Uncertainty Quantification Problems Research Articles

Machine-Learning-Based Hybrid Random-Fuzzy Uncertainty Quantification for EMC and SI Assessment

Modeling the effects of uncertainty is of crucial importance in the signal integrity and Electromagnetic Compatibility assessment of electronic products. In this article, a novel machine-learning-based approach for uncertainty quantification problems involving both random and epistemic variables is presented. The proposed methodology leverages evidence theory to represent probabilistic and epistemic uncertainties in a common framework. Then, Bayesian optimization is used to efficiently propagate this hybrid uncertainty on the performance of the system under study. Two suitable application examples validate the accuracy and efficiency of the proposed method.

Evaluation of ensemble methods for quantifying uncertainties in steady-state CFD applications with small ensemble sizes

Bayesian uncertainty quantification (UQ) is of interest to industry and academia as it provides a framework for quantifying and reducing the uncertainty in computational models by incorporating available data. For systems with very high computational costs, for instance, the computational fluid dynamics (CFD) problem, the conventional, exact Bayesian approach such as Markov chain Monte Carlo is intractable. To this end, the ensemble-based Bayesian methods have been used for CFD applications. However, their applicability for UQ has not been fully analyzed and understood thus far. Here, we evaluate the performance of three widely used iterative ensemble-based data assimilation methods, namely ensemble Kalman filter, ensemble randomized maximum likelihood method, and ensemble Kalman filter with multiple data assimilation for UQ problems. We present the derivations of the three ensemble methods from an optimization viewpoint. Further, a scalar case is used to demonstrate the performance of the three different approaches with emphasis on the effects of small ensemble sizes. Finally, we assess the three ensemble methods for quantifying uncertainties in steady-state CFD problems involving turbulent mean flows. Specifically, the Reynolds averaged Navier–Stokes (RANS) equation is considered the forward model, and the uncertainties in the propagated velocity are quantified and reduced by incorporating observation data. The results show that the ensemble methods cannot accurately capture the true posterior distribution, but they can provide a good estimation of the uncertainties even when very limited ensemble sizes are used. Based on the overall performance and efficiency from the comparison, the ensemble randomized maximum likelihood method is identified as the best choice of approximate Bayesian UQ approach among the three ensemble methods evaluated here.

Surrogate modeling for fast uncertainty quantification: Application to 2D population balance models

Surrogate modeling is a useful tool for enabling uncertainty quantification (UQ) tasks that require many expensive model evaluations, as it replaces expensive high-fidelity models with cheap-to-evaluate surrogates. This paper investigates sparse polynomial chaos and Kriging methods for surrogate modeling of first-principles models with probabilistic uncertainty in parameters and initial conditions. The surrogate modeling methods are demonstrated on a 2-dimensional population balance (2D-PB) model for batch cooling crystallization of ibuprofen with 20 uncertain parameters. Our analysis indicates that not only sparse polynomial chaos expansions are powerful for probabilistic UQ, but also the approximation accuracy of Kriging surrogate models can be significantly improved when polynomial chaos expansions are used to describe their trend. A basis-adaptive least-angle-regression strategy is shown to be particularly useful for inducing sparsity in polynomial chaos expansions, allowing for dealing with problems with a relatively large number of uncertain inputs. The utility of sparse polynomial chaos- and Kriging-based surrogate models is illustrated for various forward and inverse UQ problems, including global sensitivity analysis as well as Bayesian and maximum a posteriori parameter estimation of the 2D-PB model, where massive savings in computational cost (up to 30,000-fold) are observed.

A novel active learning-based Gaussian process metamodelling strategy for estimating the full probability distribution in forward UQ analysis

This paper proposes an active learning-based Gaussian process (AL-GP) metamodelling method to estimate the cumulative as well as complementary cumulative distribution function (CDF/CCDF) for forward uncertainty quantification (UQ) problems. Within the field of UQ, previous studies focused on developing AL-GP approaches for reliability (rare event probability) analysis of expensive black-box solvers. A naive iteration of these algorithms with respect to different CDF/CCDF threshold values would yield a discretized CDF/CCDF. However, this approach inevitably leads to a trade off between accuracy and computational efficiency since both depend (in opposite way) on the selected discretization. In this study, a specialized error measure and a learning function are developed such that the resulting AL-GP method is able to efficiently estimate the CDF/CCDF for a specified range of interest without an explicit dependency on discretization. Particularly, the proposed AL-GP method is able to simultaneously provide accurate CDF and CCDF estimation in their median-low probability regions. Three numerical examples are introduced to test and verify the proposed method.

