Polynomial Chaos Surrogate Research Articles

Polynomial Chaos–Based Bayesian Inference of K-Profile Parameterization in a General Circulation Model of the Tropical Pacific

Abstract The authors present a polynomial chaos (PC)–based Bayesian inference method for quantifying the uncertainties of the K-profile parameterization (KPP) within the MIT general circulation model (MITgcm) of the tropical Pacific. The inference of the uncertain parameters is based on a Markov chain Monte Carlo (MCMC) scheme that utilizes a newly formulated test statistic taking into account the different components representing the structures of turbulent mixing on both daily and seasonal time scales in addition to the data quality, and filters for the effects of parameter perturbations over those as a result of changes in the wind. To avoid the prohibitive computational cost of integrating the MITgcm model at each MCMC iteration, a surrogate model for the test statistic using the PC method is built. Because of the noise in the model predictions, a basis-pursuit-denoising (BPDN) compressed sensing approach is employed to determine the PC coefficients of a representative surrogate model. The PC surrogate is then used to evaluate the test statistic in the MCMC step for sampling the posterior of the uncertain parameters. Results of the posteriors indicate good agreement with the default values for two parameters of the KPP model, namely the critical bulk and gradient Richardson numbers; while the posteriors of the remaining parameters were barely informative.

Uncertainty quantification and global sensitivity analysis of complex chemical processes with a large number of input parameters using compressive polynomial chaos

Uncertainties are ubiquitous and unavoidable in process design and modeling while they can significantly affect safety, reliability, and economic decisions. The large number of uncertainties in complex chemical processes make the well-known Monte-Carlo and polynomial chaos approaches for uncertainty quantification computationally expensive and even infeasible. This study focused on the uncertainty quantification and sensitivity analysis of complex chemical processes with a large number of uncertainties. An efficient method was proposed using a compressed sensing technique to overcome the computational limitations for complex and large scale systems. In the proposed method, compressive sparse polynomial chaos surrogates were constructed and applied to quantify the uncertainties and reflect their propagation effect on process design. Rigorous case studies were provided by the interface between MATLAB™ and Aspen HYSYS™ for a propylene glycol production process and lean dry gas processing plant. The proposed methodology was compared with traditional Monte-Carlo/Quasi Monte-Carlo sampling-based and standard polynomial chaos approaches to highlight its advantages in terms of computational efficiency. The proposed approach could mitigate the simulation costs significantly using an accurate, efficient-to-evaluate polynomial chaos that can be used in place of expensive simulations. In addition, the global sensitivity indices, which show the relative importance of uncertain inputs on the process output, could be derived analytically from the obtained polynomial chaos surrogate model.

Single-site Lennard-Jones models via polynomial chaos surrogates of Monte Carlo molecular simulation.

In this work, two Polynomial Chaos (PC) surrogates were generated to reproduce Monte Carlo (MC) molecular simulation results of the canonical (single-phase) and the NVT-Gibbs (two-phase) ensembles for a system of normalized structureless Lennard-Jones (LJ) particles. The main advantage of such surrogates, once generated, is the capability of accurately computing the needed thermodynamic quantities in a few seconds, thus efficiently replacing the computationally expensive MC molecular simulations. Benefiting from the tremendous computational time reduction, the PC surrogates were used to conduct large-scale optimization in order to propose single-site LJ models for several simple molecules. Experimental data, a set of supercritical isotherms, and part of the two-phase envelope, of several pure components were used for tuning the LJ parameters (ε, σ). Based on the conducted optimization, excellent fit was obtained for different noble gases (Ar, Kr, and Xe) and other small molecules (CH4, N2, and CO). On the other hand, due to the simplicity of the LJ model used, dramatic deviations between simulation and experimental data were observed, especially in the two-phase region, for more complex molecules such as CO2 and C2 H6.

