Hyper-differential sensitivity analysis with respect to model discrepancy: Posterior Optimal Solution Sampling
Joseph Hart, Bart van Bloemen Waanders
TL;DR
The paper develops a Bayesian extension of hyper-differential sensitivity analysis to quantify and propagate uncertainty when updating a low-fidelity optimization with limited high-fidelity evaluations. By modeling the model discrepancy δ(z, θ) in an affine, Kronecker-structured form and using a Gaussian prior and likelihood, the authors derive closed-form expressions for posterior samples of the discrepancy and the updated optimal solution, enabling efficient computation even for high-dimensional problems. A Hessian-projection step focuses inference on directions with the greatest impact on the objective, improving robustness and reducing computational cost. The approach is demonstrated on three numerical examples (diffusion–reaction, mass–spring, and advection–diffusion) showing significant improvements over the low-fidelity solution with only a handful of high-fidelity evaluations. This framework offers a principled, scalable method to fuse low-fidelity models, high-fidelity data, and optimization objectives in PDE-constrained settings, with broad implications for design and control under uncertainty.
Abstract
Optimization constrained by high-fidelity computational models has potential for transformative impact. However, such optimization is frequently unattainable in practice due to the complexity and computational intensity of the model. An alternative is to optimize a low-fidelity model and use limited evaluations of the high-fidelity model to assess the quality of the solution. This article develops a framework to use limited high-fidelity simulations to update the optimization solution computed using the low-fidelity model. Building off a previous article [22], which introduced hyper-differential sensitivity analysis with respect to model discrepancy, this article provides novel extensions of the algorithm to enable uncertainty quantification of the optimal solution update via a Bayesian framework. Specifically, we formulate a Bayesian inverse problem to estimate the model discrepancy and propagate the posterior model discrepancy distribution through the post-optimality sensitivity operator for the low-fidelity optimization problem. We provide a rigorous treatment of the Bayesian formulation, a computationally efficient algorithm to compute posterior samples, a guide to specify and interpret the algorithm hyper-parameters, and a demonstration of the approach on three examples which highlight various types of discrepancy between low and high-fidelity models.