Generalized polynomial chaos based surrogate models for acoustics and vibrations

The Monte Carlo (MC) method is most exact but computationally expensive for uncertainty quantification (UQ). Instead, surrogate models can approximate UQ at a lower computational cost. However, as the input uncertainty dimensionality increases, the number of samples required to construct the surrogate models also increases rapidly; thus, the advantage of using surrogate models disappears. Therefore, it is necessary to combine the surrogate models with dimensionality reduction for effective UQ. This study compares the performance of UQ for the direct MC method and four different combinations of surrogate models and dimensionality reduction methods in benchmark test functions. The results clarify the effectiveness of training a surrogate model and a dimensionality reduction technique simultaneously rather than separately.

In the framework of process safety studies, consequence assessment of accidental scenarios is a crucial step affecting the eventual risk profile associated with the facilities under analysis. Conventional models used for consequence assessment are based on integral models, and may not be adequate to cope with the dynamic evolution of accidental scenarios and their three‐dimensional features. On the other hand, consequence assessment models based on computational fluid dynamics (CFD) approaches are promising to cope with complex scenarios and environments, but setting the simulation introduces relevant uncertainties associated with both the input data, assumptions, and with the modelling of physical effects involved. In the present study, uncertainty quantification (UQ) techniques are applied to support advanced safety studies based on CFD simulations of hazardous gas dispersion. Firstly, the accidental scenarios are characterized by defining release scenarios and conditions and quantifying source terms using integral models. At the same time, input meteorological data are gathered. This enables the development of high‐fidelity CFD simulations of gas dispersion based on different input sets and eventually the implementation of UQ techniques. The generalized polynomial chaos (gPC) expansion is employed to obtain hazardous gas concentration based on the variation of wind direction and speed. The present method is applied for the analysis of a real plant featuring a complex layout. The results show the advantages of the present approach by quantifying the influence of meteorological conditions and providing indications for supporting the development of protection systems and emergency measures.

Discretization of a random field by the generalized polynomial chaos (GPC) begins with selecting a specific type of orthogonal polynomial (e.g. Legendre and Hermite polynomials) (e.g. Ghanem and Spanos, 1991). This selection of a type of orthogonal polynomial can be performed based on the reported experiences (e.g. Xiu and Karniadakis, 2003) or data revealing the distribution of a random field to be discretized. If such data or reported experiences are unavailable, a third way may be generating some pilot tests to study the performance of a specific type of orthogonal polynomial in discretizing this random field. This study tries to develop an evolutionary algorithm-based auxiliary tool for the implementation of such pilot tests. A similar tool (Allaix and Carbone, 2009), which is based on the singleobjective evolutionary algorithm, had been developed for constructing the Karhunen-Loeve (KL) representation of a random field. Both KL and GPC expansions are two of the popular random field discretization methods (Ghanem and Spanos, 1991). But, the KL expansion should be applied under a prerequisite of knowing the covariance matrix of a random field to be discretized (e.g. Ghanem and Spanos, 1991); while, the GPC expansions can be applied without similar prerequisites (e.g. Xiu and Karniadakis, 2003). Therefore, the development of an auxiliary tool for constructing a GPC representation of a random field would be necessary. The succeeding research considers the derivation of an GPC representation of a random field as a multi-objective (MO) problem having two goals: (a) limiting the computational efforts spent in applying the resulting GPC representation; and (b) keeping the resulting GPC representation satisfying all accuracy standards (e.g. getting the sufficiently accurate prediction of statistical parameters of a random field). The former goal will be attained by limiting the highest order of polynomial term and total number of uncorrelated random variables used to construct a GPC representation; while the latter goal will be attained by minimizing multiple error estimators. Since there are multiple goals to be attained, a multiple objective evolutionary algorithm (MOEA) is required. Among all available MOEAs, the strength Pareto evolutionary algorithm II (SPEA 2) (Zitzler, et al., 2001) is chosen. The highest order of polynomial term and total number of uncorrelated random variables used to construct an GPC representation are considered as two parameters to be identified. The remainder of this study is organized into four sections. In Sec. 2, the theoretical backgrounds of GPC expansions (e.g. Xiu and Karniadakis, 2003) are briefly reviewed. In Sec. 3, the SPEA 2 (Zitzler, et al., 2001) is used to construct a parameter identification procedure to identify the aforementioned highest order of orthogonal polynomial and total

Uncertainty quantification in a resonant nonlinear MEMS structure

Uncertainty quantification has proven to be an indispensable study for enhancing reliability and robustness of engineering systems in the early design phase. Single and multi-fidelity surrogate modelling methods have been used to replace the expensive high fidelity analyses which must be repeated many times for uncertainty quantification. However, since the number of analyses required to build an accurate surrogate model increases exponentially with the number of random input variables, most surrogate modelling methods suffer from the curse of dimensionality. As an alternative approach, the Low-Rank Approximation method can be applied to high-dimensional uncertainty quantification studies with a low computational cost, where the number of coefficients for building the surrogate model increases only linearly with the number of random input variables. In this study, the Low-Rank Approximation method is implemented for multi-fidelity applications with additive and multiplicative correction approaches to make the high-dimensional uncertainty quantification analysis more efficient and accurate. The developed uncertainty quantification methodology is tested on supersonic aircraft design problems and its predictions are compared with the results of single- and multi-fidelity Polynomial Chaos Expansion and Monte Carlo methods. For the same computational cost, the Low-Rank Approximation method outperformed both in surrogate modeling and uncertainty quantification cases for all the benchmarks and real-world engineering problems addressed in the present study.

