Hessian-based model reduction with applications to initial-condition inverse problems

Abstract

(cont.) Reduced-order models that are able to approximate output quantities of interest of high-fidelity computational models over a wide range of input parameters play an important role in making tractable large-scale optimal design, optimal control, and inverse problem applications. We consider the problem of determining a reduced model of an initial value problem that spans all important initial conditions, and pose the task of determining appropriate training sets for reduced-basis construction as a sequence of optimization problems. We show that, under certain assumptions, these optimization problems have an explicit solution in the form of an eigenvalue problem, yielding an efficient Hessian based model reduction algorithm that scales well to systems with states of high dimension. Furthermore, tight upper bounds are given for the error in the outputs of the reduced models. The reduction methodology is demonstrated for several linear systems, including a large-scale contaminant transport problem. Models constructed with the Hessian-based approach are used to solve an initial condition inverse problem, and the resulting initial condition estimates compare favorably to those computed with high-fidelity models and low-rank approximations. Initial condition estimates are then formed with limited observational data to demonstrate that predictions of system state using reduced models are possible given relatively short measurement time windows. We show that reduced state can be used to approximate full state given an appropriate reduced basis, meaning that approximate forward simulations of large-scale systems can be computed in reduced space.