  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.
