Polynomial Chaos Surrogate Research Articles

Real-Time Hybrid Simulation Using Nonlinear Autoregressive with Exogenous Input Model through Data-Driven Arbitrary Polynomial Chaos Surrogate

The current practice of real-time hybrid simulation (RTHS) often requires specialized finite element programs for computational modeling of the analytical substructures. Considering the limited nonlinear modeling capacity or the increasing computation cost for complex modeling, surrogate models of the analytical substructure provide novel alternatives for RTHS to avoid finite element analysis with fast computation. This study explores the use of arbitrary polynomial chaos expansion (APC) and nonlinear autoregressive with exogenous input (NARX) model to emulate the dynamic behavior of analytical substructures in RTHS. The NARX model training can be conducted numerically in an off-line mode using existing general purpose finite element analysis software, and its implementation presents minimum computational demands on the RTHS equipment. RTHS of a single-degree-of-freedom structure with a self-centering viscous damper is conducted as proof of concept to experimentally demonstrate the effectiveness and superiority of the proposed APC-NARX-based approach. The APC is further compared with other metamodeling techniques including polynomial chaos expansion (PCE) and Kriging to surrogate NARX model coefficients to account for ground motion uncertainties in RTHS. It is demonstrated that APC-NARX modeling with optimal order enables better accuracy of RTHS results than those of Kriging- and PCE-NARX.

Verification of polynomial chaos surrogates in the framework of structural vibrations with uncertainties

Surface response models, such as polynomial chaos Expansion, are commonly used to deal with the case of uncertain input parameters. Such models are only surrogates, so it is necessary to develop tools to assess the level of error between the reference solution (unknown in general), and the value provided by the surrogate. This is called a posteriori model verification. In most works, people usually search for the mean quadratic error between the reference problem and the surrogate. They use statistical approaches such as resampling or cross-fold validation, residual based approaches, or properties of the surrogate such as variance decay. Here, we propose a new approach for the specific framework of structural vibrations. Our proposition consists of a residual-based approach combined with a polynomial chaos expansion to evaluate the error as a full random variable, not only its mean square. We propose different variants for evaluating the error. Simple polynomial interpolation gives good results, but introducing a modal basis makes it possible to obtain the error with good accuracy and very low cost.

Polynomial chaos surrogate and bayesian learning for coupled hydro-mechanical behavior of soil slope

As rainfall infiltrates into soil slopes, the hydraulic and mechanical behaviors of soils are interacted. In this study, an efficient probabilistic parameter estimation method for coupled hydro-mechanical behavior in soil slope is proposed. This method integrates the Polynomial Chaos Expansion (PCE) method, the coupled hydro-mechanical modeling, and the Bayesian learning method. A coupled hydro-mechanical numerical model is established for the simulation of behaviors of unsaturated soil slope under rainfall infiltration, following by training a cheap-to-run PCE surrogate to replace it. Probabilistic estimation of soil parameters is conducted based on the Bayesian learning technique with the Markov Chain Monte Carlo (MCMC) simulation. A numerical example of an unsaturated slope under rainfall infiltration is presented to illustrate the proposed method. The effects of measurement durations and response types on parameter estimation are addressed. The result shows that with the increase of measurement duration, the uncertainties of soil parameters are significantly reduced. The uncertainties of hydraulic properties are reduced significantly using the pore water pressure data, while the uncertainties of soil strength parameters are reduced greatly using the measured displacement data.

