An LP empirical quadrature procedure for reduced basis treatment of parametrized nonlinear PDEs

We present a model reduction formulation for parametrized nonlinear partial differential equations (PDEs). Our approach builds on two ingredients: reduced basis (RB) spaces which provide rapidly convergent approximations to the parametric manifold; sparse empirical quadrature rules which provide rapid evaluation of the nonlinear residual and output forms associated with the RB spaces. We identify both the RB spaces and the sparse quadrature rules in the offline stage through a greedy training procedure over the parameter domain; the procedure requires the dual norm of the finite element (FE) residual at many training points in the parameter domain, but only very few FE solutions—the snapshots retained in the RB space. The quadrature rules are identified by a linear program (LP) empirical quadrature procedure (EQP) which (i) admits efficient solution by a simplex method, and (ii) directly controls the solution error induced by the approximate quadrature. We demonstrate the formulation for a parametrized neo-Hookean beam: the dimension of the approximation space and the number of quadrature points are both reduced by two orders of magnitude relative to FE treatment, with commensurate savings in computational cost.