Multifidelity Uncertainty Quantification Research Articles

Deep learning enhanced cost-aware multi-fidelity uncertainty quantification of a computational model for radiotherapy

Deep learning enhanced cost-aware multi-fidelity uncertainty quantification of a computational model for radiotherapy

Multi-fidelity Uncertainty Quantification for Homogenization Problems in Structure-Property Relationships from Crystal Plasticity Finite Elements

Multi-fidelity Uncertainty Quantification for Homogenization Problems in Structure-Property Relationships from Crystal Plasticity Finite Elements

PyApprox: A software package for sensitivity analysis, Bayesian inference, optimal experimental design, and multi-fidelity uncertainty quantification and surrogate modeling

PyApprox is a Python-based one-stop-shop for probabilistic analysis of numerical models such as those used in the earth, environmental and engineering sciences. Easy to use and extendable tools are provided for constructing surrogates, sensitivity analysis, Bayesian inference, experimental design, and forward uncertainty quantification. The algorithms implemented represent a wide range of methods for model analysis developed over the past two decades, including recent advances in multi-fidelity approaches that use multiple model discretizations and/or simplified physics to significantly reduce the computational cost of various types of analyses. An extensive set of Benchmarks from the literature is also provided to facilitate the easy comparison of new or existing algorithms for a wide range of model analyses. This paper introduces PyApprox and its various features, and presents results demonstrating the utility of PyApprox on a benchmark problem modeling the advection of a tracer in groundwater.

Multifidelity uncertainty quantification with models based on dissimilar parameters

Multifidelity uncertainty quantification (MF UQ) sampling approaches have been shown to significantly reduce the variance of statistical estimators while preserving the bias of the highest-fidelity model, provided that the low-fidelity models are well correlated. However, maintaining a high level of correlation can be challenging, especially when models depend on different input uncertain parameters, which drastically reduces the correlation. Existing MF UQ approaches do not adequately address this issue. In this work, we propose a new sampling strategy that exploits a shared space to improve the correlation among models with dissimilar parameterization. We achieve this by transforming the original coordinates onto an auxiliary manifold using the adaptive basis (AB) method (Tipireddy and Ghanem, 2014). The AB method has two main benefits: (1) it provides an effective tool to identify the low-dimensional manifold on which each model can be represented, and (2) it enables easy transformation of polynomial chaos representations from high- to low-dimensional spaces. This latter feature is used to identify a shared manifold among models without requiring additional evaluations. We present two algorithmic flavors of the new estimator to cover different analysis scenarios, including those with legacy and non-legacy high-fidelity (HF) data. We provide numerical results for analytical examples, a direct field acoustic test, and a finite element model of a nuclear fuel assembly. For all examples, we compare the proposed strategy against both single-fidelity and MF estimators based on the original model parameterization.

Context-aware learning of hierarchies of low-fidelity models for multi-fidelity uncertainty quantification

Multi-fidelity Monte Carlo methods leverage low-fidelity and surrogate models for variance reduction to make tractable uncertainty quantification even when numerically simulating the physical systems of interest with high-fidelity models is computationally expensive. This work proposes a context-aware multi-fidelity Monte Carlo method that optimally balances the costs of training low-fidelity models with the costs of Monte Carlo sampling. It generalizes the previously developed context-aware bi-fidelity Monte Carlo method to hierarchies of multiple models and to more general types of low-fidelity models. When training low-fidelity models, the proposed approach takes into account the context in which the learned low-fidelity models will be used, namely for variance reduction in Monte Carlo estimation, which allows it to find optimal trade-offs between training and sampling to minimize upper bounds of the mean-squared errors of the estimators for given computational budgets. This is in stark contrast to traditional surrogate modeling and model reduction techniques that construct low-fidelity models with the primary goal of approximating well the high-fidelity model outputs and typically ignore the context in which the learned models will be used in upstream tasks. The proposed context-aware multi-fidelity Monte Carlo method applies to hierarchies of a wide range of types of low-fidelity models such as sparse-grid and deep-network models. Numerical experiments with the gyrokinetic simulation code Gene show speedups of up to two orders of magnitude compared to standard estimators when quantifying uncertainties in small-scale fluctuations in confined plasma in fusion reactors. This corresponds to a runtime reduction from 72 days to four hours on one node of the Lonestar6 supercomputer at the Texas Advanced Computing Center.

