Imprecise random field analysis for non-linear concrete damage analysis

Dannert, Mona M.; Faes, Matthias; Fleury, Rodolfo M.N.; Fau, Amélie; Nackenhorst, Udo; Moens, David

doi:10.1016/j.ymssp.2020.107343

Cited by 14 publications

(7 citation statements)

References 42 publications

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“…Nonprobabilistic uncertainty theories include fuzzy sets (Zadeh, 1965), interval methods (Weichselberger, 2000), convex models (Ben‐Haim & Elishakoff, 2013), and Dempster‐Schafer evidence theory (Dempster, 2008). Probabilistic approaches include probability boxes (p‐boxes) (Dannert et al, 2021; Ferson & Hajagos, 2004), Bayesian (Sankararaman & Mahadevan, 2013; Wei, Liu, Valdebenito, & Beer, 2021), random sets (Fetz & Oberguggenberger, 2004, 2016), and frequentist theories (Walley & Fine, 1982). Walley (1991, 2000)) developed a unified theory of imprecise probabilities, but there are still many methods to investigate the imprecision.…”

Section: Modern MC Methods For Uqmentioning

confidence: 99%

Modern Monte Carlo methods for efficient uncertainty quantification and propagation: A survey

Zhang

2020

WIREs Computational Stats

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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 corresponding 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 UQ 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 data sets. 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 MC methods. The extension and application of these methods are discussed in detail. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Monte Carlo Methods Statistical and Graphical Methods of Data Analysis > Sampling

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Section: Modern MC Methods For Uqmentioning

confidence: 99%

Modern Monte Carlo methods for efficient uncertainty quantification and propagation: A survey

Zhang

2020

WIREs Computational Stats

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“…Although all L i /l fulfil the same mean error (z) ≈ 1.3 %, the local error still varies significantly when T is small. Depending on the quantity of interest, this can lead to localisation effects [4].…”

Section: Influence Of the Truncation Ordermentioning

confidence: 99%

“…However, by introducing a second loop over the epistemic uncertainties the sampling process can become very expensive. Discretising the individual random fields for each correlation length by Karhunen-Loève (KL) expansion furthermore leads to high-dimensional problems, especially when small correlation lengths are involved [3,4].…”

Section: Introductionmentioning

confidence: 99%

“…For this purpose, the concept of imprecise random fields and the KL expansion as a method to discretise random fields are introduced in Section 2. It is important to ensure that the imprecise response is not affected by the local or global error arising from the truncation of the KL expansion [4]. Therefore, an imprecise random field input is carefully investigated in Section 3 in terms of the affect of the correlation length and truncation order.…”

Section: Introductionmentioning

confidence: 99%

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Limit Representations of Imprecise Random Fields

Dannert¹,

Häufler²,

Nackenhorst³

2021

4th International Conference on Uncertainty Quantification in Computational Sciences and Engineering

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In order to describe spatially uncertain parameters by random fields, the underlying autocorrelation structure in engineering structures is usually not known.. The idea of imprecise random fields is to acknowledge this lack of knowledge by adding epistemic uncertainties. Within this contribution the influence of the correlation length is studied. In particular, it is shown that there exist bounds that limit the case of having no idea at all. This "absolutely no idea p-box" is defined by white noise and the random variable corresponding to the mean value and standard deviation of the imprecise random field. By this, the limits of having "absolutely no idea" can be described without the need of Karhunen-Loève expansion and random field propagation. Then, at least for linear problems, every response in between can be estimated by linear interpolation without any need for sampling.

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“…However, interval finite element and fuzzy finite element methods, as they pertain to the analysis of spatially and time-variable uncertainty fields, are still areas of active research since they are accompanied by high computational complexities (Schietzold et al, 2019). For example, efficient and convenient constructions of the interval fields are still posing a challenge (Ni and Jiang, 2020) since they are typically constructed following some basis function expansion approach, such as Karhunen-Loève (KL)-like decompositions (Moens et al, 2011;Verhaeghe et al, 2013;Muscolino et al, 2013;Sofi and Muscolino, 2015;Ni and Jiang, 2020;Dannert et al, 2021). However these approaches are intrinsically dependent on knowledge of the spatial correlation fields of the input, which might be assumed but are in general (without experimental data) not known (Dannert et al, 2018).…”

mentioning

confidence: 99%

Interval and fuzzy physics-informed neural networks for uncertain fields

Fuhg¹,

Kalogeris²,

Fau³

et al. 2021

Preprint

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Temporally and spatially dependent uncertain parameters are regularly encountered in engineering applications. Commonly these uncertainties are accounted for using random fields and processes which require knowledge about the appearing probability distributions functions which is not readily available. In these cases non-probabilistic approaches such as interval analysis and fuzzy set theory are helpful uncertainty measures. Partial differential equations involving fuzzy and interval fields are traditionally solved using the finite element method where the input fields are sampled using some basis function expansion methods. This approach however is problematic, as it is reliant on knowledge about the spatial correlation fields. In this work we utilize physics-informed neural networks (PINNs) to solve interval and fuzzy partial differential equations. The resulting network structures termed interval physics-informed neural networks (iPINNs) and fuzzy physics-informed neural networks (fPINNs) show promising results for obtaining bounded solutions of equations involving spatially uncertain parameter fields. In contrast to finite element approaches, no correlation length specification of the input fields as well as no averaging via Monte-Carlo simulations are necessary. In fact, information about the input interval fields is obtained directly as a byproduct of the presented solution scheme. Furthermore, all major advantages of PINNs are retained, i.e. meshfree nature of the scheme, and ease of inverse problem set-up.

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Imprecise random field analysis for non-linear concrete damage analysis

Cited by 14 publications

References 42 publications

Modern Monte Carlo methods for efficient uncertainty quantification and propagation: A survey

Modern Monte Carlo methods for efficient uncertainty quantification and propagation: A survey

Limit Representations of Imprecise Random Fields

Interval and fuzzy physics-informed neural networks for uncertain fields

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