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This paper introduces reduced order model (ROM) based Hessian approximations for use in inexact Newton methods for the solution of optimization problems implicitly constrained by a large-scale system, typically a discretization of a partial differential equation (PDE). The direct application of an inexact Newton method to this problem requires the solution of many PDEs per optimization iteration. To reduce the computational complexity, a ROM Hessian approximation is proposed. Since only the Hessian is approximated, but the original objective function and its gradient is used, the resulting inexact Newton method maintains the first-order global convergence property, under suitable assumptions. Thus even computationally inexpensive lower fidelity ROMs can be used, which is different from ROM approaches that replace the original optimization problem by a sequence of ROM optimization problem and typically need to accurately approximate function and gradient information of the original problem. In the proposed approach, the quality of the ROM Hessian approximation determines the rate of convergence, but not whether the method converges. The projection based ROM is constructed from state and adjoint snapshots, and is relatively inexpensive to compute. Numerical examples on semilinear parabolic optimal control problems demonstrate that the proposed approach can lead to substantial savings in terms of overall PDE solves required. 

The Loewner framework is extended to compute reduced order models (ROMs) for systems governed by the incompressible Navier-Stokes (NS) equations. For quadratic ordinary differential equations (ODEs) it constructs a ROM directly from measurements of transfer function components derived from an expansion of the system’s input-to-output map. Given measurements, no explicit access to the system is required to construct the ROM. To extend the Loewner framework, the NS equations are transformed into ODEs by projecting onto the subspace defined by the incompressibility condition. This projection is used theoretically, but avoided computationally. This paper presents the overall approach. Currently, transfer function measurements are obtained via computational simulations; obtaining them from experiments is an open issue. Numerical results show the potential of the Loewner framework, but also reveal possible lack of stability of the ROM. A possible approach, which currently requires access to the NS system, to deal with these instabilities is outlined. 

This paper introduces a modified version of the recent data-driven Loewner framework to compute reduced order models (ROMs) for a class of semi-explicit differential algebraic equation (DAE) systems, which include the semi-discretized linearized Navier-Stokes /Oseen equations. The modified version estimates the polynomial part of the original transfer function from data and incorporate this estimate into the Loewner ROM construction. Without this proposed modification the transfer function of the Loewner ROM is strictly proper, i.e., goes to zero as the magnitude of the frequency goes to infinity, and therefore may have a different behavior for large frequencies than the transfer function of the original system. The modification leads to a Loewner ROM with a transfer function that has a strictly proper and a polynomial part, just as the original model. This leads to better approximations for transfer functioncomponents in which the coefficients in the polynomial part are not too small. The construction of the improved Loewner ROM is described and the improvement is demonstrated on a large-scale system governed by the semi-discretized Oseen equations.