Primal-Dual Reduced Basis Methods for Convex Minimization Variational Problems: Robust True Solution a Posteriori Error Certification and Adaptive Greedy Algorithms

Abstract

The a posteriori error estimate and greedy algorithms play central roles in the reduced basis method (RBM). In [M. Yano, Comput. Methods Appl. Mech. Engrg., 287 (2015), pp. 290--309; ESAIM Math. Model. Numer. Anal., 50 (2016), pp. 163--185; SIAM J. Sci. Comput., 40 (2018), pp. A388--A420], several versions of RBMs based on exact error certifications and greedy algorithms with spatio-parameter adaptivities are developed. In this paper, with the parametric symmetric coercive elliptic boundary value problem as an example of the primal-dual variational problems satisfying the strong duality, we develop primal-dual RBMs (PD-RBM) with robust true error certifications and discuss three versions of greedy algorithms to balance the finite element error, the exact RB error, and the adaptive mesh refinements. For a class of convex minimization variational problems which has corresponding dual problems satisfying the strong duality, the primal-dual gap between the primal and dual functionals can be used as an a posteriori error estimator. This primal-dual gap error estimator is robust with respect to the parameters of the problem, and it can be used for both mesh refinements of finite element methods and the true RB error certification. With the help of an integration by parts formula, the primal-dual variational theory is developed for the symmetric coercive elliptic boundary value problems with nonhomogeneous boundary conditions by both the conjugate function and Lagrangian theories. A generalized Prager--Synge identity, which is the primal-dual gap error representation for this specific problem, is developed. RBMs for both the primal and dual problems with robust error estimates are developed. The dual variational problem often can be viewed as a constraint optimization problem. In the paper, different from the standard saddle-point finite element approximation, the dual RBM is treated as a Galerkin projection by constructing RB spaces satisfying the homogeneous constraint. Inspired by the greedy algorithm with spatio-parameter adaptivity of [M. Yano, SIAM J. Sci. Comput., 40 (2018), pp. A388--A420], adaptive balanced greedy algorithms with primal-dual finite element and RB error estimators are discussed. Numerical tests are presented to test the PD-RBM with adaptive balanced greedy algorithms.