CONTROL VARIATE POLYNOMIAL CHAOS: OPTIMAL FUSION OF SAMPLING AND SURROGATES FOR MULTIFIDELITY UNCERTAINTY QUANTIFICATION

We present a multifidelity uncertainty quantification numerical method that leverages the benefits of both sampling and surrogate modeling, while mitigating their downsides, for enabling rapid computation in complex dynamical systems such as automotive propulsion systems. In particular, the proposed method utilizes intrusive generalized polynomial chaos to quickly generate additional information that is highly correlated with the original nonlinear dynamical system. We then leverage a Monte Carlo-based control variate to correct the bias caused by the surrogate approximation. In contrast to related works merging adaptive surrogate approximation and sampling in a multifidelity setting, the intrusive generalized polynomial chaos (gPC) surrogate is selected because it avoids statistical errors by design by providing analytical estimates of output statistics. Moreover, it enables theoretical contributions that provide an estimator design strategy that optimally balances the computational efforts allocated to sampling and to gPC construction. We deploy our approach to multiple numerical examples including simulations of hybrid-electric propulsion systems, where the proposed estimator is shown to achieve orders-of-magnitude reduction in mean squared error of statistics estimation under comparable costs of purely sampling or purely surrogate approaches.

Implementation of Multidisciplinary and Multifidelity Uncertainty Quantification Methods for Sonic Boom Prediction

To advance a supersonic aircraft design process, an uncertainty quantification study is conducted for sonic boom prediction while considering uncertainties associated with flight and atmospheric conditions. The uncertainty quantification process is implemented within a multidisciplinary analyses framework and assisted with a multifidelity surrogate model based approach. The sonic boom prediction framework requires input from the flowfield pressure distribution solution to generate the near-field pressure signature of the aircraft, which is then propagated throughout the atmosphere to the ground by using aeroacoustic methods. The open-source SU2 suite is employed as a high-fidelity flow solver tool to obtain the aerodynamic solution, while in-house postprocessing scripts are developed to generate the required near-field pressure signature. For low-fidelity flow analysis, A502 PAN AIR, a higher-order panel code which solves flows around slender bodies in low angles of attack for subsonic and supersonic regimes, is used. For nonlinear aeroacoustic propagation, NASA Langley Research Center’s code sBOOM exploits the near-field pressure signature for both high-fidelity and low-fidelity sonic boom calculations. Efficient uncertainty quantification tools are developed in house by implementing multifidelity polynomial chaos expansion and multifidelity Monte Carlo methods. Several flight and atmospheric parameters are selected to include randomness where these uncertainties are propagated into the sonic boom loudness prediction of the JAXA wing-body model, which is a low boom aircraft. Finally, an overall assessment of the multifidelity uncertainty quantification methods is presented in terms of efficiency and numerical accuracy.

Ensemble Approximate Control Variate Estimators: Applications to MultiFidelity Importance Sampling

The recent growth in multifidelity uncertainty quantification has given rise to a large set of variance reduction techniques that leverage information from model ensembles to provide variance reduction for estimates of the statistics of a high-fidelity model. In this paper we provide two contributions: (1) we utilize an ensemble estimator to account for uncertainties in the optimal weights of approximate control variate (ACV) approaches and derive lower bounds on the number of samples required to guarantee variance reduction; and (2) we extend an existing multifidelity importance sampling (MFIS) scheme to leverage control variates. Our approach directly addresses a limitation of many multifidelity sampling strategies that require the usage of pilot samples to estimate covariances. As such we make significant progress towards both increasing the practicality of approximate control variates—for instance, by accounting for the effect of pilot samples—and using multifidelity approaches more effectively for estimating low-probability events. The numerical results indicate our hybrid MFIS-ACV estimator achieves up to 50% improvement in variance reduction over the existing state-of-the-art MFIS estimator, which had already shown an outstanding convergence rate compared to the Monte Carlo method, on several problems of computational mechanics.

