Parametric Quantification Research Articles

Polynomial Chaos Expansion for Parametric Problems in Engineering Systems: A Review

In engineering systems with uncertain parameters, it is crucial for system analysis and control to analyze the relationship between these uncertain parameters and system outputs (or states). Nevertheless, the acquisition of this parameter-output function, termed parametric problem in this article, is not easy because it is usually a complicated implicit nonlinear function. Polynomial chaos expansion (PCE), based on the generalized Fourier expansion, is a state-of-the-art method for both parametric problems and uncertainty quantification. Its basic idea is to approximate the implicit parameter-output function with a globally optimal explicit polynomial function, which remains accurate in the case of strong nonlinearity compared with the local Taylor expansion. This article provides a review of PCE and its applications in parametric problems. In terms of PCE theory, a rigorous and complete framework of PCE is established and a detailed review of typical PCE methods is presented. In terms of applications, two kinds of general parametric problems, namely parametric nonlinear algebraic equations and parametric differential equations, as well as their examples in some engineering fields, are introduced. Furthermore, some worthwhile future works are presented at the end of this article, which may facilitate the development of PCE.

Antenna Modeling Using Variable-Fidelity EM Simulations and Constrained Co-Kriging

Utilization of fast surrogate models has become a viable alternative to direct handling of full-wave electromagnetic (EM) simulations in EM-driven design. Their purpose is to alleviate the difficulties related to high computational cost of multiple simulations required by the common numerical procedures such as parametric optimization or uncertainty quantification. Yet, conventional data-driven (or approximation) modeling techniques are severely affected by the curse of dimensionality. This is a serious limitation when it comes to modeling of highly nonlinear antenna characteristics. In practice, general-purpose surrogates can be rendered for the structures described by a few parameters within limited ranges thereof, which is grossly insufficient from the utility point of view. This paper proposes a novel modeling approach involving variable-fidelity EM simulations incorporated into the recently reported nested kriging modeling framework. Combining the information contained in the densely sampled low- and sparsely sampled high-fidelity models is realized using co-kriging. The resulting surrogate exhibits the predictive power comparable to the model constructed using exclusively high-fidelity data while offering significantly reduced setup cost. The advantages over conventional surrogates are pronounced even further. The presented modeling procedure is demonstrated using two antenna examples and further validated through the application case studies.

Design-Oriented Two-Stage Surrogate Modeling of Miniaturized Microstrip Circuits With Dimensionality Reduction

Contemporary microwave design heavily relies on full-wave electromagnetic (EM) simulation tools. This is especially the case for miniaturized devices where EM cross-coupling effects cannot be adequately accounted for using equivalent network models. Unfortunately, EM analysis incurs considerable computational expenses, which becomes a bottleneck whenever multiple evaluations are required. Common simulation-based design tasks include parametric optimization and uncertainty quantification. These can be accelerated using fast replacement models, among which the data-driven surrogates are the most popular. Notwithstanding, a construction of approximation models for microwave components is hindered by the dimensionality issues as well as high nonlinearity of system characteristics. A partial alleviation of the mentioned difficulties can be achieved with the recently reported performance-driven modeling methods, including the nested kriging framework. Therein, the computational benefits are obtained by appropriate confinement of the surrogate model domain, spanned by a set of pre-optimized reference designs, and by focusing on the parameter space region that contains high quality designs with respect to the considered performance figures. This paper presents a methodology that incorporates the concept of nested kriging and enhances it by explicit dimensionality reduction based on spectral decomposition of the reference design set. Extensive verification studies conducted for a compact rat-race coupler and a three-section impedance matching transformer demonstrate superiority of the presented approach over both the conventional techniques and the nested kriging in terms of modeling accuracy. Design utility of our surrogates is corroborated through application cases studies.