Does Information Theory Provide a New Paradigm for Earth Science? Hypothesis Testing

AbstractModel evaluation and hypothesis testing are fundamental to any field of science. We propose here that by changing slightly the way we think and communicate about inference—from being fundamentally a problem of uncertainty quantification to being a problem of information quantification—allows us to avoid certain problems related to testing models as hypotheses. We propose that scientists are typically interested in assessing the information provided by models, not the truth value or likelihood of a model. Information theory allows us to formalize this perspective.

Uncertainty quantification of hydrodynamic forces on the DTC model in shallow water waves using CFD and non-intrusive polynomial chaos method

In this paper, uncertainty quantification of hydrodynamic forces on the DTC model advancing in shallow water waves is studied. First, verification and validation (V&V) and uncertainty quantification (UQ) in uncertainty analysis for CFD simulation are compared. A state-of-the-art method for UQ problems, the non-intrusive polynomial chaos (NIPC) method is introduced and validated to be effective by studying a stochastic function, together with the Monte-Carlo (MC) method. Then NIPC method is applied to quantify the uncertainty of the resistance, heave motion and pitch motion of the DTC model in shallow water waves. Computations are performed with the CFD software STAR-CCM+. The uncertainty induced by waves treated as stochastic variables with Gaussian distribution is studied. UQ results obtained by MC method and NIPC method are compared with the benchmark data. The results obtained by NIPC method show better agreement with the benchmark data than those by MC method.

Sensitivity Analysis of Burgers' Equation with Shocks

The generalized polynomial chaos (gPC) method has been extensively used in uncertainty quantification problems where equations contain random variables. For gPC to achieve high accuracy, PDE solutions need to have high regularity in the random space, but this is what hyperbolic type problems cannot provide. We provide a counterargument in this paper and show that even though the solution profile develops singularities in the random space, which destroys the spectral accuracy of gPC, the physical quantities (such as the shock emergence time, the shock location, and the shock strength) are all smooth functions of the uncertainties coming from both initial data and the wave speed. With proper shifting, the solution's polynomial interpolation approximates the real solution accurately, and the error decays as the order of the polynomial increases. Therefore this work provides a new perspective to “quantify uncertainties" and significantly improves the accuracy of the gPC method with a slight reformulation. We use the Burgers' equation as an example for thorough analysis, and the analysis could be extended to general conservation laws with convex fluxes.

Gradient-Based Dimension Reduction of Multivariate Vector-Valued Functions

Multivariate functions encountered in high-dimensional uncertainty quantification problems often vary most strongly along a few dominant directions in the input parameter space. We propose a gradient-based method for detecting these directions and using them to construct ridge approximations of such functions, in the case where the functions are vector-valued (e.g., taking values in $\mathbb{R}^n$). The methodology consists of minimizing an upper bound on the approximation error, obtained by subspace Poincar\'e inequalities. We provide a thorough mathematical analysis in the case where the parameter space is equipped with a Gaussian probability measure. The resulting method generalizes the notion of active subspaces associated with scalar-valued functions. A numerical illustration shows that using gradients of the function yields effective dimension reduction. We also show how the choice of norm on the codomain of the function has an impact on the function's low-dimensional approximation.

Characterizing Impacts of Model Uncertainties in Quantitative Photoacoustics

This work is concerned with uncertainty quantification problems for image reconstructions in quantitative photoacoustic imaging (PAT), a recent hybrid imaging modality that utilizes the photoacoustic effect to achieve high-resolution imaging of optical properties of tissue-like heterogeneous media. We quantify mathematically and computationally the impact of uncertainties in various model parameters of PAT on the accuracy of reconstructed optical properties. We derive, via sensitivity analysis, analytical bounds on error in image reconstructions in some simplified settings and develop a computational procedure, based on the method of polynomial chaos expansion, for such error characterization in more general settings. Numerical simulations based on synthetic data are presented to illustrate the main ideas.