Quantitative uncertainty estimation in biophysical models of fish larval connectivity in the Florida Keys

Abstract The impacts of seven uncertain biological parameters on simulated larval connectivity in the Florida Keys are investigated using Polynomial chaos surrogates. These parameters describe biological traits and behaviours—such as mortality, swimming abilities, and orientation—and modulate larval settlement as well as dispersal forecasts. However, these parameters are poorly constrained by observations and vary naturally between individual larvae. The present investigation characterizes these input uncertainties with probability density functions informed by previous studies of Abudefduf saxatilis. The parametric domain is sampled via ensemble calculations, then a polynomial-based surrogate is built to explicitly approximate the dependence of the model outputs on the uncertain model inputs, which enables a robust statistical analysis of uncertainties. This approach allows the computation of probabilistic dispersal kernels that are further analyzed to understand the impact of the parameter uncertainties. We find that the biological input parameters influence the connectivity differently depending on dispersal distance and release location. The global sensitivity analysis shows that the interactions between detection distance threshold, orientation ontogeny, and orientation accuracy, are the dominant contributors to the uncertainty in settlement abundance in the Florida Keys. Uncertainties in swimming speed and mortality, on the other hand, seem to contribute little to dispersal uncertainty.

Polynomial surrogates for Bayesian traveltime tomography

This paper tackles the issue of the computational load encountered in seismic imaging by Bayesian traveltime inversion. In Bayesian inference, the exploration of the posterior distribution of the velocity model parameters requires extensive sampling. The computational cost of this sampling step can be prohibitive when the first arrival traveltime prediction involves the resolution of an expensive number of forward models based on the eikonal equation. We propose to rely on polynomial chaos surrogates of the traveltimes between sources and receivers to alleviate the computational burden of solving the eikonal equation during the sampling stage. In an offline stage, the approach builds a functional approximation of the traveltimes from a set of solutions of the eikonal equation corresponding to a few values of the velocity model parameters selected in their prior range. These surrogates then substitute the eikonal-based predictions in the posterior evaluation, enabling very efficient extensive sampling of Bayesian posterior, for instance, by a Markov Chain Monte Carlo algorithm. We demonstrate the potential of the approach using numerical experiments on the inference of two-dimensional domains with layered velocity models and different acquisition geometries (microseismic and seismic refraction contexts). The results show that, in our experiments, the number of eikonal model evaluations required to construct accurate surrogates of the traveltimes is low. Further, an accurate and complete characterization of the posterior distribution of the velocity model is possible, thanks to the generation of large sample sets at a low cost. Finally, we discuss the extension of the current approach to more realistic velocity models and operational situations.

Bayesian calibration of order and diffusivity parameters in a fractional diffusion equation

This work focuses on parameter calibration of a variable-diffusivity fractional diffusion model. A random, spatially-varying diffusivity field with log-normal distribution is considered. The variance and correlation length of the diffusivity field are considered uncertain parameters, and the order of the fractional sub-diffusion operator is also taken uncertain and uniformly distributed in the range . A Karhunen-Loève (KL) decomposition of the random diffusivity field is used, leading to a stochastic problem defined in terms of a finite number of canonical random variables. Polynomial chaos (PC) techniques are used to express the dependence of the stochastic solution on these random variables. A non-intrusive methodology is used, and a deterministic finite-difference solver of the fractional diffusion model is utilized for this purpose. The PC surrogates are first used to assess the sensitivity of quantities of interest (QoIs) to uncertain inputs and to examine their statistics. In particular, the analysis indicates that the fractional order has a dominant effect on the variance of the QoIs considered, followed by the leading KL modes. The PC surrogates are further exploited to calibrate the uncertain parameters using a Bayesian methodology. Different setups are considered, including distributed and localized forcing functions and data consisting of either noisy observations of the solution or its first moments. In the broad range of parameters addressed, the analysis shows that the uncertain parameters having a significant impact on the variance of the solution can be reliably inferred, even from limited observations.

