Initial-Condition Inverse Problems

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 di-mension. 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