Uncertainty quantification of MEMS devices with correlated random parameters

The concern about process deviations rises because that the performance uncertainty they cause are strengthened with the miniaturization and complication of Microelectromechanical System (MEMS) devices. To predict the statistic behavior of devices, Monte Carlo method is widely used, but it is limited by the low efficiency. The recently emerged generalized polynomial chaos expansion method, though highly efficient, cannot solve uncertainty quantification problems with correlated deviations, which is common in MEMS applications. In this paper, a Gaussian mixture model (GMM) and Nataf transformation based polynomial chaos method is proposed. The distribution of correlated process deviations is estimated using GMM, and modified Nataf transformation is applied to convert the correlated random vectors of GMM into mutually independent ones. Then polynomial chaos expansion and stochastic collection can be implemented. The effectiveness of our proposed method is demonstrated by the simulation results of V-beam thermal actuator, and its computation speed is faster compared with the Monte Carlo technique without loss of accuracy. This method can be served as an efficient analysis technique for MEMS devices which are sensitive to correlated process deviations.

Jointly Robust Prior for Gaussian Stochastic Process in Emulation, Calibration and Variable Selection

Gaussian stochastic process (GaSP) has been widely used in two fundamental problems in uncertainty quantification, namely the emulation and calibration of mathematical models. Some objective priors, such as the reference prior, are studied in the context of emulating (approximating) computationally expensive mathematical models. In this work, we introduce a new class of priors, called the jointly robust prior, for both the emulation and calibration. This prior is designed to maintain various advantages from the reference prior. In emulation, the jointly robust prior has an appropriate tail decay rate as the reference prior, and is computationally simpler than the reference prior in parameter estimation. Moreover, the marginal posterior mode estimation with the jointly robust prior can separate the influential and inert inputs in mathematical models, while the reference prior does not have this property. We establish the posterior propriety for a large class of priors in calibration, including the reference prior and jointly robust prior in general scenarios, but the jointly robust prior is preferred because the calibrated mathematical model typically predicts the reality well. The jointly robust prior is used as the default prior in two new R packages, called "RobustGaSP" and "RobustCalibration", available on CRAN for emulation and calibration, respectively.

Fast uncertainty quantification for dynamic flux balance analysis using non-smooth polynomial chaos expansions.

We present a novel surrogate modeling method that can be used to accelerate the solution of uncertainty quantification (UQ) problems arising in nonlinear and non-smooth models of biological systems. In particular, we focus on dynamic flux balance analysis (DFBA) models that couple intracellular fluxes, found from the solution of a constrained metabolic network model of the cellular metabolism, to the time-varying nature of the extracellular substrate and product concentrations. DFBA models are generally computationally expensive and present unique challenges to UQ, as they entail dynamic simulations with discrete events that correspond to switches in the active set of the solution of the constrained intracellular model. The proposed non-smooth polynomial chaos expansion (nsPCE) method is an extension of traditional PCE that can effectively capture singularities in the DFBA model response due to the occurrence of these discrete events. The key idea in nsPCE is to use a model of the singularity time to partition the parameter space into two elements on which the model response behaves smoothly. Separate PCE models are then fit in both elements using a basis-adaptive sparse regression approach that is known to scale well with respect to the number of uncertain parameters. We demonstrate the effectiveness of nsPCE on a DFBA model of an E. coli monoculture that consists of 1075 reactions and 761 metabolites. We first illustrate how traditional PCE is unable to handle problems of this level of complexity. We demonstrate that over 800-fold savings in computational cost of uncertainty propagation and Bayesian estimation of parameters in the substrate uptake kinetics can be achieved by using the nsPCE surrogates in place of the full DFBA model simulations. We then investigate the scalability of the nsPCE method by utilizing it for global sensitivity analysis and maximum a posteriori estimation in a synthetic metabolic network problem with a larger number of parameters related to both intracellular and extracellular quantities.

A new surrogate modeling method combining polynomial chaos expansion and Gaussian kernel in a sparse Bayesian learning framework

SummarySurrogate modeling techniques have been increasingly developed for optimization and uncertainty quantification problems in many engineering fields. The development of surrogates requires modeling high‐dimensional and nonsmooth functions with limited information. To this end, the hybrid surrogate modeling method, where different surrogate models are combined, offers an effective solution. In this paper, a new hybrid modeling technique is proposed by combining polynomial chaos expansion and kernel function in a sparse Bayesian learning framework. The proposed hybrid model possesses both the global characteristic advantage of polynomial chaos expansion and the local characteristic advantage of the Gaussian kernel. The parameterized priors are utilized to encourage the sparsity of the model. Moreover, an optimization algorithm aiming at maximizing Bayesian evidence is proposed for parameter optimization. To assess the performance of the proposed method, a detailed comparison is made with the well‐established PC‐Kriging technique. The results show that the proposed method is superior in terms of accuracy and robustness.