Yield-Constrained Optimization Design Using Polynomial Chaos for Microwave Filters

Yield optimization aims at finding microwave filter designs with high yield under fabrication tolerance. The electromagnetic (EM) simulation-based yield optimization methods are computationally expensive because a large number of EM simulations is required. Moreover, the microwave filter design usually requires several performance objectives to be met, which is not considered by the current yield optimization methods for microwave filters. In this paper, an efficient yield-constrained optimization using polynomial chaos surrogates (YCOPCS) is employed for microwave filters considering multiple objectives. In the YCOPCS method, the low-cost and high-accuracy of polynomial chaos is used as a surrogate. An efficient yield-constrained design framework is implemented to obtain the optimal design solution. Two numerical examples demonstrate the performance of the YCOPCS method, including a coupling matrix model of a fourth-order filter with cascaded quadruplet topology and an EM simulation model of a microwave waveguide bandpass filter. The numerical results show that the YCOPCS method can obtain the filter designs with higher yield and reduce EM simulations by 80% compared to Monte Carlo-based yield optimization in all testing examples.

Efficient Bayesian inversion for shape reconstruction of lithography masks

Background: Scatterometry is a fast, indirect and non-destructive optical method for quality control in the production of lithography masks. To solve the inverse problem in compliance with the upcoming need for improved accuracy, a computationally expensive forward model has to be defined which maps geometry parameters to diffracted light intensities. Aim: To quantify the uncertainties in the reconstruction of the geometry parameters, a fast to evaluate surrogate for the forward model has to be introduced. Approach: We use a non-intrusive polynomial chaos based approximation of the forward model which increases speed and thus enables the exploration of the posterior through direct Bayesian inference. Additionally, this surrogate allows for a global sensitivity analysis at no additional computational overhead. Results: This approach yields information about the complete distribution of the geometry parameters of a silicon line grating, which in return allows to quantify the reconstruction uncertainties in the form of means, variances and higher order moments of the parameters. Conclusion: The use of a polynomial chaos surrogate allows to quantify both parameter influences and reconstruction uncertainties. This approach is easy to use since no adaptation of the expensive forward model is required.

A Data-Driven Nonparametric Approach for Probabilistic Load-Margin Assessment Considering Wind Power Penetration

A modern power system is characterized by an increasing penetration of wind power, which results in large uncertainties in its states. These uncertainties must be quantified properly; otherwise, the system security may be threatened. Facing this challenge, we propose a cost-effective, data-driven approach to assessing a power system's load margin probabilistically. Using actual wind data, a kernel density estimator is applied to infer the nonparametric wind speed distributions, which are further merged into the framework of a vine copula. The latter enables us to simulate complex multivariate and highly dependent model inputs with a variety of bivariate copulae that precisely represent the tail dependence in the correlated samples. Furthermore, to reduce the prohibitive computational time of traditional Monte-Carlo simulations that process a large amount of samples, we propose to use a nonparametric, Gaussian-process-emulator-based reduced-order model to replace the original complicated continuation power-flow model through a Bayesian-learning framework. To accelerate the convergence rate of this Bayesian algorithm, a truncated polynomial chaos surrogate, which serves as a highly efficient, parametric Bayesian prior, is developed. This emulator allows us to execute the time-consuming continuation power-flow solver at the sampled values with a negligible computational cost. Results of simulations that are performed on several test systems reveal the impressive performance of the proposed method in the probabilistic load-margin assessment.

Leveraging Adapted Polynomial Chaos Metamodels for Real-Time Bayesian Updating

Abstract As engineered systems face an ever-increasing array of threats, the ability to perform real-time predictions and decision-making is central for coping with potential menaces. For thorough real-time predictions, uncertainty measures need to be incorporated, albeit conventional uncertainty quantification (UQ) efforts are often time-consuming and thus go against the essence of real-time decision-making. A fast, metamodel-based Bayesian updating scheme within a decision-making under uncertainty framework is proposed. The latter combines data harnessing, assimilation, and parameter updating in an online setting. For fast parameter updating in the online phase, the adaptation property of polynomial chaos (PC) metamodels is exploited. In the offline phase, a global Gaussian approximation of the response surface is obtained through a first-order PC surrogate, from which, a rotated basis of the underlying Gaussian Hilbert space is identified. In the subsequent online phase, a cheap, locally adapted solution embedded in a lower dimensional subspace is leveraged within a Bayesian updating setting to perform fast data assimilation. Incorporating uncertainty into real-time decision-making paves the way for higher confidence in online predictions, increased system resilience, and adaptivity to surroundings. The aforementioned fast parameter updating scheme based on adapted PC metamodels is evaluated on a set of problems with varying complexity.

