Sparse data-driven quadrature rules via $\ell^p$-quasi-norm minimization

In this paper we show the use of the focal underdetermined system solver to recover sparse empirical quadrature rules for parametrized integrals from existing data, consisting of the values of given parametric functions sampled on a discrete set of points. This algorithm, originally proposed for image and signal reconstruction, relies on an approximated $\ell^p$-quasi-norm minimization. The choice of $0<p<1$ fits the nature of the constraints to which quadrature rules are subject, thus providing a more natural formulation for sparse quadrature recovery compared to the one based on $\ell^1$-norm minimization. We also extend an a priori error estimate available for the $\ell^1$-norm formulation by considering the error resulting from data compression. Finally, we present two numerical examples to illustrate some practical applications. The first concerns the fundamental solution of the linear 1D Schr\"odinger equation, the second example deals with the hyper-reduction of a partial differential equation modelling a nonlinear diffusion process in the framework of the reduced basis method. For both the examples we compare our method with the one based on $\ell^1$-norm minimization and the one relaying on the use of the non-negative least square method. Matlab codes related to the numerical examples and the algorithms described are provided.