Comparison of intrusive and nonintrusive polynomial chaos expansion-based approaches for high dimensional parametric uncertainty quantification and propagation

We present an uncertainty quantification (UQ) algorithm using the intrusive generalized polynomial chaos (gPC) expansion in combination with dimension reduction techniques and compare the UQ accuracy and computational efficiency of the intrusive gPC-based UQ algorithm to other sampling-based nonintrusive methods. The successful application of intrusive gPC-based UQ is associated with the stochastic Galerkin (SG) projection, which yields a family of models described by several coupled equations of gPC coefficients. Using these coefficients, the evolution of uncertainty in a dynamic system can be quickly determined when there is probabilistic uncertainty in the system. While elegant, when dealing with models that involve complex functions (e.g., nonpolynomial terms) and larger numbers of uncertainties, SG projection becomes computationally intractable and cannot be applied directly to solve gPC coefficients in real-time. To address this issue, the generalized dimension reduction method (gDRM) is used to convert a high-dimensional integral involved in the SG projection into several lower-dimensional integrals that can be easily solved. To show the accuracy of UQ, the algorithm in this work is compared to sampling-based approaches such as the nonintrusive stochastic collocation (SC) and Monte Carlo (MC) simulations using three cases: a nonlinear algebraic benchmark, a penicillin manufacturing process, and autocrine signalling networks of living cells.

Test–retest repeatability of [18F]Flortaucipir PET in Alzheimer’s disease and cognitively normal individuals

The aim of this study was to investigate the test–retest (TRT) repeatability of various parametric quantification methods for [18F]Flortaucipir positron emission tomography (PET). We included eight subjects with dementia or mild cognitive impairment due to Alzheimer’s disease and six cognitively normal subjects. All underwent two 130-min dynamic [18F]Flortaucipir PET scans within 3 ± 1 weeks. Data were analyzed using reference region models receptor parametric mapping (RPM), simplified reference tissue method 2 (SRTM2) and reference logan (RLogan), as well as standardized uptake value ratios (SUVr, time intervals 40–60, 80–100 and 110–130 min post-injection) with cerebellar gray matter as reference region. We obtained distribution volume ratio or SUVr, first for all brain regions and then in three tau-specific regions-of-interest (ROIs). TRT repeatability (%) was defined as |retest–test|/(average (test + retest)) × 100. For all methods and across ROIs, TRT repeatability ranged from (median (IQR)) 0.84% (0.68–2.15) to 6.84% (2.99–11.50). TRT repeatability was good for all reference methods used, although semi-quantitative models (i.e. SUVr) performed marginally worse than quantitative models, for instance TRT repeatability of RPM: 1.98% (0.78–3.58) vs. SUVr80–100: 3.05% (1.28–5.52), p < 0.001. Furthermore, for SUVr80–100 and SUVr110–130, with higher average SUVr, more variation was observed. In conclusion, while TRT repeatability was good for all models used, quantitative methods performed slightly better than semi-quantitative methods.

Advances in Parametric and Model-Form Uncertainty Quantification in Canonical Aeroelastic Systems

Uncertainty quantification is going to play a crucial role in the aeroelastic design and optimization of aircraft. Stochastic aeroelastic models are currently being considered to account for manufacturing tolerance in material properties, variability in flight conditions or uncertainty in the aeroelastic model itself. In this paper, some challenging issues in the development of efficient and robust stochastic solvers are reported within the framework of canonical aeroelastic systems. First, independent or correlated parametric uncertainties are propagated to compute the probability density function of the critical flutter velocity or the limit cycle oscillations in the presence of discontinuous responses. Secondly, inverse stochastic aeroelastic problems are addressed, in which experimental data are used to calibrate several stochastic aerodynamic models within a Bayesian framework. Studied configurations concern linear and non-linear pitching and plunging airfoils, and the stochastic flutter of a cantilevered straight composite wing subject to ply angle and thickness uncertainties.

Online identification of large-scale chaotic system

The ensemble prediction system (EPS) is an approach employed in meteorology to estimate forecast uncertainty of dynamical systems. In EPS, an ensemble of auxiliary simulations is launched along with the main prediction. Recently, an application with the EPS framework was proposed as a method that enables algorithmic tuning of parameters of large-scale models, in cases where high-CPU demands make usual iterative optimization impractical. The approach was aimed and tested for operational numerical weather prediction models, with a relatively small number of parameters and well-tuned initial values. Here, we present a new version of the approach as a general-purpose parameter estimation method for situations where effective parallel computing is available, but high-CPU requirements exclude the use of standard sequential approaches. We treat the problem as a stochastic optimization task and employ an evolutionary approach, the differential evolution as the optimizer. We demonstrate improved convergence properties, especially for strongly biased initial values or higher number of parameters. For parametric uncertainty quantification, the approach can be considered as a heuristic sampler of the parameter distributions.

