journal of Approximate Reasoning*, 2020 (122):24-46.

### Significance and Impact

In many cases across computational science and engineering, uncertainty quantification is playing an increasingly important role in computationally evaluating the performance of complex mathematical, physical, and engineering systems. Typically, a computationally expensive high-fidelity model characterizes the system with high accuracy but high costs. Thus the standard Monte Carlo method is often very time-consuming because it relies on a large number of random samples (model evaluations) to estimate the statistical quantities of 29 response outputs. Several efficient Monte Carlo methods are therefore proposed to address the computational challenges. Multilevel Monte Carlo method (MLMC) utilizes control variates technique to reduce the computational cost by performing most simulations at a relatively low cost and only a few simulations at a high-cost. Similar to the MLMC method, the multifidelity Monte Carlo method (MFMC), as a variant of the control variates, aims to combine high-fidelity models and low-fidelity models to speed up the statistical estimation. In the context of imprecise probabilities, typically arising from small data issues, the multimodel Monte Carlo method (MMMC) is developed to quantify the uncertainties using multimodel inference, which combines the model-form and model parameter uncertainties, and then efficiently propagate an ensemble of probability models through the optimal importance sampling reweighting scheme. These efficient modern Monte Carlo methods can be employed to address many UQ challenges, not only for forward UQ problems, but also more general UQ related issues, e.g., optimization with uncertainty, robust design with uncertainty, and UQ in artificial intelligence and machine learning.

Last Updated: January 14, 2021 - 11:16 am