期刊
SIAM JOURNAL ON SCIENTIFIC COMPUTING
卷 40, 期 1, 页码 A22-A51出版社
SIAM PUBLICATIONS
DOI: 10.1137/16M1109084
关键词
optimal control; reduced order models; Hessian approximation; initial-condition problems
资金
- NSF [DMS-1522798]
- DARPA EQUiPS Program [UTA15-001068]
- Excellence Initiative of the German Federal Government
- Institutional Strategy: DFG [ZUK 49/2]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1522798] Funding Source: National Science Foundation
This paper presents a new reduced order model (ROM) Hessian approximation for linear-quadratic optimal control problems where the optimal control is the initial value. Such problems arise in parameter identification and data assimilation, where the parameters to be identified appear in the initial data, and also as subproblems in multiple shooting formulations of more general optimal control problems. The new ROM Hessians can provide a substantially better approximation than the underlying basic ROM approximation, and thus can substantially reduce the computing time needed to solve these optimal control problems. The computation of a Hessian vector product requires the solution of the linearized state equation with initial value given by the vector to which the Hessian is applied, followed by the solution of the second order adjoint equation. Projection-based ROMs of these two linear differential equations are used to generate the Hessian approximation while the objective function and gradient are computed exactly using the full model. The challenge is that in general no fixed ROM well-approximates the application of the Hessian to all possible vectors of initial data. The new approach, after having selected a basic ROM, augments this basic ROM by one vector. This vector is either the right-hand side or the vector of initial data to which the Hessian is applied. It is shown that although the size of the ROM increases only by one, this new augmented ROM produces substantially better
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