Compressed Sensing with Sparse Corruptions: Fault-Tolerant Sparse Collocation Approximations

The recovery of approximately sparse or compressible coefficients in a Polynomial Chaos Expansion is a common goal in modern parametric uncertainty quantification (UQ). However, relatively little effort in UQ has been directed toward theoretical and computational strategies for addressing the sparse corruptions problem, where a small number of measurements are highly corrupted. Such a situation has become pertinent today since modern computational frameworks are sufficiently complex with many interdependent components that may introduce hardware and software failures, some of which can be difficult to detect and result in a highly polluted simulation result. In this paper we present a novel compressive sampling-based theoretical analysis for a regularized $\ell^1$ minimization algorithm that aims to recover sparse expansion coefficients in the presence of measurement corruptions. Our recovery results are uniform, and prescribe algorithmic regularization parameters in terms of a user-defined a priori estimate on the ratio of measurements that are believed to be corrupted. We also propose an iteratively reweighted optimization algorithm that automatically refines the value of the regularization parameter, and empirically produces superior results. Our numerical results test our framework on several medium-to-high dimensional examples of solutions to parameterized differential equations, and demonstrate the effectiveness of our approach.

Power Electronics Modeling and Design: Using Parametric and Model-Form Uncertainty Quantification to Assess Predictive Accuracy of Power Converter Models

This article presents an enhanced modeling framework for the design of power electronics converters and systems. This methodology provides the means to assess and quantify the predictive accuracy of power converter models, a capability that is attained by the proper identification, characterization, and quantification of the different, and often numerous, sources of uncertainty in the modeling and simulation process. For illustrative purposes, an in-depth example is presented using a power electronics building block (PEBB) as the modeling test bed, demonstrating how, through this formal modeling procedure-and taking into consideration the uncertainties in the system-the model in question can predict the variable of interest within a known range with a given level of confidence, without requiring heuristic safety factors. This predicted range can be used as the required design margin when the selected output is a design variable, evincing the inherent potential of having a less conservative and more reliable power converter using the design methodology with uncertainty quantification.

Tuning without over-tuning: parametric uncertainty quantification for the NEMO ocean model

Abstract. In this paper we discuss climate model tuning and present an iterative automatic tuning method from the statistical science literature. The method, which we refer to here as iterative refocussing (though also known as history matching), avoids many of the common pitfalls of automatic tuning procedures that are based on optimisation of a cost function, principally the over-tuning of a climate model due to using only partial observations. This avoidance comes by seeking to rule out parameter choices that we are confident could not reproduce the observations, rather than seeking the model that is closest to them (a procedure that risks over-tuning). We comment on the state of climate model tuning and illustrate our approach through three waves of iterative refocussing of the NEMO (Nucleus for European Modelling of the Ocean) ORCA2 global ocean model run at 2° resolution. We show how at certain depths the anomalies of global mean temperature and salinity in a standard configuration of the model exceeds 10 standard deviations away from observations and show the extent to which this can be alleviated by iterative refocussing without compromising model performance spatially. We show how model improvements can be achieved by simultaneously perturbing multiple parameters, and illustrate the potential of using low-resolution ensembles to tune NEMO ORCA configurations at higher resolutions.

