We will present in this talk a finite element formulation of boundary-value problems that aims at constructing approximations tailored towards the calculation of quantities of interest. The main idea is based on a reformulation of a boundary-value problem as a minimization problem that involves inequality constraints on the error in the goal functionals so that the resulting model is capable of delivering quantities of interest within some prescribed tolerance. Chaudhry et al. have proposed in [1] a similar method in which constraints are enforced via a penalization approach. However, an issue with that approach is concerned with the selection of suitable penalization parameters. Our goal in this work aims at circumventing this difficulty by imposing the inequality constraints through Lagrange multipliers using the Karush-Kuhn-Tucker (KKT) conditions. We will also show how to  design an adaptive strategy to construct adapted meshes based on a posteriori error estimates. Such a paradigm represents a departure from classical goal-oriented approaches in which one computes first the finite element solution and then adapts the mesh by controlling the error with respect to quantities of interest using dual-based error estimates [2]. Numerical examples will be presented in order to demonstrate the efficiency of the proposed approach. We will also show how such a formulation can be applied to the construction of reduced models using the so-called proper generalized decomposition (or low-rank approximation) method.

[1] J.H. Chaudhry, E.C. Cyr, K. Liu, T.A. Manteuffel, L.N. Olson, and L. Tang, Enhancing least-squares finite element methods through a quantity-of-interest, SIAM Journal of Numerical Analysis,  52(6), pp. 3085-3105, 2014.

[2] J.T. Oden and S. Prudhomme, Goal-oriented error estimation and adaptivity for the finite element method'', Computers and Mathematics with Applications, 41, 735-756, 2001.