Sparse Data-Driven Quadrature Rules via lp-Quasi-Norm Minimization
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We present the focal underdetermined system solver as numerical tool 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 lp- quasi-norm minimization. Compared to l1-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 l1-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 l1-norm minimization and nonnegative least squares method.