Data assimilation and sampling in Banach spaces

This paper studies the problem of approximating a function f in a Banach space $$\mathcal{X}$$ from measurements $$l_j(f)$$ , $$j=1,\ldots ,m$$ , where the $$l_j$$ are linear functionals from $$\mathcal{X}^*$$ . Quantitative results for such recovery problems require additional information about the sought after function f. These additional assumptions take the form of assuming that f is in a certain model class $$K\subset \mathcal{X}$$ . Since there are generally infinitely many functions in K which share these same measurements, the best approximation is the center of the smallest ball B, called the Chebyshev ball, which contains the set $$\bar{K}$$ of all f in K with these measurements. Therefore, the problem is reduced to analytically or numerically approximating this Chebyshev ball. Most results study this problem for classical Banach spaces $$\mathcal{X}$$ such as the $$L_p$$ spaces, $$1\le p\le \infty $$ , and for K the unit ball of a smoothness space in $$\mathcal{X}$$ . Our interest in this paper is in the model classes $$K=\mathcal{K}(\varepsilon ,V)$$ , with $$\varepsilon >0$$ and V a finite dimensional subspace of $$\mathcal{X}$$ , which consists of all $$f\in \mathcal{X}$$ such that $$\mathrm{dist}(f,V)_\mathcal{X}\le \varepsilon $$ . These model classes, called approximation sets, arise naturally in application domains such as parametric partial differential equations, uncertainty quantification, and signal processing. A general theory for the recovery of approximation sets in a Banach space is given. This theory includes tight a priori bounds on optimal performance and algorithms for finding near optimal approximations. It builds on the initial analysis given in Maday et al. (Int J Numer Method Eng 102:933–965, 2015) for the case when $$\mathcal{X}$$ is a Hilbert space, and further studied in Binev et al. (SIAM UQ, 2015). It is shown how the recovery problem for approximation sets is connected with well-studied concepts in Banach space theory such as liftings and the angle between spaces. Examples are given that show how this theory can be used to recover several recent results on sampling and data assimilation.

On the parametric uncertainty quantification of the Rothermel's rate of spread model

Parametric uncertainty quantification of the Rothermel's fire spread model is presented using the Polynomial Chaos expansion method under a Non-Intrusive Spectral Projection (NISP) approach. Several Rothermel's model input parameters have been considered random with an associated prescribed probability density function. Two different vegetation fire scenarios are considered and NISP method results and performance are compared with four other stochastic methodologies: Sensitivity Derivative Enhance Sampling; two Monte Carlo techniques; and Global Sensitivity Analysis. The stochastic analysis includes a sensitivity analysis study to quantify the direct influence of each random parameter on the solution. The NISP approach achieved performance three orders of magnitude faster than the traditional Monte Carlo method. The NISP capability to perform uncertainty quantification associated with fast convergence makes it well suited to be applied for stochastic prediction of fire spread.

Bayesian estimation of parametric uncertainties, quantification and reduction using optimal design of experiments for CO2 adsorption on amine sorbents

Uncertainty quantification plays a significant role in establishing reliability of mathematical models, while applying to process optimization or technology feasibility studies. Uncertainties, in general, could occur either in mathematical model or in model parameters. In this work, process of CO2 adsorption on amine sorbents, which are loaded in hollow fibers is studied to quantify the impact of uncertainties in the adsorption isotherm parameters on the model prediction. The process design variable that is most closely related to the process economics is the CO2 sorption capacity, whose uncertainty is investigated. We apply Bayesian analysis and determine a utility function surface corresponding to the value of information gained by the respective experimental design point. It is demonstrated that performing an experiment at a condition with a higher utility has a higher reduction of design variable prediction uncertainty compared to choosing a design point at a lower utility.

Extended Reconstructed Sea Surface Temperature Version 4 (ERSST.v4): Part II. Parametric and Structural Uncertainty Estimations

Abstract Described herein is the parametric and structural uncertainty quantification for the monthly Extended Reconstructed Sea Surface Temperature (ERSST) version 4 (v4). A Monte Carlo ensemble approach was adopted to characterize parametric uncertainty, because initial experiments indicate the existence of significant nonlinear interactions. Globally, the resulting ensemble exhibits a wider uncertainty range before 1900, as well as an uncertainty maximum around World War II. Changes at smaller spatial scales in many regions, or for important features such as Niño-3.4 variability, are found to be dominated by particular parameter choices. Substantial differences in parametric uncertainty estimates are found between ERSST.v4 and the independently derived Hadley Centre SST version 3 (HadSST3) product. The largest uncertainties are over the mid and high latitudes in ERSST.v4 but in the tropics in HadSST3. Overall, in comparison with HadSST3, ERSST.v4 has larger parametric uncertainties at smaller spatial and shorter time scales and smaller parametric uncertainties at longer time scales, which likely reflects the different sources of uncertainty quantified in the respective parametric analyses. ERSST.v4 exhibits a stronger globally averaged warming trend than HadSST3 during the period of 1910–2012, but with a smaller parametric uncertainty. These global-mean trend estimates and their uncertainties marginally overlap. Several additional SST datasets are used to infer the structural uncertainty inherent in SST estimates. For the global mean, the structural uncertainty, estimated as the spread between available SST products, is more often than not larger than the parametric uncertainty in ERSST.v4. Neither parametric nor structural uncertainties call into question that on the global-mean level and centennial time scale, SSTs have warmed notably.

