Epistemic Uncertainty Quantification in Scientific Models

In the field of uncertainty quantification (UQ), epistemic uncertainty often refers to the kind of uncertainty whose complete probabilistic description is not available, largely due to our lack of knowledge about the uncertainty. Quantification of the impacts of epistemic uncertainty is naturally difficult, because most of the existing stochastic tools rely on the specification of the probability distributions and thus do not readily apply to epistemic uncertainty. And there have been few studies and methods to deal with epistemic uncertainty. A recent work can be found in [J. Jakeman, M. Eldred, D. Xiu, Numerical approach for quantification of epistemic uncertainty, J. Comput. Phys. 229 (2010) 4648-4663], where a framework for numerical treatment of epistemic uncertainty was proposed. In this paper, firstly, we present a new method, similar to that of Jakeman et al. but significantly extending its capabilities. Most notably, the new method (1) does not require the encapsulation problem to be in a bounded domain such as a hypercube; (2) does not require the solution of the encapsulation problem to converge point-wise. In the current formulation, the encapsulation problem could reside in an unbounded domain, and more importantly, its numerical approximation could be sought in Lp norm. These features thus make the new approach more flexible and amicable to practical implementation. Both the mathematical framework and numerical analysis are presented to demonstrate the effectiveness of the new approach. And then, we apply this methods to work with one of the more restrictive uncertainty models, i.e., the fuzzy logic, where the p-distance, the weighted expected value and variance are defined to assess the accuracy of the solutions. At last, we give a brief introduction to our future work, which is epistemic uncertainty quantification using evidence theory

Influence of Slenderness on the Evaluation of Epistemic Uncertainty Related to Non-Linear Numerical Analysis of RC Columns

This investigation is devoted to quantify the epistemic uncertainty related to the nonlinear analysis of reinforced concrete columns characterized by high slenderness using numerical codes. The adoption of refined numerical tools, which are able to consider both mechanical and geometric non linearities, implies to perform assumptions and approximations with respect to reality. Whit reference to reliability analysis, these simplifications lead, inevitably, to additional uncertainties which are of epistemic nature. In fact, these uncertainties may be reduced by the engineers/analysts by increasing the level of refinement of the numerical model and/or increasing knowledge about parameters associated to material models. However, also numerical model established by expert engineers/analysts are affected by this kind of epistemic uncertainty. Accepting that the level of uncertainty associated to the experimental tests set are minimized, the epistemic uncertainty associated to non-linear numerical simulations can be quantified characterizing the model uncertainty random variable comparing the outcomes of numerical results to the associated experimental ones. The present investigation proposes the quantification of the model uncertainty related to non-linear numerical simulations of slender RC columns. A total number of 40 experimental results known from literature are herein selected in coherence with current Eurocodes specifications. The experiments are reproduced adopting non-linear numerical analysis differentiating between several modelling hypotheses (i.e., numerical code; materials models). The comparison between experimental and numerical results is adopted to characterize the most suitable probabilistic model for the model uncertainty random variable associated to non-linear numerical simulations of RC columns subjected to significant slenderness. The outcomes of the research are useful to provide background to the characterization of partial safety factor for model uncertainty in non-linear numerical analysis using the approach of the global resistance format for safety verifications

Quantification of uncertainty in aerodynamic heating of a reentry vehicle due to uncertain wall and freestream conditions

The primary focus of this study is to demonstrate an efficient approach for uncertainty quantification of surface heat flux to the spherical non-ablating heatshield of a generic reentry vehicle due to epistemic and aleatory uncertainties that may exist in various parameters used in the numerical solution of hypersonic, viscous, laminar blunt-body flows with thermo-chemical non-equilibrium. Two main uncertainty sources were treated in the computational fluid dynamics (CFD) simulations: (1) aleatory uncertainty in the freestream velocity and (2) epistemic uncertainty in the recombination efficiency for a partially catalytic wall boundary condition. The Second-Order Probability utilizing a stochastic response surface obtained with Point-Collocation Non-Intrusive Polynomial Chaos was used for the propagation of mixed (aleatory and epistemic) uncertainties. The uncertainty quantication approach was validated on a stochastic model problem with mixed uncertainties for the prediction of stagnation point heat transfer with Fay-Riddell relation, which included the comparison with direct Monte Carlo sampling results. In the stochastic CFD problem, the uncertainty in surface heat transfer was obtained in terms of intervals at different probability levels at various locations including the stagnation point and the shoulder region. The mixed uncertainty results were compared to the results obtained with a purely aleatory uncertainty analysis to show the difference between two uncertainty quantication approaches. A global sensitivity analysis indicated that the velocity has a stronger contribution to the overall uncertainty in the stagnation point heat transfer for the range of input uncertainties considered in this study --Abstract, page iii

Uncertainty Quantification of CFD Data Generated for a Model Scramjet Isolator Flowfield