A generalized probabilistic learning approach for multi-fidelity uncertainty quantification in complex physical simulations

Two of the most significant challenges in uncertainty quantification pertain to the high computational cost for simulating complex physical models and the high dimension of the random inputs. In applications of practical interest, both of these problems are encountered, and standard methods either fail or are not feasible. To overcome the current limitations, we present a generalized formulation of a Bayesian multi-fidelity Monte-Carlo (BMFMC) framework that can exploit lower-fidelity model versions in a small data regime. The goal of our analysis is an efficient and accurate estimation of the complete probabilistic response for high-fidelity models. BMFMC circumvents the curse of dimensionality by learning the relationship between the outputs of a reference high-fidelity model and potentially several lower-fidelity models. While the continuous formulation is mathematically exact and independent of the low-fidelity model’s accuracy, we address challenges associated with the small data regime (i.e., only a small number of 50 to 300 high-fidelity model runs can be performed). Specifically, we complement the formulation with a set of informative input features at no extra cost. Despite the inaccurate and noisy information that some low-fidelity models provide, we demonstrate that accurate and certifiable estimates for the quantities of interest can be obtained for uncertainty quantification problems in high stochastic dimensions, with significantly fewer high-fidelity model runs than state-of-the-art methods for uncertainty quantification. We illustrate our approach by applying it to challenging numerical examples such as Navier–Stokes flow simulations and fluid–structure interaction problems.

Multi-fidelity uncertainty quantification of particle deposition in turbulent pipe flow

Particle deposition in fully-developed turbulent pipe flow is quantified taking into account uncertainty in electric charge, van der Waals strength, and temperature effects. A framework is presented for obtaining variance-based sensitivity in multiphase flow systems via a multi-fidelity Monte Carlo approach that optimally manages model evaluations for a given computational budget. The approach combines a high-fidelity model based on direct numerical simulation and a lower-order model based on a one-dimensional Eulerian description of the two-phase flow. Significant speedup is obtained compared to classical Monte Carlo estimation. Deposition is found to be most sensitive to electrostatic interactions and exhibits largest uncertainty for mid-sized (i.e., moderate Stokes number) particles.

Multifidelity uncertainty quantification and model validation of large-scale multidisciplinary systems

A simulation-based framework for multifidelity uncertainty quantification is presented, which informs and guides the design process of complex, large-scale, multidisciplinary systems throughout their life cycle. In this framework, uncertainty in system models is identified, characterized, and propagated in an integrated manner through the analysis cycles needed to quantify the effects of uncertainty on the quantities of interest. This is part of the process to design systems and verify their compliance to performance requirements. Uncertainty quantification is performed through mean and variance estimators as well as global sensitivity analyses. These computational analyses are made tractable by the use of multifidelity methods, which leverage a variety of low-fidelity models to obtain speed-ups, while keeping the main high-fidelity model in the loop to guarantee convergence to the correct result. This framework was applied to the James Webb Space Telescope observatory integrated model used to calculate the wavefront error caused by thermal distortions. The framework proved to reduce the time required to perform global sensitivity analyses from more than 2 months to less than 2 days, while reducing the error in the final estimates of the quantities of interest, including model uncertainty factors. These technical performance improvements are crucial to the optimization of project resources such as schedule and budget and ultimately mission success.

