A bi-fidelity surrogate modeling approach for uncertainty propagation in three-dimensional hemodynamic simulations

Gao, Han; Zhu, Xueyu; Wang, Jianxun

doi:10.1016/j.cma.2020.113047

Cited by 25 publications

(22 citation statements)

References 98 publications

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“…The application of Bi-Fidelity techniques to various problems is an active area of research [26,27]. In the setting of uncertainty quantification for PDE models, it is frequently described in the context of uncertainty quantification via Stochastic Collocation methods; see, e.g., [28,29] for the general procedure or [30][31][32][33] for applications. The combination with the stochastic Galerkin method works similarly; however, it is not very common in literature.…”

Section: A Bi-fidelity Approach For Calculating the Stochastic Galerk...mentioning

confidence: 99%

Computing Black Scholes with Uncertain Volatility—A Machine Learning Approach

Hellmuth

Klingenberg

2022

Mathematics

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In financial mathematics, it is a typical approach to approximate financial markets operating in discrete time by continuous-time models such as the Black Scholes model. Fitting this model gives rise to difficulties due to the discrete nature of market data. We thus model the pricing process of financial derivatives by the Black Scholes equation, where the volatility is a function of a finite number of random variables. This reflects an influence of uncertain factors when determining volatility. The aim is to quantify the effect of this uncertainty when computing the price of derivatives. Our underlying method is the generalized Polynomial Chaos (gPC) method in order to numerically compute the uncertainty of the solution by the stochastic Galerkin approach and a finite difference method. We present an efficient numerical variation of this method, which is based on a machine learning technique, the so-called Bi-Fidelity approach. This is illustrated with numerical examples.

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Section: A Bi-fidelity Approach For Calculating the Stochastic Galerk...mentioning

confidence: 99%

Computing Black Scholes with Uncertain Volatility—A Machine Learning Approach

Hellmuth

Klingenberg

2022

Mathematics

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“…Finally, τ S , τ I , τ R represent the relaxation frequencies playing the role of the Knudsen number ε introduced in Section 2.3 when discussing the linear transport equation in the diffusive scaling. In this model, the reproduction number for the system (32) is given by…”

Section: 2mentioning

confidence: 99%

“…In realistic applications, a priori assessment of the model quality and prediction errors is very important. In this direction, a previous study [32] introduced a novel empirical error bound estimation approach with ease of implementation to evaluate the performance of the bi-fidelity surrogates a priori. We will briefly describe the methodology here.…”

Section: 1mentioning

confidence: 99%

“…We will briefly describe the methodology here. In [32], two important quantities have been identified which are useful to build the approximation quality indicator for the BFSC approximation:…”

Section: 1mentioning

confidence: 99%

“…However, the choice of the samples for the high fidelity solver remains arbitrary. The second methodology, is based on a simpler bi-fidelity setting where a single low-fidelity model is used to effectively inform the selection of representative points in the parameter space and then employ this information to construct accurate approximations to high-fidelity solutions [32,55,63,[105][106][107]. As a result, it does not necessarily require that the low-fidelity and high-fidelity to reside in the same physical space.…”

Section: Introductionmentioning

confidence: 99%

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Multi-fidelity methods for uncertainty propagation in kinetic equations

Dimarco¹,

Liu²,

Pareschi³

et al. 2021

Preprint

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The construction of efficient methods for uncertainty quantification in kinetic equations represents a challenge due to the high dimensionality of the models: often the computational costs involved become prohibitive. On the other hand, precisely because of the curse of dimensionality, the construction of simplified models capable of providing approximate solutions at a computationally reduced cost has always represented one of the main research strands in the field of kinetic equations. Approximations based on suitable closures of the moment equations or on simplified collisional models have been studied by many authors. In the context of uncertainty quantification, it is therefore natural to take advantage of such models in a multi-fidelity setting where the original kinetic equation represents the high-fidelity model, and the simplified models define the low-fidelity surrogate models. The scope of this article is to survey some recent results about multi-fidelity methods for kinetic equations that are able to accelerate the solution of the uncertainty quantification process by combining high-fidelity and low-fidelity model evaluations with particular attention to the case of compressible and incompressible hydrodynamic limits. We will focus essentially on two classes of strategies: multifidelity control variates methods and bi-fidelity stochastic collocation methods. The various approaches considered are analyzed in light of the different surrogate models used and the different numerical techniques adopted. Given the relevance of the specific choice of the surrogate model, an application-oriented approach has been chosen in the presentation.

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A bi‐fidelity inverse analysis method for deep excavations considering three‐dimensional effects

Tao,

Pan,

Sun

et al. 2024

Num Anal Meth Geomechanics

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Inverse analysis methods are commonly used in braced excavations for improved deformation predictions. This paper proposes a bi‐fidelity ensemble randomized maximum likelihood (BF‐EnRML) method for efficient inverse analyses of deep excavations considering the three‐dimensional effects. The bi‐fidelity (BF) model is developed by the low‐fidelity model (i.e., two‐dimensional finite element model, 2D FEM) and the high‐fidelity model (i.e., 3D FEM) for a balance between efficiency and accuracy. A large number of 2D FEMs are first used to explore the relationship between soil parameters and wall deflections. A few 3D FEMs are then performed to calibrate the discrepancy between 2D‐3D deflections caused by the inability of 2D FEM to consider the three‐dimensional effects. The constructed BF model serves as the forward model in inverse analysis. The soil parameters are updated by incorporating the monitoring data based on EnRML and further used to predict wall deflections in later stages. A hypothetical excavation and a real project are studied to evaluate the performance of the proposed method. The results show that the BF model can provide wall deflection predictions close to those calculated from 3D FEM while using a computational cost of 2D FEM. The BF‐EnRML method can efficiently update the soil parameters and improve the wall deflection predictions. Moreover, factors affecting the accuracy of the BF model are studied, including the number of required 3D FEMs, the distance from the evaluated wall section to the excavation corner, the number of data points along the wall depth, and the number of excavation stages.

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A bi-fidelity surrogate modeling approach for uncertainty propagation in three-dimensional hemodynamic simulations

Cited by 25 publications

References 98 publications

Computing Black Scholes with Uncertain Volatility—A Machine Learning Approach

Computing Black Scholes with Uncertain Volatility—A Machine Learning Approach

Multi-fidelity methods for uncertainty propagation in kinetic equations

A bi‐fidelity inverse analysis method for deep excavations considering three‐dimensional effects

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