High-Dimensional Uncertainty Quantification of Electronic and Photonic IC With Non-Gaussian Correlated Process Variations

Uncertainty quantification based on generalized polynomial chaos has been used in many applications. It has also achieved great success in variation-aware design automation. However, almost all existing techniques assume that the parameters are mutually independent or Gaussian correlated, which is rarely true in real applications. For instance, in chip manufacturing, many process variations are actually correlated. Recently, some techniques have been developed to handle non-Gaussian correlated random parameters, but they are time-consuming for high-dimensional problems. We present a new framework to solve uncertainty quantification problems with many non-Gaussian correlated uncertainties. First, we propose a set of smooth basis functions to well capture the impact of non-Gaussian correlated process variations. We develop a tensor approach to compute these basis functions in a high-dimension setting. Second, we investigate the theoretical aspect and practical implementation of a sparse solver to compute the coefficients of all basis functions. We provide some theoretical analysis for the exact recovery condition and error bound of this sparse solver in the context of uncertainty quantification. We present three adaptive sampling approaches to improve the performance of the sparse solver. Finally, we validate our methods by synthetic and practical electronic/photonic ICs with 19 to 57 non-Gaussian correlated variation parameters. Our approach outperforms Monte Carlo by thousands of times in terms of efficiency. It can also accurately predict the output density functions with multiple peaks caused by non-Gaussian correlations, which are hard to capture by existing methods.

Analysis of quasi-Monte Carlo methods for elliptic eigenvalue problems with stochastic coefficients

We consider the forward problem of uncertainty quantification for the generalised Dirichlet eigenvalue problem for a coercive second order partial differential operator with random coefficients, motivated by problems in structural mechanics, photonic crystals and neutron diffusion. The PDE coefficients are assumed to be uniformly bounded random fields, represented as infinite series parametrised by uniformly distributed i.i.d. random variables. The expectation of the fundamental eigenvalue of this problem is computed by (a) truncating the infinite series which define the coefficients; (b) approximating the resulting truncated problem using lowest order conforming finite elements and a sparse matrix eigenvalue solver; and (c) approximating the resulting finite (but high dimensional) integral by a randomly shifted quasi-Monte Carlo lattice rule, with specially chosen generating vector. We prove error estimates for the combined error, which depend on the truncation dimension s, the finite element mesh diameter h, and the number of quasi-Monte Carlo samples N. Under suitable regularity assumptions, our bounds are of the particular form $${\mathcal {O}}(h^2 + N^{-1 + \delta })$$ , where $$\delta > 0$$ is arbitrary and the hidden constant is independent of the truncation dimension, which needs to grow as $$h\rightarrow 0$$ and $$N \rightarrow \infty $$ . As for the analogous PDE source problem, the conditions under which our error bounds hold depend on a parameter $$p \in (0, 1)$$ representing the summability of the terms in the series expansions of the coefficients. Although the eigenvalue problem is nonlinear, which means it is generally considered harder than the source problem, in almost all cases ( $$p \ne 1$$ ) we obtain error bounds that converge at the same rate as the corresponding rate for the source problem. The proof involves a detailed study of the regularity of the fundamental eigenvalue as a function of the random parameters. As a key intermediate result in the analysis, we prove that the spectral gap (between the fundamental and the second eigenvalues) is uniformly positive over all realisations of the random problem.

Effect of mesh refinement on the estimation of model input parameters using Inverse Uncertainty Quantification

