An empirical interpolation approach to reduced basis approximations for variational inequalities

Variational inequalities (VIs) are pervasive in mathematical modelling of equilibrium and optimization problems in engineering and science. Examples of applications include traffic network equilibrium problems, financial equilibrium, obstacle problems, lubrication phenomena and many others. Since these problems are computationally expensive to solve, we focus here on the development of model order reduction techniques, in particular the reduced basis technique. Reduced basis techniques for the approximation of solutions to elliptic VIs have been developed in the last few years. These methods apply to VIs of the so-called first kind, i.e. problems that can be equivalently described by a minimization of a functional over a convex set. However, these recent approaches are inapplicable to VIs of the so-called second kind, i.e. problems that involve minimization of a functional containing non-differentiable terms. In this article, we evaluate the feasibility of using the reduced basis method (RBM) combined with the empirical interpolation method (EIM) to treat VIs. In the proposed approach, the problem is approximated using a penalty or barrier method, and EIM is then applied to the penalty or barrier term. Numerical examples are presented to assess the performance of the proposed method, in particular the accuracy and computational efficiency of the approximation. Although the numerical examples involve only VIs of the first kind, we also evaluate the feasibility of using the RBM combined with the EIM to treat VIs of the second kind.