A Surrogate-Based Variance Reduction Approach to Multifidelity Uncertainty Quantification -- With Applications in Automotive Systems

A Surrogate-Based Variance Reduction Approach to Multifidelity Uncertainty Quantification -- With Applications in Automotive Systems

Multi-Fidelity Uncertainty Quantification of Particle Deposition in Turbulent Pipe Flow

Multi-Fidelity Uncertainty Quantification of Particle Deposition in Turbulent Pipe Flow

Co-kriging based multi-fidelity uncertainty quantification of beam vibration using coarse and fine finite element meshes

Multi-fidelity models have exploded in popularity as they promise to circumvent the computational complexity of a high-fidelity model without sacrificing accuracy. In this paper, we demonstrate the process of building a multi-fidelity model and illustrate its advantage through an uncertainty quantification study using the beam vibration problem. A multi-fidelity co-kriging model is built with data from low- and high-fidelity models, which are finite element models with coarse and fine discretization, respectively. The co-kriging model’s predictive capabilities are excellent, achieving accuracy within 1% of the high-fidelity model while providing 98% computational savings over the high-fidelity model in the uncertainty quantification study.

Multi-fidelity surrogate model-assisted fatigue analysis of welded joints

In this study, Kriging based multi-fidelity (MF) surrogate models are constructed to accelerate the fatigue analysis of welded joints. The influence of leg length, leg height, the width of the specimen, and load in the fatigue test are taken into consideration. In the construction of the MF surrogate model, the finite element model that is calibrated with the experiment is chosen as the high-fidelity (HF) model, while the finite element model that is not calibrated with the experiment is considered as the low-fidelity (LF) model, aiming to capture the trend of the HF model. The Leave-one-out (LOO) verification method is used to evaluate the benefits of the three types of Kriging-based MF surrogate models comparing to the single-fidelity one. The results show that the accuracy improvement of MF surrogate models compared with the HF Kriging surrogate model is between 49.17 and 79.92%, while it is between 53.4 and 87.99% compared with the LF Kriging surrogate model. To determine the most suitable MF surrogate models for different responses of the welded single lap joint, three different MF uncertainty quantification (UQ) metrics are used to evaluate the prediction errors of the MF surrogate models. Based on the results of the UQ, a comprehensive ranking for the MF surrogate models is provided by introducing the entropy weighting-based technique for order preference by similarity to ideal solution (EW-TOPSIS). The developed methods can also be generalized to the selection of the MF surrogate model for other engineering applications.

Multi-fidelity uncertainty quantification of film cooling flow under random operational and geometrical conditions

In this paper, the effects of operational and geometrical uncertainties on flow and thermal characteristics of film cooling are presented and discussed. The computational cost of Uncertainty Quantification (UQ) using classical Polynomial Chaos Expansion (PCE) increments exponentially with increasing the number of uncertain variables. To tackle this deficiency, UQ analysis is performed via the Multi-Fidelity (MF) PCE method. The MF PCE method is first validated using the analytical test functions and 2D film cooling flow and then applied to the turbulent flow and heat transfer of film cooling jet in cross-flow. Combination of three operational (U∞, M and DR) and four geometrical uncertain variables (L/D, W/D, α and β) are considered in the film cooling. Results demonstrated that the proposed MF PCE method reduces the computational cost by more than 50% in comparison to the classical PCE. Results of sensitivity analysis show that the uncertainty in mainstream velocity, U∞ has a negligible effect (less than 0.5%) on the film cooling adiabatic effectiveness, η. In the vicinity of coolant injection, the nondimensional domain width (W/D) mostly influences (by 39%) the cooling effectiveness. While, the remaining of random parameters have more or less similar effects. In far downstream of the injection hole, the nondimensional width (W/D) and blowing ratio (M) are the most influential parameters on the film cooling effectiveness (with a total contribution of 77%).

