We present the Continuous Empirical Cubature Method (CECM), a novel algorithm for empirically devising efficient integration rules. The CECM aims to improve existing cubature methods by producing rules that are close to the optimal, featuring far less points than the number of functions to integrate. The CECM consists on a two-stage strategy. First, a point selection strategy is applied for obtaining an initial approximation to the cubature rule, featuring as many points as functions to integrate. The second stage consists in a sparsification strategy in which, alongside the indexes and corresponding weights, the spatial coordinates of the points are also considered as design variables. The positions of the initially selected points are changed to render their associated weights to zero, and in this way, the minimum number of points is achieved. Although originally conceived within the framework of hyper-reduced order models (HROMs), we present the method's formulation in terms of generic vector-valued functions, thereby accentuating its versatility across various problem domains. To demonstrate the extensive applicability of the method, we conduct numerical validations using univariate and multivariate Lagrange polynomials. In these cases, we show the method's capacity to retrieve the optimal Gaussian rule. We also asses the method for an arbitrary exponential-sinusoidal function in a 3D domain, and finally consider an example of the application of the method to the hyperreduction of a multiscale finite element model, showcasing notable computational performance gains. A secondary contribution of the current paper is the Sequential Randomized SVD (SRSVD) approach for computing the Singular Value Decomposition (SVD) in a column-partitioned format. The SRSVD is particularly advantageous when matrix sizes approach memory limitations.

翻译：我们提出了连续经验求积方法（CECM），这是一种通过经验方式设计高效积分规则的新型算法。CECM旨在改进现有求积方法，通过生成接近最优的积分规则，使其积分点数量远少于被积函数个数。该算法采用两阶段策略：第一阶段应用点选择策略，获得包含与被积函数数量相等的积分点的初始近似求积规则；第二阶段采用稀疏化策略，将点的空间坐标、索引及对应权重均视为设计变量，通过调整初始选取点的位置使其关联权重归零，从而实现最少积分点数。尽管该方法最初提出于超缩阶模型（HROMs）框架中，本文给出了基于通用向量值函数的公式表述，从而突显其跨领域的广泛适用性。为验证方法的普适性，我们使用单变量和多变量拉格朗日多项式进行数值验证，展示了该方法恢复最优高斯求积规则的能力。进一步在三维域中对任意指数-正弦函数进行评估，最后将该方法应用于多尺度有限元模型的超缩比实例，展示了显著的计算性能提升。本文的次要贡献是提出了序贯随机奇异值分解（SRSVD）方法，用于以列分区格式计算奇异值分解（SVD），该方案在矩阵规模接近内存限制时具有显著优势。