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1. Modified Polynomial Chaos Expansion for Efficient Uncertainty Quantification in Biological Systems

   https://doi.org/10.3390/applmech1030011  

   Son, Jeongeun; Du, Dongping; Du, Yuncheng (September 2020, Applied Mechanics)

   null (Ed.)

   Uncertainty quantification (UQ) is an important part of mathematical modeling and simulations, which quantifies the impact of parametric uncertainty on model predictions. This paper presents an efficient approach for polynomial chaos expansion (PCE) based UQ method in biological systems. For PCE, the key step is the stochastic Galerkin (SG) projection, which yields a family of deterministic models of PCE coefficients to describe the original stochastic system. When dealing with systems that involve nonpolynomial terms and many uncertainties, the SG-based PCE is computationally prohibitive because it often involves high-dimensional integrals. To address this, a generalized dimension reduction method (gDRM) is coupled with quadrature rules to convert a high-dimensional integral in the SG into a few lower dimensional ones that can be rapidly solved. The performance of the algorithm is validated with two examples describing the dynamic behavior of cells. Compared to other UQ techniques (e.g., nonintrusive PCE), the results show the potential of the algorithm to tackle UQ in more complicated biological systems. 

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2. Quantification and propagation of Aleatoric uncertainties in topological structures

   https://doi.org/10.1016/j.ress.2023.109122  

   Wang, Zihan; Daeipour, Mohamad; Xu, Hongyi (May 2023, Reliability Engineering & System Safety)

   Quantification and propagation of aleatoric uncertainties distributed in complex topological structures remain a challenge. Existing uncertainty quantification and propagation approaches can only handle parametric uncertainties or high dimensional random quantities distributed in a simply connected spatial domain. There lacks a systematic method that captures the topological characteristics of the structural domain in uncertainty analysis. Therefore, this paper presents a new methodology that quantifies and propagates aleatoric uncertainties, such as the spatially varying local material properties and defects, distributed in a topological spatial domain. We propose a new random field-based uncertainty representation approach that captures the topological characteristics using the shortest interior path distance. Parameterization methods like PPCA and β-Variational Autoencoder (βVAE) are employed to convert the random field representation of uncertainty to a small set of independent random variables. Then non-intrusive uncertainties propagation methods such as polynomial chaos expansion and univariate dimension reduction are employed to propagate the parametric uncertainties to the output of the problem. The effectiveness of the proposed methodology is demonstrated by engineering case studies. The accuracy and computational efficiency of the proposed method is confirmed by comparing with the reference values of Monte Carlo simulations with a sufficiently large number of samples. 

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3. Human Uncertainty-Aware MPC for Enhanced Human-Robot Collaborative Manipulation

   https://doi.org/10.1109/ICPS59941.2024.10640020  

   Mahmud, Al Jaber; Nguyen, Duc M; Veiga, Filipe; Xiao, Xuesu; Wang, Xuan (May 2024, IEEE)

   This paper presents the development of a novel control algorithm designed for tasks involving human-robot collaboration. By using an 8-DOF robotic arm, our approach aims to counteract human-induced uncertainties added to the robot's nominal trajectory. To address this challenge, we incorporate a variable within the regular Model Predictive Control (MPC) framework to account for human uncertainties, which are modeled as following a normal distribution with a non-zero mean and variance. Our solution involves formulating and solving an uncertainty-aware Discrete Algebraic Ricatti Equation (ua-DARE), which yields the optimal control law for all joints to mitigate the impact of these uncertainties. We validate our methodology through theoretical analysis, demonstrating the effectiveness of the ua-DARE in providing an optimal control strategy. Our approach is further validated through simulation experiments using a Fetch robot model, where the results highlight a significant improvement in performance over a baseline algorithm that does not consider human uncertainty while solving for optimal control law. 

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4. Integrated Space Mission Planning Under Uncertainty via Stochastic and Decomposition-Based Optimization

   https://doi.org/10.2514/6.2024-4816  

   Isaji, Masafumi; Ho, Koki (July 2024, AIAA AVIATION FORUM AND ASCEND 2024)

   As we strive to establish a long-term presence in space, it is crucial to plan large-scale space missions and campaigns with future uncertainties in mind. However, integrated space mission planning, which simultaneously considers mission planning and spacecraft design, faces significant challenges when dealing with uncertainties; this problem is formulated as a stochastic mixed integer nonlinear program (MINLP), and solving it using the conventional method would be computationally prohibitive for realistic applications. Extending a deterministic decomposition method from our previous work, we propose a novel and computationally efficient approach for integrated space mission planning under uncertainty. The proposed method effectively combines the Alternating Direction Method of Multipliers (ADMM)-based decomposition framework from our previous work, robust optimization, and two-stage stochastic programming (TSSP).This hybrid approach first solves the integrated problem deterministically, assuming the worst scenario, to precompute the robust spacecraft design. Subsequently, the two-stage stochastic program is solved for mission planning, effectively transforming the problem into a more manageable mixed-integer linear program (MILP). This approach significantly reduces computational costs compared to the exact method, but may potentially miss solutions that the exact method might find. We examine this balance through a case study of staged infrastructure deployment on the lunar surface under future demand uncertainty. When comparing the proposed method with a fully coupled benchmark, the results indicate that our approach can achieve nearly identical objective values (no worse than 1% in solved problems) while drastically reducing computational costs. 

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5. A Stochastic Galerkin Method for the Boltzmann Equation with Multi-Dimensional Random Inputs Using Sparse Wavelet Bases

   https://doi.org/10.4208/nmtma.2017.s12  

   Shu, Ruiwen; Hu, Jingwei; Jin, Shi (May 2017, Numerical Mathematics: Theory, Methods and Applications)

   Abstract We propose a stochastic Galerkin method using sparse wavelet bases for the Boltzmann equation with multi-dimensional random inputs. Themethod uses locally supported piecewise polynomials as an orthonormal basis of the random space. By a sparse approach, only a moderate number of basis functions is required to achieve good accuracy in multi-dimensional random spaces. We discover a sparse structure of a set of basis-related coefficients, which allows us to accelerate the computation of the collision operator. Regularity of the solution of the Boltzmann equation in the random space and an accuracy result of the stochastic Galerkin method are proved in multi-dimensional cases. The efficiency of the method is illustrated by numerical examples with uncertainties from the initial data, boundary data and collision kernel. 

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