论文标题
稀疏数据驱动的正交规则通过$ \ ell^p $ -quasi-norm最小化
Sparse data-driven quadrature rules via $\ell^p$-quasi-norm minimization
论文作者
论文摘要
在本文中，我们显示了使用焦点不确定的系统求解器从现有数据中恢复参数化积分的稀疏经验正交规则，由在离散点集中采样的给定参数函数的值组成。该算法最初是为图像和信号重建而提出的，依赖于近似的$ \ ell^p $ -quasi-norm最小化。 $ 0 <p <1 $的选择符合正交规则对约束的性质的性质，因此与基于$ \ ell^1 $ norm最小化的稀疏正交恢复提供了更自然的配方。我们还通过考虑数据压缩导致的误差，扩展了$ \ ell^1 $ norm公式的先验错误估计。最后，我们提出了两个数值示例，以说明一些实际应用。第一个涉及线性1DSchrödinger方程的基本解决方案，第二个示例涉及在还原基本方法框架中非线性扩散过程建模的部分微分方程对非线性扩散过程的过度还原。对于这两个示例，我们将我们的方法与基于$ \ ell^1 $ norm最小化的方法进行了比较，而一个传达的方法是使用非负平方法的使用。提供了与数值示例和所述算法相关的MATLAB代码。
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.
