Sparse Data-Driven Quadrature Rules via ℓ p -Quasi-Norm Minimization

This paper is concerned with the use of the focal underdetermined system solver to recover sparse empirical quadrature rules for parametrized integrals from existing data. This algorithm, originally proposed for image and signal reconstruction, relies on an approximated ℓ p -quasi-norm minimization. Compared to ℓ 1 -norm minimization, the choice of 0 < p < 1 provides a natural framework to accommodate usual constraints which quadrature rules must fulfil. We also extend an a priori error estimate available for the ℓ 1 -norm formulation by considering the error resulting from data compression. Finally, we present numerical examples to investigate the numerical performance of our method and compare our results to both ℓ 1 -norm minimization and nonnegative least squares method. Matlab codes related to the numerical examples and the algorithms described are provided.