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Tikhonov regularization is one of the most commonly used methods ofregularization of ill-posed problems. In the setting of finite elementsolutions of elliptic partial differential control problems, Tikhonovregularization amounts to adding suitably weighted least squares terms of thecontrol variable, or derivatives thereof, to the Lagrangian determining theoptimality system. In this note we show that stabilization methods fordiscretely ill--posed problems developed in the setting ofconvection--dominated convection--diffusion problems, can be highly suitablefor stabilizing optimal control problems, and that Tikhonov regularization willlead to less accurate discrete solutions. We consider data assimilationproblems for Poisson's equation as illustration and derive new error estimatesboth for the the reconstruction of the solution from measured data andreconstruction of the source term from measured data. These estimates includeboth the effect of discretization error and error in measurements.

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