A Certified Trust Region Reduced Basis Approach to PDE-Constrained Optimization

Abstract

Abstract. Parameter optimization problems constrained by partial differential equations (PDEs) appear in many science and engineering applications. Solving these optimization problems may require a prohibitively large number of computationally expensive PDE solves, especially if the dimension of the design space is large. It is therefore advantageous to replace expensive high-dimensional PDE solvers (e.g., finite element) with lower-dimensional surrogate models. In this paper, the reduced basis (RB) model reduction method is used in conjunction with a trust region optimization framework to accelerate PDE-constrained parameter optimization. Novel a posteriori error bounds on the RB cost and cost gradient for quadratic cost functionals (e.g., least squares) are presented and used to guarantee convergence to the optimum of the high-fidelity model. The proposed certified RB trust region approach uses high-fidelity solves to update the RB model only if the approximation is no longer sufficiently accurate, reducing the number of full-fidelity solves required. We consider problems governed by elliptic and parabolic PDEs and present numerical results for a thermal fin model problem in which we are able to reduce the number of full solves necessary for the optimization by up to 86%.

Feedback Control of Nonlinear PDEs Using Data-Efficient Reduced Order Models Based on the Koopman Operator

“…In the latter case, there exist different approaches to ensure convergence towards the true optimum. The two most popular are to derive error bounds based on the singular values associated with the POD modes [5,6,7,8] and to adapt classical trust-region approaches to surrogate modeling [9,10,11].…”

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

“…We refer e.g. to [52] and the references therein for RB-methods in PDE-constrained optimization in general, and to [53] or the survey [29] for POD in particular.…”

“…[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]. A different method, so-called optimality system POD, has been proposed in [41].…”

“…Finite element discretization errorestimates for the state equation from [10] are obtained in [11]. Having in mind a possible coupling of numerical optimization and MOR à la [52,53], we start in the present paper with deriving a-posteriori reduced basis errors for the corresponding state equation. Herein, the presence of a non-monotone nonlinearity is the main difference to earlier publications concerned with RB-a-posteriori error-estimates for nonlinear PDEs [33,34,53], and also poses the main difficulty in our analysis.…”

A solution for fractional PDE constrained optimization problems using reduced basis method

“…A reduced basis surrogate model was proposed for PCOP in Dihlmann and Haasdonk (2015). Authors in Qian et al (2017) used RB method in conjunction with a trust region optimization framework to accelerate PCOP. RB method was applied to solve quadratic multi-objective optimal control problems governed by linear parameterized variational equations in Lapichino et al (2017).…”