Sequential Reduced-Order Modeling for Time-Dependent Optimization Problems with Initial Value Controls

Abstract

This paper introduces an efficient sequential application of reduced order models (ROMs) to solve linear quadratic optimal control problems with initial value controls. The numerical solution of such a problem requires Hessian-times-vector multiplications, each of which requires solving a linearized state equation with initial value given by the vector and solving a second-order adjoint equation. Projection-based ROMs are applied to these differential equations to generate a Hessian approximation. However, in general, no fixed ROM well-approximates the application of the Hessian to all possible vectors of initial data. To improve a basic ROM, Heinkenschloss and Jando: Reduced-Order Modeling for Time-Dependent Optimization Problems with Initial Value Controls (SIAM Journal on Scientific Computing, 40(1), A22–A51, 2018, https://doi.org/10.1137/16M1109084) introduce an augmentation of the basic ROM by the right-hand side of the optimality system. This augmented ROM substantially improves the accuracy of the computed control, but this accuracy may still not be enough. The proposed sequential application of the augmented ROM can compute an approximate control with the same accuracy as the one obtained using only the expensive full-order model, but at a fraction of the cost.