POD basis updates for nonlinear PDE control

Abstract

In the present paper a semilinear boundary control problem is considered. For its numerical solution proper orthogonal decomposition (POD) is applied. POD is based on a Galerkin type discretization with basis elements created from the evolution problem itself. In the context of optimal control this approach may suffer from the fact that the basis elements are computed from a reference trajectory containing features which are quite different from those of the optimally controlled trajectory. Therefore, different POD basis update strategies which avoid this problem of unmodelled dynamics are compared numerically.

2 Model order reduction by proper orthogonal decomposition

“…One of the most mature methods developed in this context is trust-region POD, proposed in [9], which since then has successfully been applied in many applications. We also refer to the work [38], where strategies for updating the POD bases are compared.…”

A non-conforming dual approach for adaptive Trust-Region Reduced Basis approximation of PDE-constrained optimization

“…For µ ∈ P and 1 ≤ i ≤ P , let d µi u h,µ ∈ V h be the solution of the discrete version of (2) and d µi u r,µ ∈ V pr,dµ i r be the solution of (18). We then have using the coercivity of a µ in the first inequality, the definition d µi e pr h,µ in the first equality, Propostion 1.4 applied to u h,µ in the second equality, the definition of the discrete sensitivity primal residual (23) in the third equality and the continuity of d µi a µ in the last inequality.…”

“…In [54] a posteriori error bounds are utilized to monitor the approximation quality of the gradient. We also refer to [23], where the authors utilize basis update strategies to improve the reduced order approximation scheme with respect to the optimization goal. The TR strategy can be combined with second-order methods for nonlinear optimization: with the Newton method to solve the reduced problem and with the SQP method for the all-at-once approach; cf.…”

A-posteriori reduced basis error-estimates for a semi-discrete in space quasilinear parabolic PDE

“…Appropriate coupling between numerical optimization and MOR is an active area of research, see e.g. [26]. For approaches based on a-posteriori error estimation and trust-region type-algorithms, respectively, we refer to [7,57], and in particular [52,53].…”