Proper quantification of the uncertainties in the input parameters of a thermal-hydraulics code (e.g. physical models) is an essential part of the forward Uncertainty Quantification (UQ) problem. Such quantification can be systematically achieved by solving the Inverse Uncertainty Quantification (IUQ) problem using data from code predictions and experimental measurements. The IUQ problem is highly dependent on several factors such as geometry and discretization. In this paper, we study the effect of mesh refinement on the statistical parameters of three uncertain physical models (interfacial friction coefficient, wall to liquid friction coefficient and critical heat flux). A mathematical framework based on MLE and MAP algorithms is implemented to perform IUQ for the physical models of the thermal-hydraulics code RSTART based on experimental data from the OECD/NEA BWR Full-size Fine-mesh Bundle Test (BFBT) benchmark. Sensitivity analysis required for the IUQ problem was achieved by implementing discrete adjoint methods in the thermal-hydraulics code. The statistical parameters of the three physical models were obtained by implementing the MLE and MAP methods using code predictions that correspond to three different mesh sizes. Results from MLE and MAP showed a noticeable effect of mesh refinement on the estimates of the mean and a negligible effect on the estimates of the variance of the physical models. The compensation of error due to mesh refinement was studied by running the code with perturbation values calculated using the three different mesh sizes. It was shown that the results based on the most refined mesh led to best agreement between experimental data and code prediction.

Global sensitivity analysis of static characteristics of tilting-pad journal bearing to manufacturing tolerances

A tiny difference between nominal dimension and actual processing dimension in the range of manufacturing tolerance can vary the performance of journal bearing. For minimizing this variation, sensitivity analysis is usually conducted. In present paper, the variance-based global sensitivity analysis was introduced to deal with the uncertainty quantification problem for a five-pad journal bearing. By using Fourier Amplitude Sensitivity Test (FAST) method, the first-order sensitivity indices of static characteristics including capacity, power loss and inlet flow were analyzed and evaluated for five normal distributed processing parameters: pad internal radius, shaft radius, pivot radius, pivot position and pad angular extent. The FAST method was proved perfectly suited to the bearing model. The quantitative sensitivities of output to each input parameter were obtained.

IGA-based multi-index stochastic collocation for random PDEs on arbitrary domains

This paper proposes an extension of the Multi-Index Stochastic Collocation (MISC) method for forward uncertainty quantification (UQ) problems in computational domains of shape other than a square or cube, by exploiting isogeometric analysis (IGA) techniques. Introducing IGA solvers to the MISC algorithm is very natural since they are tensor-based PDE solvers, which are precisely what is required by the MISC machinery. Moreover, the combination-technique formulation of MISC allows the straightforward reuse of existing implementations of IGA solvers. We present numerical results to showcase the effectiveness of the proposed approach.

An inverse method for distributed dynamic load identification of structures with interval uncertainties

A time-domain based method for distributed dynamic load identification is proposed in this study considering unknown-but-bounded uncertainties in structural systems. The spatiotemporal dynamic load is further approximated by Chebyshev orthogonal polynomial in time history. Thus, the problem of distributed load reconstruction may be converted into the issue of polynomial coefficient calculation at each sampling time by utilizing a series of dynamic analysis. In accordance with the practical engineering, the acceleration response is used as the system input. In terms of the uncertainty quantification problems, the interval analysis method based on Taylor expansion (IAMBTE) is systematically developed to accomplish the envelope interval of identified load. To facilitate the analysis, two kinds of load are required to be identified, among which the nominal value of the identified load may be straightforwardly achieved through the inverse system, whereas the interval boundaries should be settled by the interval propagation analysis. Eventually, two numerical examples are investigated to demonstrate the efficiency and precision of the developed methodology.

Assessment of Deterministic Shape Optimizations Within a Stochastic Framework for Supersonic Organic Rankine Cycle Nozzle Cascades

The design of converging–diverging blades for organic Rankine cycle (ORC) applications widely relies on automated shape-optimization processes. As a result, the optimization produces an adapted-nozzle cascade at the design conditions. However, only few works account for the uncertainties in those conditions and their consequences on cascade performance. The proposed solution, i.e., including uncertainties within the optimization routine, demands an overall huge computational cost to estimate the target output statistic at each iteration of the optimization algorithm. With the aim of understanding if this computational cost is avoidable, we study the impact of uncertainties in the design conditions on the robustness of deterministically optimized profiles. Several optimized blades, obtained with different objective functions, constraints, and design variables, are considered in the present numerical analysis, which features a turbulent compressible flow solver and a state-of-the-art uncertainty-quantification (UQ) method. By including measured field variations in the formulation of the UQ problem, we show that a deterministic shape optimization already improves the robustness of the profile with respect to the baseline configuration. Guidelines about objective functions and blade parametrizations for deterministic optimizations are also provided. Finally, a novel methodology to estimate the mass-flow-rate probability density function (PDF) for choked supersonic turbines is proposed, along with a robust reformulation of the constraint problem without increasing the computational cost.