Hyper-differential sensitivity analysis with respect to model discrepancy: mathematics and computation

Model discrepancy (the difference between model predictions and reality) is ubiquitous in computational models for physical systems. It is common to derive partial differential equations (PDEs) from first principles physics, but make simplifying assumptions to produce tractable expressions for the governing equations or closure models. These PDEs are then used for analysis and design to achieve desirable performance. The end goal in many such cases is solving a PDE-constrained optimization (PDECO) problem. This article considers the sensitivity of PDECO problems with respect to model discrepancy. We introduce a general representation of discrepancy and apply post-optimality sensitivity analysis to derive an expression for the sensitivity of the optimal solution with respect to it. An efficient algorithm is presented which combines the PDE discretization, post-optimality sensitivity operator, adjoint-based derivatives, and a randomized generalized singular value decomposition to enable scalable computation of the sensitivity of the optimal solution with respect to model discrepancy. Kronecker product structure in the underlying linear algebra and infrastructure investment in PDECO is exploited to yield a general purpose algorithm which is computationally efficient and portable across applications. Known physics and problem specific characteristics of discrepancy are imposed softly through user specified weighting matrices. We demonstrate our proposed framework on two nonlinear PDECO problems to highlight its computational efficiency and rich insight.