Adaptive multi-fidelity polynomial chaos approach to Bayesian inference in inverse problems

The polynomial chaos (PC) expansion has been widely used as a surrogate model in the Bayesian inference to speed up the Markov chain Monte Carlo (MCMC) calculations. However, the use of a PC surrogate introduces the modeling error, that may severely distort the estimate of the posterior distribution. This error can be corrected by increasing the order of the PC expansion, but the cost for building the surrogate may increase dramatically. In this work, we seek to address this challenge by proposing an adaptive procedure to construct a multi-fidelity PC surrogate. This new strategy combines (a large number of) low-fidelity surrogate model evaluations and (a small number of) high-fidelity model evaluations, yielding a locally adaptive multi-fidelity approach. Here the low-fidelity surrogate is chosen as the prior-based PC surrogate, while the high-fidelity model refers to the true forward model. The key idea is to construct and refine the multi-fidelity approach over a sequence of samples adaptively determined from data so that the approximation can eventually concentrate on the posterior distribution. We illustrate the performance of the proposed strategy through two nonlinear inverse problems. It is shown that the proposed adaptive multi-fidelity approach can improve significantly the accuracy, yet without a dramatic increase in computational complexity. The numerical results also indicate that our new algorithm can enhance the efficiency by several orders of magnitude compared to a standard MCMC approach using only the true forward model.

Nonintrusive Global Sensitivity Analysis for Linear Systems With Process Noise

The focus of this paper is on the global sensitivity analysis (GSA) of linear systems with time-invariant model parameter uncertainties and driven by stochastic inputs. The Sobol' indices of the evolving mean and variance estimates of states are used to assess the impact of the time-invariant uncertain model parameters and the statistics of the stochastic input on the uncertainty of the output. Numerical results on two benchmark problems help illustrate that it is conceivable that parameters, which are not so significant in contributing to the uncertainty of the mean, can be extremely significant in contributing to the uncertainty of the variances. The paper uses a polynomial chaos (PC) approach to synthesize a surrogate probabilistic model of the stochastic system after using Lagrange interpolation polynomials (LIPs) as PC bases. The Sobol' indices are then directly evaluated from the PC coefficients. Although this concept is not new, a novel interpretation of stochastic collocation-based PC and intrusive PC is presented where they are shown to represent identical probabilistic models when the system under consideration is linear. This result now permits treating linear models as black boxes to develop intrusive PC surrogates.

AN ADAPTIVE MULTIFIDELITY PC-BASED ENSEMBLE KALMAN INVERSION FOR INVERSE PROBLEMS

The ensemble Kalman inversion (EKI), as a derivative-free methodology, has been widely used in the parameter estimation of inverse problems. Unfortunately, its cost may become moderately large for systems described by high dimensional nonlinear PDEs, as EKI requires a relatively large ensemble size to guarantee its performance. In this paper, we propose an adaptive multi-fidelity polynomial chaos (PC) based EKI technique to address this challenge. Our new strategy combines a large number of low-order PC surrogate model evaluations and a small number of high-fidelity forward model evaluations, yielding a multi-fidelity approach. Especially, we present a new approach that adaptively constructs and refines a multi-fidelity PC surrogate during the EKI simulation. Since the forward model evaluations are only required for updating the low-order multi-fidelity PC model, whose number can be much smaller than the total ensemble size of the classic EKI, the entire computational costs are thus significantly reduced. The new algorithm was tested through the two-dimensional time fractional inverse diffusion problems and demonstrated great effectiveness in comparison with PC based EKI and classic EKI.

