A Certified Reduced Basis Approach for Parametrized Linear-Quadratic Optimal Control Problems with Control Constraints

Abstract

In this talk, we consider the efficient and reliable solution of distributed optimal control problems governed by parametrized elliptic partial differential equations involving constraints on the control. The reduced basis method is used as a low-dimensional surrogate model to solve the optimal control problem. To this end, we introduce reduced basis spaces not only for the state and adjoint variable but also for the distributed control variable and propose rigorous error bounds for the error in the optimal control. The reduced basis optimal control problem and associated a posteriori error bounds can be efficiently evaluated in an offline-online computational procedure, thus making our approach relevant in the many-query or real-time context. We present numerical results for a model problem to show the validity of our approach.Many problems in science and engineering can be modeled in terms of optimal control problems governed by parametrized partial differential equations (PDEs). While the PDE describes the underlying system or component behavior, the parameters often serve to identify a particular configuration of the component -such as boundary and initial conditions, material properties, and geometry. In such cases -in addition to solving the optimal control problem itself -one is often interested in exploring many different parameter configurations and thus in speeding up the solution of the optimal control problem. However, using classical discretization techniques such as finite elements or finite volumes even a single solution is often computationally expensive and time-consuming, a parameter-space exploration thus prohibitive. One way to decrease the computational burden is the surrogate model approach, where the original high-dimensional model is replaced by a reduced order approximation. These ideas have received a lot of attention in the past and various model order reduction techniques have been used in this context. However, the solution of the reduced order optimal conThis work was supported by the Excellence Initiative of the German federal and state governments and the German Research Foundation through Grant GSC 111. trol problem is generally sub-optimal and reliable error estimation is thus crucial. Besides serving as a certificate of fidelity for the sub-optimal solution, our a posteriori error bounds are also a crucial ingredient in generating the reduced basis with greedy algorithms. A new approach for efficient computation of error bounds for unconstrained distributed control problems was proposed in Kärcher et al. (2014). This approach, however, and all other existing approaches in the literature, see e.g. Negri et al. (2013); Negri (2011); Rozza et al. (2012), are not directly applicable to the important case with additional constraints on the control. In this work we extend the methodology presented in Zhang et al. (2014) to consider PDE-constrained optimal control problems. The authors in Zhang et al. (2014) develop a certified Reduced Basis (RB) method that provides sharp and inexpens...

A Certified Reduced Basis Method for Linear Parametrized Parabolic Optimal Control Problems in Space-Time Formulation

“…It is well-known in literature that optimal control problems solved through Lagrangian formulation result in saddle point structures, see e.g [4,10,12,23,24,25,33,34] for steady problems and for time dependent cases, see e.g. [17,19,18,43,44,46].…”

“…Finally, we will briefly present the reduced saddle point optimization problem and the aggregated space strategy following the approach already exploited in previous literature, see e.g. [4,10,12,23,24,25,33,34]. All the concepts will be presented for the time dependent scenario: indeed, the steady case complies with the more general formulation.…”

“…where Q N is a space to be determined. In particular, we employ the aggregated space technique, an approach exploited for saddle point problem arising in PDE(µ)-constrained optimization [4,10,12,23,24,25,33,34]. It consists in the definition of the reduced space (59)…”

“…The goal of this work is to propose a reduced basis (RB) approach to deal with the study for several values of µ ∈ P in a low-dimensional framework reducing the computational costs of multiple simulations [16,36,40,41]. Concerning the employment of reduced techniques to OCP(µ)s, the interested reader may refer to several papers as [3,4,10,12,22,23,24,25,33,34,38,46], where the reduction has been implemented through different methodology for a wide range of governing equations. In particular, we seek to extend the approach presented in [34] for steady OCP(µ)s to quadratic optimization models constrained to linear time dependent PDE(µ)s in a space-time framework.…”

POD-Galerkin Model Order Reduction for Parametrized Nonlinear Time Dependent Optimal Flow Control: an Application to Shallow Water Equations

“…In general, reduction methods for parametrized nonlinear time dependent OCP(µ)s are very complex to analyse both theoretically and numerically. Although the literature is quite consolidated for steady constraints, see for example [7,8,20,25,34,36,37,47,46,50], where the interested reader may find theoretical and numerical analysis for different linear models, there is very small knowledge about time dependency [31,35,61,63]. Another difficulty to be overcome is the treatment and the reduction of nonlinear OCP(µ)s, see for example [38,54,61,63,71].…”