This paper presents an approach for efficient uncertainty analysis (UA) using an intrusive generalized polynomial chaos (gPC) expansion. The key step of the gPC‐based uncertainty quantification
- PAR ID:
- 10280793
- Date Published:
- Journal Name:
- Applied Mechanics
- Volume:
- 1
- Issue:
- 3
- ISSN:
- 2673-3161
- Page Range / eLocation ID:
- 153 to 173
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Summary ( UQ) is the stochastic Galerkin (SG) projection, which can convert a stochastic model into a set of coupled deterministic models. The SG projection generally yields a high‐dimensional integration problem with respect to the number of random variables used to describe the parametric uncertainties in a model. However, when the number of uncertainties is large and when the governing equation of the system is highly nonlinear, the SG approach‐based gPC can be challenging to derive explicit expressions for the gPC coefficients because of the low convergence in the SG projection. To tackle this challenge, we propose to use a bivariate dimension reduction method (BiDRM) in this work to approximate a high‐dimensional integral in SG projection with a few one‐ and two‐dimensional integrations. The efficiency of the proposed method is demonstrated with three different examples, including chemical reactions and cell signaling. As compared to other UA methods, such as the Monte Carlo simulations and nonintrusive stochastic collocation (SC), the proposed method shows its superior performance in terms of computational efficiency and UA accuracy. -
null (Ed.)Uncertainty is a common feature in first-principles models that are widely used in various engineering problems. Uncertainty quantification (UQ) has become an essential procedure to improve the accuracy and reliability of model predictions. Polynomial chaos expansion (PCE) has been used as an efficient approach for UQ by approximating uncertainty with orthogonal polynomial basis functions of standard distributions (e.g., normal) chosen from the Askey scheme. However, uncertainty in practice may not be represented well by standard distributions. In this case, the convergence rate and accuracy of the PCE-based UQ cannot be guaranteed. Further, when models involve non-polynomial forms, the PCE-based UQ can be computationally impractical in the presence of many parametric uncertainties. To address these issues, the Gram–Schmidt (GS) orthogonalization and generalized dimension reduction method (gDRM) are integrated with the PCE in this work to deal with many parametric uncertainties that follow arbitrary distributions. The performance of the proposed method is demonstrated with three benchmark cases including two chemical engineering problems in terms of UQ accuracy and computational efficiency by comparison with available algorithms (e.g., non-intrusive PCE).more » « less
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Quantifying the impact of parametric and model-form uncertainty on the predictions of stochastic models is a key challenge in many applications. Previous work has shown that the relative entropy rate is an effective tool for deriving path-space uncertainty quantification (UQ) bounds on ergodic averages. In this work we identify appropriate information-theoretic objects for a wider range of quantities of interest on path-space, such as hitting times and exponentially discounted observables, and develop the corresponding UQ bounds. In addition, our method yields tighter UQ bounds, even in cases where previous relative-entropy-based methods also apply, e.g. , for ergodic averages. We illustrate these results with examples from option pricing, non-reversible diffusion processes, stochastic control, semi-Markov queueing models, and expectations and distributions of hitting times.more » « less
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Abstract Uncertainty quantification (UQ) in metal additive manufacturing (AM) has attracted tremendous interest in order to dramatically improve product reliability. Model-based UQ, which relies on the validity of a computational model, has been widely explored as a potential substitute for the time-consuming and expensive UQ solely based on experiments. However, its adoption in the practical AM process requires overcoming two main challenges: (1) the inaccurate knowledge of uncertainty sources and (2) the intrinsic uncertainty associated with the computational model. Here, we propose a data-driven framework to tackle these two challenges by combining high throughput physical/surrogate model simulations and the AM-Bench experimental data from the National Institute of Standards and Technology (NIST). We first construct a surrogate model, based on high throughput physical simulations, for predicting the three-dimensional (3D) melt pool geometry and its uncertainty with respect to AM parameters and uncertainty sources. We then employ a sequential Bayesian calibration method to perform experimental parameter calibration and model correction to significantly improve the validity of the 3D melt pool surrogate model. The application of the calibrated melt pool model to UQ of the porosity level, an important quality factor, of AM parts, demonstrates its potential use in AM quality control. The proposed UQ framework can be generally applicable to different AM processes, representing a significant advance toward physics-based quality control of AM products.
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This paper is concerned with developing an efficient numerical algorithm for the fast implementation of the sparse grid method for computing the d-dimensional integral of a given function. The new algorithm, called the MDI-SG (multilevel dimension iteration sparse grid) method, implements the sparse grid method based on a dimension iteration/reduction procedure. It does not need to store the integration points, nor does it compute the function values independently at each integration point; instead, it reuses the computation for function evaluations as much as possible by performing the function evaluations at all integration points in a cluster and iteratively along coordinate directions. It is shown numerically that the computational complexity (in terms of CPU time) of the proposed MDI-SG method is of polynomial order O(d3Nb)(b≤2) or better, compared to the exponential order O(N(logN)d−1) for the standard sparse grid method, where N denotes the maximum number of integration points in each coordinate direction. As a result, the proposed MDI-SG method effectively circumvents the curse of dimensionality suffered by the standard sparse grid method for high-dimensional numerical integration.
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3390/applmech1030011