Multifidelity Uncertainty Quantification of a Commercial Supersonic Transport

The objective of this work was to develop a multifidelity uncertainty quantification approach for efficient analysis of a commercial supersonic transport concept. An approach based on point-collocation, nonintrusive polynomial chaos was formulated in which a low-fidelity model could be corrected using multiple higher-fidelity models. The formulation and methodology also allows for the addition of uncertainty sources not present in the lower-fidelity models. To demonstrate the applicability and potential computational savings of the multifidelity polynomial chaos approach, two model problems were explored. The first was a supersonic airfoil with three levels of modeling fidelity, each capturing a gradual increase in modeling of the underlying flow physics. As much as 50% computational cost reduction was observed using the multifidelity approach, while predicting nearly the same amount of uncertainty in drag. The second problem was a commercial supersonic transport. This model had three levels of fidelity that included two different modeling approaches and the addition of physics between the fidelity levels. Results of this analysis yielded nearly a 70% computational savings to predict a comparable amount of uncertainty in ground noise. Both problems illustrate the applicability and significant computational savings of the multifidelity method for efficient and accurate uncertainty quantification.

A generalized approximate control variate framework for multifidelity uncertainty quantification

We describe and analyze a variance reduction approach for Monte Carlo (MC) sampling that accelerates the estimation of statistics of computationally expensive simulation models using an ensemble of models with lower cost. These lower cost models — which are typically lower fidelity with unknown statistics — are used to reduce the variance in statistical estimators relative to a MC estimator with equivalent cost. We derive the conditions under which our proposed approximate control variate framework recovers existing multifidelity variance reduction schemes as special cases. We demonstrate that existing recursive/nested strategies are suboptimal because they use the additional low-fidelity models only to efficiently estimate the unknown mean of the first low-fidelity model. As a result, they cannot achieve variance reduction beyond that of a control variate estimator that uses a single low-fidelity model with known mean. However, there often exists about an order-of-magnitude gap between the maximum achievable variance reduction using all low-fidelity models and that achieved by a single low-fidelity model with known mean. We show that our proposed approach can exploit this gap to achieve greater variance reduction by using non-recursive sampling schemes. The proposed strategy reduces the total cost of accurately estimating statistics, especially in cases where only low-fidelity simulation models are accessible for additional evaluations. Several analytic examples and an example with a hyperbolic PDE describing elastic wave propagation in heterogeneous media are used to illustrate the main features of the methodology.

Multi-fidelity uncertainty quantification method with application to nonlinear structural response analysis

• A multi-fidelity method is proposed for statistical moment calculations of nonlinear structural responses. • The proposed formulation requires only small numbers of high fidelity solution and simple statistical techniques. • High-cost material damage analysis can be limited in local areas without biasing results of uncertainty quantification . • A computational cost reduction of 52.7% has been achieved in the numerical example. The application of uncertainty quantification (UQ) in complex structural response analysis is limited by solution efficiency. A multi-fidelity (MF) method for UQ is proposed in this paper, in which statistical moments are first evaluated using low cost low-fidelity (LF) model first, and then calibrated with a small number of high-fidelity (HF) samples. Only the error distribution of LF solutions and the covariance between the errors and the LF solutions are employed to derive a simple and straight forward MF formulation. The proposed method is demonstrated in the UQ of damage analysis of a C/SiC plate with a hole, where the HF model is a nonlinear global model considering C/SiC material damage, and the LF model is a linear global model driven nonlinear sub model. Uncertainty propagation is carried out using a sparse polynomial chaos expansion method. Evaluations of the MF method based on four factors: correctness, efficiency, precision and reliability, are carried out. The results show that the MF method can estimate the statistical moments of nonlinear strain responses unbiasedly. Computational cost is reduced by 52.7% compared to that utilizing HF model alone. MF methods can reduce computational cost significantly while maintaining accuracy and can be used for wide range of applications.

Multifidelity approaches for uncertainty quantification

The aim of this paper is to give an overview of different multifidelity uncertainty quantification (UQ) schemes. Therefore, different views on multifidelity UQ approaches from a frequentist, Bayesian, and possibilistic perspective are provided and recent developments are discussed. Differences as well as similarities between the methods are highlighted and strategies to construct low‐fidelity models are explained. In addition, two state‐of‐the‐art examples to showcase the capabilities of these methods and the tremendous reduction of computational costs that can be achieved when using these approaches are provided.