We present our recent efforts towards uncertainty quantification for large-scale computational mechanics. The talk has three parts. In the first part,we present a reduce-then-sample approach to efficiently study the probabilistic response of turbomachinery flow due to random geometric variation of bladed disk. The governing equation is the Euler equation of gas dynamics, which is discretized using the discontinuous Galerkin method. We first propose a model-constrained adaptive sampling approach that explores the physics of the problem under consideration to build reduced-order models. Monte Carlo simulation is then performed using the cost-effective reduced model, which is three orders of magnitude less expensive than using the high-fidelity counterpart. We demonstrate the effectiveness of our approach in predicting the work per cycle with quantifiable uncertainty.
In the second part of the talk, we consider the shape inverse problem of electromagnetic scattering, which finds applications in nation security and scientific exploration. We address this large-scale inverse problems in a Bayesian inference framework to systematically account for all types of randomness: given noise-corrupted data, a computational model (discontinuous Galerkin discretization of the Maxwell equation), and prior information, find the posterior pdf describing the shape parameter uncertainty. Since exploring the Bayesian posterior is intractable for high dimensional parameter space and/or expensive computational model, we propose a Hessian-informed adaptive Gaussian process response surface to approximate the posterior. The Monte Carlo simulation task, which is impossible using the original posterior, is then carried out on the approximate posterior to predict the shape and its associated uncertainty with negligible cost.
In the last part of the talk, we address the problem of solving globally large-scale seismic inversion governed by the linear elasticity equation. We begin by putting our inverse problem into an infinite dimensional Bayesian framework, which is then linearized and discretized using an hp-nonconforming discontinuous Galerkin (DG) spectral element method. Next, consistency, stability, and hp-convergence of the DG method as well as its scalability up to 262 thousand cores are presented. Finally, we discuss a mesh-independent uncertainty quantification method for full wave form seismic inversion exploiting the compactness of the misfit Hessian and low rank approximation.