A Surrogate-Based Variance Reduction Approach to Multifidelity Uncertainty Quantification -- With Applications in Automotive Systems

Abstract

An increasing number of science and engineering applications demand highly efficient Uncertainty Quantification (UQ} capabilities in order to account for the presence of significant uncertainties that are often unavoidable in real-world operating conditions. For complex nonlinear systems, the task of quantifying the effect of uncertainties on system behaviors can pose major challenges as closed-form solutions to the stochastic nonlinear differential equations often do not exist. Sampling-based algorithms, most notably the Monte Carlo (MC) method, are considered the default approach when it comes to UQ for complex nonlinear systems. However, MC relies on repeated random sampling and simulations to obtain statistical estimations, which has very slow convergence. The resulting computational cost is often prohibitively high for complex nonlinear systems, hindering the adoption of such uncertainty analyses in many applications where computational resources are constrained.

This dissertation research investigates surrogate-based single-fidelity and multifidelity methods for highly efficient UQ. We first demonstrate the performance improvement obtained by the employment of generalized Polynomial Chaos (gPC) as a surrogate in the UQ of automotive propulsion system simulations. The adoption of gPC enables the exploitation of the smoothness property of the solution to the stochastic problem for significant computational cost reductions. Based upon this surrogate approach, we further develop a novel multifidelity variance reduction method for highly efficient forward UQ for systems with limited computational budgets. Our proposed method — Control Variate Polynomial Chaos (CVPC) — utilizes Control Variate (CV) as an information fusion framework to explore synergies between the flexible MC sampling and the efficient gPC expansion, while minimizing the drawbacks of both individual techniques. Specifically, a high-fidelity model of the system of interest is utilized in a MC sampling scheme to form the high-fidelity component of a CVPC estimator. A surrogate model obtained using a low-degree gPC is employed in the low-fidelity component of the CVPC estimator. The gPC coefficients are then fused with random samples of the polynomial bases to complete the low-fidelity component of the CVPC estimator. The same set of random samples are shared between the high- and low-fidelity components to generate correlations. Finally, the high- and low-fidelity components are combined through CV to achieve variance reductions. The primary theoretical contribution of this dissertation is the rigorous development of an estimator design strategy that optimally balances the computational effort needed to adapt a surrogate compared with sampling the original expensive nonlinear system. While previous works in the literature have similarly combined surrogates and sampling, to the author's best knowledge, this dissertation work is the first to provide rigorous analysis of estimator design. Numerical results from multiple application examples show that CVPC can produce significant, often by orders of magnitude, computational efficiency improvement over existing UQ methods.

The main contributions of this dissertation include: 1) application of gPC to efficient UQ and global sensitivity analysis for the online simulations of automotive propulsion systems; 2) development of theoretical foundations for the CVPC method; 3) development of the optimal estimator design theory for CVPC; 4) numerical applications of optimal CVPC. The results show that CVPC can provide significant, often by orders of magnitude, computational cost reductions over conventional UQ methods without scarifying accuracy.