A Hybrid Methodology for Jitter and Eye Estimation in High-Speed Serial Channels Using Polynomial Chaos Surrogate Models

Estimation of jitter and eye diagram in high-speed serial channels can be challenging. The existing methods might fail to show inter-symbol interference (ISI) and data dependent jitter because they are either excessively time consuming or only applicable to linear time-invariant (LTI) systems. Therefore, this paper suggests a hybrid methodology to develop surrogate models of the system from a short transient simulation and using them to efficiently determine the eye diagram and related parameters. This paper is based on uncertainty quantification techniques that use polynomial chaos (PC) expansions to estimate the output. Necessary modifications are applied to PC modeling to make it applicable to high-speed channels. Finally, the performance of the proposed approach is shown through numerical examples.

Uncertainty Quantification for Subsurface Flow and Transport: Coping With Nonlinearity/Irregularity via Polynomial Chaos Surrogate and Machine Learning

AbstractSubsurface flow and transport problems usually involve some degree of uncertainty. Polynomial chaos expansion can be used as surrogate of physical models for uncertainty quantification. However, a global model can hardly be found for model responses with strong nonlinearity or irregularity. In this study, we propose a novel approach by use of the classification method in machine learning, that is, supported vector machine, to cope with such nonlinearity/irregularity. Piecewise surrogate models are constructed in relatively smooth subdomains separated by the supported vector machine hyperplanes. We demonstrate the effectiveness of using the trained piecewise surrogate model in solute transport and two‐phase flow problems in homogeneous and heterogeneous porous media. The numerical results are compared with standard global polynomial chaos expansion results and the Monte Carlo benchmark. The proposed nonintrusive approach is able to accurately quantify uncertainty, with much smaller computational efforts.

Verifying and assessing the performance of the perturbation strategy in polynomial chaos ensemble forecasts of the circulation in the Gulf of Mexico

We present an analysis of two recent efforts aimed at quantifying the uncertainties in a 30-day HYbrid Coordinate Ocean Model forecast of the circulation in the Gulf of Mexico, with particular emphasis on the separation of Loop Current Eddy Franklin, using Polynomial Chaos methods. The analysis herein explores whether the model perturbations lead to realistic representation of the uncertainty in the Gulf Circulation. Comparisons of model output with Sea Surface Height and current mooring data show that the observational data generally falls within the envelope of the ensemble and that the modal decomposition delivers “realistic” perturbations in the Loop Current region. We use information theory metrics to quantify the information gain and the computational trade-offs between different wind and initial conditions perturbation modes. The relative entropy measures indicate that two modes for initial condition perturbations are enough, in our model configuration, to represent the uncertainty in the Loop Current region; while two modes for wind forcing perturbations are necessary in order to estimate the uncertainty in the coastal zone. The ensemble statistics are then explored using the Polynomial Chaos surrogate and the newly developed contour boxplot methods.

Uncertainty quantification in non-equilibrium molecular dynamics simulations of thermal transport

Bulk thermal conductivity estimates based on predictions from non-equilibrium molecular dynamics (NEMD) using the so-called direct method are known to be severely under-predicted since finite simulation length-scales are unable to mimic bulk transport. Moreover, subjecting the system to a temperature gradient by means of thermostatting tends to impact phonon transport adversely. Additionally, NEMD predictions are tightly coupled with the choice of the inter-atomic potential and the underlying values associated with its parameters. In the case of silicon (Si), nominal estimates of the Stillinger-Weber (SW) potential parameters are largely based on a constrained regression approach aimed at agreement with experimental data while ensuring structural stability. However, this approach has its shortcomings and it may not be ideal to use the same set of parameters to study a wide variety of Si-based systems subjected to different thermodynamic conditions. In this study, NEMD simulations are performed on a Si bar to investigate the impact of bar-length, and the applied temperature gradient on the discrepancy between predictions and the available measurement for bulk thermal conductivity at 300 K by constructing statistical response surfaces at different temperatures. The approach helps quantify the discrepancy, observed to be largely dependent on the system-size, with minimal computational effort. A computationally efficient approach based on derivative-based sensitivity measures to construct a reduced-order polynomial chaos surrogate for NEMD predictions is also presented. The surrogate is used to perform parametric sensitivity analysis, forward propagation of the uncertainty, and calibration of the important SW potential parameters in a Bayesian setting. It is found that only two (out of seven) parameters contribute significantly to the uncertainty in bulk thermal conductivity estimates for Si.