Numerical Continuation and SPDE Stability for the 2D Cubic-Quintic Allen--Cahn Equation

We study the Allen--Cahn equation with a cubic-quintic nonlinear term and a $Q$-trace-class stochastic forcing in two spatial dimensions. This stochastic partial differential equation (SPDE) is used as a test case to understand how numerical continuation methods can be carried over to the SPDE setting. First, we compute the deterministic bifurcation diagram for the PDE, i.e., without stochastic forcing. In this case, two locally asymptotically stable steady state solution branches exist upon variation of the linear damping term. Then we consider the Lyapunov operator equation for the locally linearized system around steady states for the SPDE. We discretize the full SPDE using a combination of finite differences and spectral noise approximation obtaining a finite-dimensional system of stochastic ordinary differential equations (SODEs). The large system of SODEs is used to approximate the Lyapunov operator equation via covariance matrices. The covariance matrices are numerically continued along the two bifurcation branches. We show that we can quantify the stochastic fluctuations along the branches. We also demonstrate scaling laws near branch and fold bifurcation points. Furthermore, we perform computational tests to show that, even with a suboptimal computational setup, we can quantify the subexponential-timescale fluctuations near the deterministic steady states upon stochastic forcing on a standard desktop computer setup. Hence, the proposed method for numerical continuation of SPDEs has the potential to allow for rapid parametric uncertainty quantification of spatio-temporal stochastic systems.

Parametric uncertainty quantification in the Rothermel model with randomised quasi-Monte Carlo methods

Rothermel's wildland surface fire model is a popular model used in wildland fire management. The original model has a large number of parameters, making uncertainty quantification challenging. In this paper, we use variance-based global sensitivity analysis to reduce the number of model parameters, and apply randomised quasi-Monte Carlo methods to quantify parametric uncertainties for the reduced model. The Monte Carlo estimator used in these calculations is based on a control variate approach applied to the sensitivity derivative enhanced sampling. The chaparral fuel model, selected from Rothermel's 11 original fuel models, is studied as an example. We obtain numerical results that improve the crude Monte Carlo sampling by factors as high as three orders of magnitude.

Parametric Quantification of Low GWP Refrigerant for Thermosyphon Driven Solar Water Heating System

Modern lifestyle, industrialization and economy thrive on energy which is getting expensive overtime. Thermosyphon based systems are getting attraction for their promising heat transfer efficiency and zero energy utilization. Refrigerants having ozone depletion potential (ODP) and high Global Warming Potential (GWP) have been banned or under time bared permission under Montreal (1987) and Kyoto (1997) protocols. We have devolved a Refrigerant Parametric Quantification (RPQ) method for the choice of optimal refrigerant for density driven solar water heaters. A set of 29 refrigerants are simulated Using REFPROP under various temperature and pressure conditions. The optimal parameters of thermosyphon system are identified from governing equations, international environment safety protocols and open literature. The proposed RPQ method shows most appropriate refrigerant for given temperature range. In second part, the proposed system is simulated in TRNSYS using forced circulation method. In the end, a glass evacuated tube collector is developed and tested on the principal of thermosyphon, employing the best refrigerant emerged from simulation study.

Parametric Quantification of Low GWP Refrigerant for Thermosyphon Driven Solar Water Heating System

Modern lifestyle, growing industrial and economy thrive on energy which is getting expensive over time. Natural convection based systems are getting attraction for their promising heat transfer efficiency and zero energy utilization. Refrigerants having ozone depletion potential (ODP) and high Global Warming Potential (GWP) have been banned or under time bared permission under the Montreal (1987) and Kyoto (1997) protocols. To retrofit and modify existing cooling and heating systems, various natural and synthetic refrigerants are being investigated worldwide. We have devolved a Refrigerant Parametric Quantification (RPQ) method for the choice of optimal refrigerant for thermosyphon driven solar water heaters. A set of 29 refrigerants are simulated Using REFPROP under various temperature and pressure conditions. The optimal parameters of thermosyphon system are identified from governing equations, international environmental and safety protocols. The proposed RPQ method shows most appropriate refrigerant for given temperature range. In the second part, a glass evacuated tube collector is designed; fabricated and tested employing the best refrigerant emerged from a simulation study.

