Reduced basis methods- an application to variational discretization of parametrized elliptic optimal control problems

TL;DR

This paper develops a reduced basis surrogate model for parametrized elliptic PDE optimal control problems using variational discretization, providing efficient estimators for the greedy sampling process and demonstrating good numerical performance.

Contribution

It introduces a novel reduced basis approach combined with variational discretization for parametrized elliptic control problems, with sharp residual-based estimators for model reduction.

Findings

Estimators are sharp and equivalent to actual errors in control, state, and adjoint state.

Numerical experiments confirm the effectiveness of the reduced basis method.

The approach simplifies the sampling process by avoiding residuals of the gradient equation.

Abstract

We consider a class of parameter-dependent optimal control problems of elliptic PDEs with constraints of general type on the control variable. Applying the concept of variational discretization, [4], together with techniques from the Reduced basis method, we construct a reduced basis surrogate model for the control problem. We establish estimators for the greedy sampling procedure which only involve the residuals of the state and the adjoint equation, but not of the gradient equation of the optimality system. The estimators are sharp up to a constant, i.e. they are equivalent to the approximation erros in control, state, and adjoint state. Numerical experiments show the performance of our approach.