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

Abstract

In this work, we propose to efficiently solve time dependent parametrized optimal control problems governed by parabolic partial differential equations through the certified reduced basis method. In particular, we will exploit an error estimator procedure, based on easy-to-compute quantities which guarantee a rigorous and efficient bound for the error of the involved variables. First of all, we propose the analysis of the problem at hand, proving its well-posedness thanks to Nečas -Babuška theory for distributed and boundary controls in a space-time formulation. Then, we derive error estimators to apply a Greedy method during the offline stage, in order to perform, during the online stage, a Galerkin projection onto a low-dimensional space spanned by properly chosen high-fidelity solutions. We tested the error estimators on two model problems governed by a Graetz flow: a physical parametrized distributed optimal control problem and a boundary optimal control problem with physical and geometrical parameters. The results have been compared to a previously proposed bound, based on the exact computation of the Babuška inf-sup constant, in terms of reliability and computational costs. We remark that our findings still hold in the steady setting and we propose a brief insight also for this simpler formulation.

A Streamline upwind Petrov-Galerkin Reduced Order Method for Advection-Dominated Partial Differential Equations under Optimal Control

“…We briefly recall results on well-posedness for time-dependent Linear-Quadratic OCP(µ)s based on [50,51]. We consider saddle-point formulation in order to prove wellposedness by using tools of the previous Sections in the case of null initial conditions.…”

“…, N t }. On the other hand, all terms involving time-derivative go through a time discretization equivalent to a classical implicit Euler approach [3,23,46,50,51,52]. The backward Euler method is used to discretize the state equation forward in time, instead the adjoint equation is discretized backward in time using the forward Euler method, which is equivalent to the backward Euler with respect to time T − t, for t ∈ (0, T ) [16,50].…”

“…the high-fidelity space. Then, for stabilized problems we define the discrete parametric solution manifold as (51) M N := y N (µ), u N (µ), p N (µ) FEM solution of the (35) | µ ∈ P .…”

“…We exploit a Streamline Upwind Petrov-Galerkin (SUPG) technique over a finite element method (FEM) [11,26,38] in a optimize-than-discretize approach, as done in [14], to provide stronglyconsistency to the discretization. When we deal with unsteady problems, a space-time discretization [21,46,50,51,52,57] will be used together with the SUPG stabilization for bilinear forms related to the derivative over time. The discretization procedure can easily request a huge amount of computational resources, especially for parametric time-dependent problems.…”

Stabilized Weighted Reduced Order Methods for Parametrized Advection-Dominated Optimal Control Problems governed by Partial Differential Equations with Random Inputs

“…∂ t y ∈ Y * }. For parabolic problems we will also consider the case of full-admissibility as X ad = Y t × U [5,54,55].…”