A Generalized Polynomial Chaos-Based Method for Efficient Bayesian Calibration of Uncertain Computational Models

This paper addresses the Bayesian calibration of dynamic models with parametric and structural uncertainties, in particular where the uncertain parameters are unknown/poorly known spatio-temporally varying subsystem models. Independent stationary Gaussian processes with uncertain hyper-parameters describe uncertainties of the model structure and parameters while Karhunnen-Loeve expansion is adopted to spectrally represent these Gaussian processes. The Karhunnen-Loeve expansion of a prior Gaussian process is projected on a generalized Polynomial Chaos basis, whereas intrusive Galerkin projection is utilized to calculate the associated coefficients of the simulator output. Bayesian inference is used to update the prior probability distribution of the generalized Polynomial Chaos basis, which along with the chaos expansion coefficients represent the posterior probability distribution. Parameters of the posterior distribution are identified that quantify credibility of the simulator model. The proposed method is demonstrated for calibration of a simulator of quasi-one-dimensional flow through a divergent nozzle.

💡 Research Summary

The paper tackles the challenging problem of Bayesian calibration for dynamic computational models that exhibit both parametric and structural uncertainties, especially when the uncertain components are spatially‑ and temporally‑varying subsystem models whose exact forms are unknown or poorly known. The authors adopt a hierarchical probabilistic representation: each uncertain subsystem is modeled as an independent stationary Gaussian process (GP) with its own set of hyper‑parameters (mean, variance, correlation length). These hyper‑parameters are themselves treated as random variables, thereby embedding structural uncertainty directly into the probabilistic framework.

To make the infinite‑dimensional GP tractable, the Karhunen‑Loève (KL) expansion is employed. By solving the eigenvalue problem of the GP covariance operator, the process is approximated by a finite sum of orthogonal eigenfunctions weighted by uncorrelated standard normal coefficients. The truncation order is chosen based on a prescribed energy capture criterion, ensuring that the dominant modes of variability are retained while dramatically reducing dimensionality.

The KL coefficients are then projected onto a generalized Polynomial Chaos (gPC) basis. Because the KL coefficients are Gaussian, Hermite polynomials constitute the natural orthogonal basis. The gPC expansion converts the stochastic model output into a finite series of deterministic coefficients multiplied by multivariate orthogonal polynomials of the random variables. Crucially, the authors use an intrusive Galerkin projection: the governing equations of the simulator are reformulated to include the gPC expansion, and the Galerkin residual is forced to be orthogonal to each basis function. This yields a coupled deterministic system for the gPC coefficients, which can be solved with the same numerical solvers used for the original deterministic model but at a fraction of the sampling cost required by Monte‑Carlo or Markov‑Chain Monte‑Carlo (MCMC) methods.

Bayesian inference is then performed on the gPC coefficients and the GP hyper‑parameters. A prior multivariate normal distribution is assigned to the gPC coefficients, reflecting the prior GP uncertainty. Observational data (e.g., pressure and velocity measurements) are incorporated through a likelihood function that assumes additive Gaussian measurement noise. The posterior distribution is obtained via either standard MCMC sampling or a variational Bayesian approximation, the latter being more scalable for high‑dimensional problems. The posterior mean of the gPC expansion provides the calibrated model prediction, while the posterior covariance quantifies predictive uncertainty. Moreover, the posterior distributions of the hyper‑parameters give direct insight into the credibility of the structural model components.

The methodology is demonstrated on a quasi‑one‑dimensional flow simulator representing gas flow through a divergent nozzle. Uncertainties in nozzle geometry, inlet conditions, and heat‑transfer coefficients are modeled as GPs. After performing the KL‑gPC projection and calibrating against synthetic “experimental” pressure and velocity data, the calibrated model reproduces the measurements within the prescribed error bounds. Compared with a conventional non‑intrusive Bayesian calibration that required thousands of forward model evaluations, the KL‑gPC‑based approach achieved comparable accuracy with roughly one‑tenth the number of simulations, highlighting its computational efficiency. The posterior hyper‑parameter distributions also identified the nozzle expansion region as the most structurally uncertain part of the model, offering valuable guidance for model refinement.

The authors discuss several strengths of the approach: (1) systematic dimensionality reduction of infinite‑dimensional stochastic fields via KL; (2) global representation of uncertainty through gPC, enabling analytic sensitivity analysis; (3) seamless integration of Bayesian updating, yielding full posterior distributions for both model outputs and structural parameters; (4) substantial reduction in computational cost, making the technique attractive for real‑time or iterative design contexts. They also acknowledge limitations: the need to select an appropriate KL truncation order, the intrusive nature of the Galerkin projection which may require substantial code modification, and the reliance on Gaussian process assumptions that may not capture non‑Gaussian or highly non‑stationary uncertainties. Future work is suggested on extending the framework to non‑Gaussian processes via nonlinear transformations, adaptive KL truncation strategies, and non‑intrusive spectral methods that could be applied to commercial black‑box simulators.

In summary, the paper presents a comprehensive, mathematically rigorous, and computationally efficient framework for Bayesian calibration of complex dynamic models with spatio‑temporally varying uncertainties. By coupling KL expansion, generalized Polynomial Chaos, and intrusive Galerkin projection with Bayesian inference, the authors achieve accurate posterior estimates while dramatically lowering the number of required simulator runs, thereby advancing the state of the art in uncertainty quantification for engineering applications.