Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization

TL;DR

This paper proposes using stabilization methods from convection-dominated problems as an alternative to Tikhonov regularization for ill-posed elliptic control problems, demonstrating improved accuracy and deriving new error estimates.

Contribution

It introduces stabilization techniques from convection-diffusion problems as a novel approach for regularizing elliptic control problems, offering better accuracy than Tikhonov regularization.

Findings

Stabilization methods improve discrete solution accuracy.

New error estimates account for discretization and measurement errors.

Stabilization outperforms Tikhonov regularization in the tested scenarios.

Abstract

Tikhonov regularization is one of the most commonly used methods of regularization of ill-posed problems. In the setting of finite element solutions of elliptic partial differential control problems, Tikhonov regularization amounts to adding suitably weighted least squares terms of the control variable, or derivatives thereof, to the Lagrangian determining the optimality system. In this note we show that stabilization methods for discretely ill--posed problems developed in the setting of convection--dominated convection--diffusion problems, can be highly suitable for stabilizing optimal control problems, and that Tikhonov regularization will lead to less accurate discrete solutions. We consider data assimilation problems for Poisson's equation as illustration and derive new error estimates both for the the reconstruction of the solution from measured data and reconstruction of the source term from measured data. These estimates include both the effect of discretization error and error in measurements.