Beyond fractional anisotropy: Extraction of bundle-specific structural metrics from crossing fiber models

Diffusion MRI (dMRI) measurements are used for inferring the microstructural properties of white matter and to reconstruct fiber pathways. Very often voxels contain complex fiber configurations comprising multiple bundles, rendering the simple diffusion tensor model unsuitable. Multi-compartment models deliver a convenient parameterization of the underlying complex fiber architecture, but pose challenges for fitting and model selection. Spherical deconvolution, in contrast, very economically produces a fiber orientation density function (fODF) without any explicit model assumptions. Since, however, the fODF is represented by spherical harmonics, a direct interpretation of the model parameters is impossible. Based on the fact that the fODF can often be interpreted as superposition of multiple peaks, each associated to one relatively coherent fiber population (bundle), we offer a solution that seeks to combine the advantages of both approaches: first the fiber configuration is modeled as fODF represented by spherical harmonics and then each of the peaks is parameterized separately in order to characterize the underlying bundle. In this work, the fODF peaks are approximated by Bingham distributions, capturing first and second-order statistics of the fiber orientations, from which we derive metrics for the parametric quantification of fiber bundles. We propose meaningful relationships between these measures and the underlying microstructural properties. We focus on metrics derived directly from properties of the Bingham distribution, such as peak length, peak direction, peak spread, integral over the peak, as well as a metric derived from the comparison of the largest peaks, which probes the complexity of the underlying microstructure. We compare these metrics to the conventionally used fractional anisotropy (FA) and show how they may help to increase the specificity of the characterization of microstructural properties. While metrics relying on the first moments of the Bingham distributions provide relatively robust results, second-order metrics representing the peak spread are only meaningful, if the SNR is very high and no fiber crossings are present in the voxel.

Rapid Parametric Mapping of the Longitudinal Relaxation Time T1 Using Two-Dimensional Variable Flip Angle Magnetic Resonance Imaging at 1.5 Tesla, 3 Tesla, and 7 Tesla

IntroductionVisual but subjective reading of longitudinal relaxation time (T1) weighted magnetic resonance images is commonly used for the detection of brain pathologies. For this non-quantitative measure, diagnostic quality depends on hardware configuration, imaging parameters, radio frequency transmission field (B1+) uniformity, as well as observer experience. Parametric quantification of the tissue T1 relaxation parameter offsets the propensity for these effects, but is typically time consuming. For this reason, this study examines the feasibility of rapid 2D T1 quantification using a variable flip angles (VFA) approach at magnetic field strengths of 1.5 Tesla, 3 Tesla, and 7 Tesla. These efforts include validation in phantom experiments and application for brain T1 mapping.MethodsT1 quantification included simulations of the Bloch equations to correct for slice profile imperfections, and a correction for B1+. Fast gradient echo acquisitions were conducted using three adjusted flip angles for the proposed T1 quantification approach that was benchmarked against slice profile uncorrected 2D VFA and an inversion-recovery spin-echo based reference method. Brain T1 mapping was performed in six healthy subjects, one multiple sclerosis patient, and one stroke patient.ResultsPhantom experiments showed a mean T1 estimation error of (-63±1.5)% for slice profile uncorrected 2D VFA and (0.2±1.4)% for the proposed approach compared to the reference method. Scan time for single slice T1 mapping including B1+ mapping could be reduced to 5 seconds using an in-plane resolution of (2×2) mm2, which equals a scan time reduction of more than 99% compared to the reference method.ConclusionOur results demonstrate that rapid 2D T1 quantification using a variable flip angle approach is feasible at 1.5T/3T/7T. It represents a valuable alternative for rapid T1 mapping due to the gain in speed versus conventional approaches. This progress may serve to enhance the capabilities of parametric MR based lesion detection and brain tissue characterization.