A near-optimal sampling strategy for sparse recovery of polynomial chaos expansions

Compressive sampling has become a widely used approach to construct polynomial chaos surrogates when the number of available simulation samples is limited. Originally, these expensive simulation samples would be obtained at random locations in the parameter space. It was later shown that the choice of sample locations could significantly impact the accuracy of resulting surrogates. This motivated new sampling strategies or design-of-experiment approaches, such as coherence-optimal sampling, which aim at improving the coherence property. In this paper, we propose a sampling strategy that can identify near-optimal sample locations that lead to improvement in local-coherence property and also enhancement of cross-correlation properties of measurement matrices. We provide theoretical motivations for the proposed sampling strategy along with several numerical examples that show that our near-optimal sampling strategy produces substantially more accurate results, compared to other sampling strategies.

The Impact of Parametric Uncertainties on Biogeochemistry in the E3SM Land Model

AbstractWe conduct a global sensitivity analysis (GSA) of the Energy Exascale Earth System Model (E3SM), land model (ELM) to calculate the sensitivity of five key carbon cycle outputs to 68 model parameters. This GSA is conducted by first constructing a Polynomial Chaos (PC) surrogate via new Weighted Iterative Bayesian Compressive Sensing (WIBCS) algorithm for adaptive basis growth leading to a sparse, high‐dimensional PC surrogate with 3,000 model evaluations. The PC surrogate allows efficient extraction of GSA information leading to further dimensionality reduction. The GSA is performed at 96 FLUXNET sites covering multiple plant functional types (PFTs) and climate conditions. About 20 of the model parameters are identified as sensitive with the rest being relatively insensitive across all outputs and PFTs. These sensitivities are dependent on PFT, and are relatively consistent among sites within the same PFT. The five model outputs have a majority of their highly sensitive parameters in common. A common subset of sensitive parameters is also shared among PFTs, but some parameters are specific to certain types (e.g., deciduous phenology). The relative importance of these parameters shifts significantly among PFTs and with climatic variables such as mean annual temperature.

Design analysis for optimal calibration of diffusivity in reactive multilayers

Calibration of the uncertain Arrhenius diffusion parameters for quantifying mixing rates in Zr–Al nanolaminate foils have been previously performed in a Bayesian setting [M. Vohra, J. Winokur, K.R. Overdeep, P. Marcello, T.P. Weihs, and O.M. Knio, Development of a reduced model of formation reactions in Zr–Al nanolaminates, J. Appl. Phys. 116(23) (2014): Article No. 233501]. The parameters were inferred in a low-temperature, homogeneous ignition regime, and a high-temperature self-propagating reaction regime. In this work, we extend the analysis to determine optimal experimental designs that would provide the best data for inference. We employ a rigorous framework that quantifies the expected information gain in an experiment, and find the optimal design conditions using Monte Carlo techniques, sparse quadrature, and polynomial chaos surrogates. For the low-temperature regime, we find the optimal foil heating rate and pulse duration, and confirm through simulation that the optimal design indeed leads to sharp posterior distributions of the diffusion parameters. For the high-temperature regime, we demonstrate the potential for increasing the expected information gain concerning the posteriors by increasing the sample size and reducing the uncertainty in measurements. Moreover, posterior marginals are also obtained to verify favourable experimental scenarios.