Uncertainty quantification and inference of Manning’s friction coefficients using DART buoy data during the Tōhoku tsunami

Tsunami computational models are employed to explore multiple flooding scenarios and to predict water elevations. However, accurate estimation of water elevations requires accurate estimation of many model parameters including the Manning’s n friction parameterization. Our objective is to develop an efficient approach for the uncertainty quantification and inference of the Manning’s n coefficient which we characterize here by three different parameters set to be constant in the on-shore, near-shore and deep-water regions as defined using iso-baths. We use Polynomial Chaos (PC) to build an inexpensive surrogate for the GeoClaw model and employ Bayesian inference to estimate and quantify uncertainties related to relevant parameters using the DART buoy data collected during the Tōhoku tsunami. The surrogate model significantly reduces the computational burden of the Markov Chain Monte-Carlo (MCMC) sampling of the Bayesian inference. The PC surrogate is also used to perform a sensitivity analysis.

Efficient Bayesian inference of subsurface flow models using nested sampling and sparse polynomial chaos surrogates

An efficient Bayesian calibration method based on the nested sampling (NS) algorithm and non-intrusive polynomial chaos method is presented. Nested sampling is a Bayesian sampling algorithm that builds a discrete representation of the posterior distributions by iteratively re-focusing a set of samples to high likelihood regions. NS allows representing the posterior probability density function (PDF) with a smaller number of samples and reduces the curse of dimensionality effects. The main difficulty of the NS algorithm is in the constrained sampling step which is commonly performed using a random walk Markov Chain Monte-Carlo (MCMC) algorithm. In this work, we perform a two-stage sampling using a polynomial chaos response surface to filter out rejected samples in the Markov Chain Monte-Carlo method. The combined use of nested sampling and the two-stage MCMC based on approximate response surfaces provides significant computational gains in terms of the number of simulation runs. The proposed algorithm is applied for calibration and model selection of subsurface flow models.

A priori testing of sparse adaptive polynomial chaos expansions using an ocean general circulation model database

This work explores the implementation of an adaptive strategy to design sparse ensembles of oceanic simulations suitable for constructing polynomial chaos surrogates. We use a recently developed pseudo-spectral algorithm that is based on a direct application of the Smolyak sparse grid formula and that allows the use of arbitrary admissible sparse grids. The adaptive algorithm is tested using an existing simulation database of the oceanic response to Hurricane Ivan in the Gulf of Mexico. The a priori tests demonstrate that sparse and adaptive pseudo-spectral constructions lead to substantial savings over isotropic sparse sampling in the present setting.

Minimal multi-element stochastic collocation for uncertainty quantification of discontinuous functions

We propose a multi-element stochastic collocation method that can be applied in high-dimensional parameter space for functions with discontinuities lying along manifolds of general geometries. The key feature of the method is that the parameter space is decomposed into multiple elements defined by the discontinuities and thus only the minimal number of elements are utilized. On each of the resulting elements the function is smooth and can be approximated using high-order methods with fast convergence properties. The decomposition strategy is in direct contrast to the traditional multi-element approaches which define the sub-domains by repeated splitting of the axes in the parameter space. Such methods are more prone to the curse-of-dimensionality because of the fast growth of the number of elements caused by the axis based splitting. The present method is a two-step approach. Firstly a discontinuity detector is used to partition parameter space into disjoint elements in each of which the function is smooth. The detector uses an efficient combination of the high-order polynomial annihilation technique along with adaptive sparse grids, and this allows resolution of general discontinuities with a smaller number of points when the discontinuity manifold is low-dimensional. After partitioning, an adaptive technique based on the least orthogonal interpolant is used to construct a generalized Polynomial Chaos surrogate on each element. The adaptive technique reuses all information from the partitioning and is variance-suppressing. We present numerous numerical examples that illustrate the accuracy, efficiency, and generality of the method. When compared against standard locally-adaptive sparse grid methods, the present method uses many fewer number of collocation samples and is more accurate.