2019/09/16 by Joseph Hart, Hart, Joseph, Bart van Bloemen Waanders +3 · 1 citation
Computer Science · Decision Sciences · Physics and Astronomy · #Advanced Multi-Objective Optimization Algorithms #FOS: Mathematics #Model Reduction and Neural Networks #Optimization and Control (math.OC) #Probabilistic and Robust Engineering Design
paper · pdf · doi:10.48550/arxiv.1909.07336
openalex publication_date 2019/09/16 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
Many problems in engineering and sciences require the solution of large scale\noptimization constrained by partial differential equations (PDEs). Though\nPDE-constrained optimization is itself challenging, most applications pose\nadditional complexity, namely, uncertain parameters in the PDEs. Uncertainty\nquantification (UQ) is necessary to characterize, prioritize, and study the\ninfluence of these uncertain parameters. Sensitivity analysis, a classical tool\nin UQ, is frequently used to study the sensitivity of a model to uncertain\nparameters. In this article, we introduce "hyper-differential sensitivity\nanalysis" which considers the sensitivity of the solution of a PDE-constrained\noptimization problem to uncertain parameters. Our approach is a goal-oriented\nanalysis which may be viewed as a tool to complement other UQ methods in the\nservice of decision making and robust design. We formally define\nhyper-differential sensitivity indices and highlight their relationship to the\nexisting optimization and sensitivity analysis literatures. Assuming the\npresence of low rank structure in the parameter space, computational efficiency\nis achieved by leveraging a generalized singular value decomposition in\nconjunction with a randomized solver which converts the computational\nbottleneck of the algorithm into an embarrassingly parallel loop. Two\nmulti-physics examples, consisting of nonlinear steady state control and\ntransient linear inversion, demonstrate efficient identification of the\nuncertain parameters which have the greatest influence on the optimal solution.\n