We show the use of the focal underdetermined system solver to recover sparse  quadrature rules for parametrized integrals from existing data  . 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 suitable formulation for sparse quadrature recovery than 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 application: the space-time evaluation of the fundamental solution of  linear 1D Schr\"odinger equation where we compare our method with the one based on \ell^1-norm minimization ; the hyper-reduction of a nonlinear diffusion partial differential equation in the framework of reduced basis method .
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 .
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