# Modern Monte Carlo Methods for Efficient Uncertainty Quantification and Propagation: A Survey

arXiv: https://arxiv.org/abs/2011.00680 · HTML: https://mlsanity.com/paper/2011.00680/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Nov 2020.

## Built on by (most cited first, 21 of 21)

- [Agentic Uncertainty Quantification](https://mlsanity.com/paper/2601.15703/) (2026)
- [Monotonicity and Double Descent in Uncertainty Estimation with Gaussian Processes](https://mlsanity.com/paper/2210.07612/) (2022)
- [Active Learning with Multifidelity Modeling for Efficient Rare Event Simulation](https://mlsanity.com/paper/2106.13790/) (2021)
- [A PAC-Bayesian Perspective on the Interpolating Information Criterion](https://mlsanity.com/paper/2311.07013/) (2023)
- [Interactive Multi-fidelity Learning for Cost-effective Adaptation of Language Model with Sparse Human Supervision](https://mlsanity.com/paper/2310.20153/) (2023)
- [Comparing Uncertainty Measurement and Mitigation Methods for Large Language Models: A Systematic Review](https://mlsanity.com/paper/2504.18346/) (2025)
- [Accelerating Inverse Learning via Intelligent Localization with Exploratory Sampling](https://mlsanity.com/paper/2212.01016/) (2022)
- [Discriminant Distance-Aware Representation on Deterministic Uncertainty Quantification Methods](https://mlsanity.com/paper/2402.12664/) (2024)
- [CogDPM: Diffusion Probabilistic Models via Cognitive Predictive Coding](https://mlsanity.com/paper/2405.02384/) (2024)
- [Quantile Regression using Random Forest Proximities](https://mlsanity.com/paper/2408.02355/) (2024)
- [From Passive Metric to Active Signal: The Evolving Role of Uncertainty Quantification in Large Language Models](https://mlsanity.com/paper/2601.15690/) (2026)
- [Concerning Uncertainty -- A Systematic Survey of Uncertainty-Aware XAI](https://mlsanity.com/paper/2603.26838/) (2026)
- [On the Quantification of Image Reconstruction Uncertainty without Training Data](https://mlsanity.com/paper/2311.09639/) (2023)
- [Physics-informed RL for Maximal Safety Probability Estimation](https://mlsanity.com/paper/2403.16391/) (2024)
- [Exploring Efficient Quantification of Modeling Uncertainties with Differentiable Physics-Informed Machine Learning Architectures](https://mlsanity.com/paper/2506.18247/) (2025)
- [Embedding Trust: Semantic Isotropy Predicts Nonfactuality in Long-Form Text Generation](https://mlsanity.com/paper/2510.21891/) (2025)
- [A personalized Uncertainty Quantification framework for patient survival models: estimating individual uncertainty of patients with metastatic brain tumors in the absence of ground truth](https://mlsanity.com/paper/2311.17173/) (2023)
- [UQ of 2D Slab Burner DNS: Surrogates, Uncertainty Propagation, and Parameter Calibration](https://mlsanity.com/paper/2411.16693/) (2024)
- [Spline Dimensional Decomposition with Interpolation-based Optimal Knot Selection for Stochastic Dynamic Analysis](https://mlsanity.com/paper/2505.12879/) (2025)
- [Quantum Learning and Estimation for Coordinated Operation between Distribution Networks and Energy Communities](https://mlsanity.com/paper/2506.11730/) (2025)
- [Conformalized Quantum DeepONet Ensembles: Towards Scalable Operator Learning with Distribution-Free Guarantees](https://mlsanity.com/paper/2605.00330/) (2026)

## Abstract (arXiv)

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 setting 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.