SENSITIVITY ANALYSIS OF IMAGE CLASSIFICATION MODELS USING GENERALIZED POLYNOMIAL CHAOS - 6th International Conference on Uncertainty Quantification in Computational Sciences and Engineering - 15-18 June 2025, Rhodes Island, Greece

Intrinsic and extrinsic techniques for quantification uncertainty of an interpretable GRU deep learning model used to predict atmospheric total suspended particulates (TSP) in Zabol, Iran during the dusty period of 120-days wind

Efficient Spectral-Chaos Methods for Uncertainty Quantification in Long-Time Response of Stochastic Dynamical Systems

The goal of this book was to introduce the reader to the recent advances in the field of uncertainty quantification and error propagation for computer vision, image processing, and image analysis that are based on partial differential equations (PDEs). Our original motivation was the observation that more and more often, measured data in the form of images are being used as part of the simulation pipeline, and that error propagation and uncertainty quantification—starting from images but then moving through the pipeline—are important to understanding the ending result of this scientific or engineering process. To analyze the image processing pipeline, we have presented concepts that enable error propagation to be formulated with a set of basic operations: the idea of stochastic images and corresponding (numerical PDE) operations on those images. From this perspective, one approach commonly used within the engineering literature is to employ Monte Carlo sampling techniques to attempt to quantify the impact of errors and variability on an engineering pipeline. Within the uncertainty quantification world, alternative approaches such as the class of generalized polynomial chaos (gPC) methods, which make assumptions about the ability to approximate the process of interest through polynomial approximations, have gained traction as a way of accelerating the convergence of the error and uncertainty quantification process in a computationally tractable way. In this book, we have shown the results of our exploration of the use of the gPC methodology for image processing and computer vision problems. We have relied on the fact that all the prerequisites of the gPC framework (e.g., finite variance, smoothness in the stochastic space, etc.) are intrinsically satisfied in the image processing context. In particular, we have shown how the generalized polynomial chaos (gPC) approach and its corresponding rules for computation (e.g., sums, products, projections on the gPC space, etc.) lead to straightforward generalizations to image processing and computer vision techniques. We have shown that the gPC methodology, when combined with image processing and computer vision techniques, provides a powerful approach to error propagation and uncertainty quantification, and to accessing system sensitivity.

High-dimensional uncertainty quantification for Mars atmospheric entry using adaptive generalized polynomial chaos

Uncertainty quantification (UQ) is the study of the effects of uncertainty on the values of analytical results and the predictions of scientific models. Sources of uncertainty include imprecise knowledge of the exact values of parameters, lack of confidence in the physical models, use of imperfectly calibrated models, and irreducible uncertainties due to physical characteristics. The Air Force Research Laboratory has undertaken the challenge of understanding, developing and analyzing the techniques of UQ as they apply to Laser Beam Control. This paper proposes a simple methodology and simple results with our first attempt of applying UQ as a new analysis tool. The software toolkit which was chosen was an analytical group of algorithms from a Sandia National Laboratory (SNL) package called DAKOTA (Design Analysis Kit for Optimization and Terascale Applications). The specific application of interest to the Air Force Research Laboratory (AFRL) is the analytical prediction of the performance of a Laser Beam Control systems under various scenarios, conditions, and missions. The application of rigorous UQ techniques to the models used to predict beam control performance could greatly improve our confidence in these predictions and also improve the acceptance of advanced Laser Beam Control systems within the science and engineering communities1,2. The proposed work would follow a multi-step approach, analyzing the more easily quantified sources of uncertainty, and then including increasingly complicated physical phenomena as the work progresses. Will present the initial results, and the first steps in the incorporation of UQ into our Laser Beam Control Modeling and Simulation environments.

As far as the preliminary thermal design of gas turbine components is concerned, 1-D codes are still widely used in standard industrial practice. Among the different components, the combustor is one of the most critical ones and its thermal design still greatly affects the reliability and life of the entire engine. During the initial phases of the design process, parameters are often roughly known. For this preliminary phase, a low-order approach is preferred instead of a high-fidelity simulation: the exploration of the whole space is extremely important to better understand the behavior of the system and to focus on the design objectives. Uncertainty quantification (UQ) methods, mainly developed in recent years and applied in many fields, are useful tools for the preliminary design phase and provide support during the whole design process. The objective of this work is to estimate the main sources of uncertainties in the design phase of an aeroengine effusion cooled combustor. The test case is based on a full annular lean-burn combustor, tested during the LEMCOTEC (Low EMissions COre-engine TEChnologies) European project. Among the test points investigated in the experimental campaign, the Approach condition is here analyzed. The inner liner is taken into consideration to investigate the metal temperature. Therm-1D, a 1-D in-house simulation code, is used to model the combustor and the open-source tool DAKOTA is adopted for the uncertainty quantification analysis. The baseline case of the combustor is studied and several uncertainty analyses are investigated. They are divided into 3 main groups: geometrical, tuning modelling parameters and thermal loads. For each group, the most relevant parameters are considered as a source of input uncertainty. In particular, a classical Monte Carlo approach is compared with four innovative polynomial-chaos approaches for each group: Gauss quadrature, total order with LHS sampling, stochastic collocation, and Smolyak. The analyses proved how the last two methods give the best results with a sensible lower amount of simulation (depending on the number of input variables). Lastly, results are compared with experimental data to achieve a better understanding of the most relevant input parameters and the propagation of their uncertainty on the results.

Surrogate SDOF models for probabilistic performance assessment of multistory buildings: Methodology and application for steel special moment frames

Uncertainty quantification of two-phase flow in porous media via the Coupled-TgNN surrogate model

Model and analysis of single event transient sensitivity based on uncertainty quantification