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 Schrödinger 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.
Stochastic collocation methods via $L_1$ minimization using randomized quadratures
Shanghai Normal University（上海师范大学）；University of Utah（犹他大学）；Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）
Multilevel quadrature for elliptic parametric partial differential equations in case of polygonal approximations of curved domains
Universität Bonn（波恩大学）；Fraunhofer Institute for Algorithms and Scientific Computing (SCAI)（弗劳恩霍夫算法与科学计算研究所）；Universität Basel（巴塞尔大学）；Università della Svizzera italiana（瑞士意大利语大学）
Numerical Integration in Multiple Dimensions with Designed Quadrature
Scientific Computing and Imaging Institute, University of Utah（科学计算与成像研究所，犹他大学）；School of Computing, University of Utah（计算机学院，犹他大学）；Department of Mathematics, University of Utah（数学系，犹他大学）
Efficient assembly based on B-spline tailored quadrature rules for the IgA-SGBEM
University of Parma（帕尔马大学）；University of Cassino and southern Lazio（卡西诺及南拉齐奥大学）；University of Siena（锡耶纳大学）；University of Pavia（帕维亚大学）；University of Florence（佛罗伦萨大学）
Convergence Rates for $\ell_p$ Norm Minimization in Convex Vector Optimization
King Fahd University of Petroleum & Minerals（国王法赫德国王石油矿物大学）；Interdisciplinary Research Center for Smart Mobility and Logistics, King Fahd University of Petroleum & Minerals（国王法赫德国王石油矿物大学智能交通与物流交叉研究中心）
Nonmonotone Barzilai-Borwein Gradient Algorithm for $\ell_1$-Regularized Nonsmooth Minimization in Compressive Sensing
Institute of Applied Mathematics, College of Mathematics and Information Science, Henan University（河南大学数学与信息科学学院应用数学研究所）；National Center for Theoretical Sciences (South), National Cheng Kung University（台湾成功大学南方理论科学中心）；Department of Applied Mathematics, Hong Kong Polytechnic University（香港理工大学应用数学系）
