An LP empirical quadrature procedure for parametrized functions

Abstract

We extend the linear program empirical quadrature procedure proposed in and subsequently to the case in which the functions to be integrated are associated with a parametric manifold. We pose a discretized linear semi-infinite program: we minimize as objective the sum of the (positive) quadrature weights, an ℓ[subscript 1] norm that yields sparse solutions and furthermore ensures stability; we require as inequality constraints that the integrals of J functions sampled from the parametric manifold are evaluated to accuracy [¯ over δ]. We provide an a priori error estimate and numerical results that demonstrate that under suitable regularity conditions, the integral of any function from the parametric manifold is evaluated by the empirical quadrature rule to accuracy [¯ over δ] as J→∞. We present two numerical examples: an inverse Laplace transform; reduced-basis treatment of a nonlinear partial differential equation.