Computational fluid dynamics is now considered to be an indispensable tool for the design and development of scramjet engine components. Unfortunately, the quantification of uncertainties is rarely addressed with anything other than sensitivity studies, so the degree of confidence associated with the numerical results remains exclusively with the subject matter expert that generated them. This practice must be replaced with a formal uncertainty quantification process for computational fluid dynamics to play an expanded role in the system design, development, and flight certification process. Given the limitations of current hypersonic ground test facilities, this expanded role is believed to be a requirement by some in the hypersonics community if scramjet engines are to be given serious consideration as a viable propulsion system. The present effort describes a simple, relatively low cost, nonintrusive approach to uncertainty quantification that includes the basic ingredients required to handle both aleatoric (random) and epistemic (lack of knowledge) sources of uncertainty. The nonintrusive nature of the approach allows the computational fluid dynamicist to perform the uncertainty quantification with the flow solver treated as a "black box". Moreover, a large fraction of the process can be automated, allowing the uncertainty assessment to be readily adapted into the engineering design and development workflow. In the present work, the approach is applied to a model scramjet isolator problem where the desire is to validate turbulence closure models in the presence of uncertainty. In this context, the relevant uncertainty sources are determined and accounted for to allow the analyst to delineate turbulence model-form errors from other sources of uncertainty associated with the simulation of the facility flow

Why Simpler Computer Simulation Models Can Be Epistemically Better for Informing Decisions

For computer simulation models to usefully inform climate risk management, uncertainties in model projections must be explored and characterized. Because doing so requires running the model many ti..

Efficient uncertainty quantification in aerospace analysis and design

The main purpose of this study is to apply a computationally efficient uncertainty quantification approach, Non-Intrusive Polynomial Chaos (NIPC) based stochastic expansions, to robust aerospace analysis and design under mixed (aleatory and epistemic) uncertainties and demonstrate this technique on model problems and robust aerodynamic optimization. The proposed optimization approach utilizes stochastic response surfaces obtained with NIPC methods to approximate the objective function and the constraints in the optimization formulation. The objective function includes the stochastic measures which are minimized simultaneously to ensure the robustness of the final design to both aleatory and epistemic uncertainties. For model problems with mixed uncertainties, Quadrature-Based and Point-Collocation NIPC methods were used to create the response surfaces used in the optimization process. For the robust airfoil optimization under aleatory (Mach number) and epistemic (turbulence model) uncertainties, a combined Point-Collocation NIPC approach was utilized to create the response surfaces used as the surrogates in the optimization process. Two stochastic optimization formulations were studied: optimization under pure aleatory uncertainty and optimization under mixed uncertainty. As shown in this work for various problems, the NIPC method is computationally more efficient than Monte Carlo methods for moderate number of uncertain variables and can give highly accurate estimation of various metrics used in robust design optimization under mixed uncertainties. This study also introduces a new adaptive sampling approach to refine the Point-Collocation NIPC method for further improvement of the computational efficiency. Two numerical problems demonstrated that the adaptive approach can produce the same accuracy level of the response surface obtained with oversampling ratio of 2 using less function evaluations. --Abstract, page iii

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. An approach based on non-intrusive polynomial chaos was formulated in which a low-fidelity model could be corrected by any number of high-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 of the multifidelity polynomial chaos approach, two model problems were explored. The first was supersonic airfoil with three levels of modeling fidelity, each capturing an additional level of physics. 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. Both problems illustrate the applicability and significant computational savings of the multifidelity polynomial chaos method

Deep Learning And Uncertainty Quantification: Methodologies And Applications

Uncertainty quantification is a recent emerging interdisciplinary area that leverages the power of statistical methods, machine learning models, numerical methods and data-driven approach to provide reliable inference for quantities of interest in natural science and engineering problems. In practice, the sources of uncertainty come from different aspects such as: aleatoric uncertainty where the uncertainty comes from the observations or is due to the stochastic nature of the problem; epistemic uncertainty where the uncertainty comes from inaccurate mathematical models, computational methods or model parametrization. Cope with the above different types of uncertainty, a successful and scalable model for uncertainty quantification requires prior knowledge in the problem, careful design of mathematical models, cautious selection of computational tools, etc. The fast growth in deep learning, probabilistic methods and the large volume of data available across different research areas enable researchers to take advantage of these recent advances to propose novel methodologies to solve scientific problems where uncertainty quantification plays important roles. The objective of this dissertation is to address the existing gaps and propose new methodologies for uncertainty quantification with deep learning methods and demonstrate their power in engineering applications. On the methodology side, we first present a generative adversarial framework to model aleatoric uncertainty in stochastic systems. Secondly, we leverage the proposed generative model with recent advances in physics-informed deep learning to learn the uncertainty propagation in solutions of partial differential equations. Thirdly, we introduce a simple and effective approach for posterior uncertainty quantification for learning nonlinear operators. Fourthly, we consider inverse problems of physical systems on identifying unknown forms and parameters in dynamical systems via observed noisy data. On the application side, we first propose an importance sampling approach for sequential decision making. Second, we propose a physics-informed neural network method to quantify the epistemic uncertainty in cardiac activation mapping modeling and conduct active learning. Third, we present an anto-encoder based framework for data augmentation and generation for data that is expensive to obtain such as single-cell RNA sequencing