We derive a Fast Multipole Method (FMM) where a low-rank approximation of the kernel is obtained using the Empirical Interpolation Method (EIM). Contrary to classical interpolation-based FMM, where the interpolation points and basis are fixed beforehand, the EIM is a nonlinear approximation method which constructs interpolation points and basis which are adapted to the kernel under consideration. The basis functions are obtained using evaluations of the kernel itself. We restrict ourselves to translation-invariant kernels, for which a modified version of the EIM approximation can be used in a multilevel FMM context; we call the obtained algorithm Empirical Interpolation Fast Multipole Method (EIFMM). An important feature of the EIFMM is a built-in error estimation of the interpolation error made by the low-rank approximation of the far-field behavior of the kernel: the algorithm selects the optimal number of interpolation points required to ensure a given accuracy for the result, leading to important gains for inhomogeneous kernels.
A kernel-independent sum-of-Gaussians method by de la Vallée-Poussin sums
在科学和工程计算的许多应用中，为了构造核求和或卷积问题的有效算法，经常需要用高斯和（SOG）逼近相互作用的核。本文通过引入de la Vallée-Poussin和和切比雪夫多项式，提出了一种与核无关的SOG方法. SOG适用于一般的相互作用核，高斯带宽的下界是可调的，因此高斯可以很容易地通过快速高斯算法求和。基于平方根法的平衡截断模型降阶可以进一步减少高斯数。数值结果表明，该方法具有良好的降阶精度和降阶效率。
Fast Ewald Summation with Prolates for Charged Systems in the NPT Ensemble
Center for Computational Mathematics, Flatiron Institute（西蒙斯基金会平顿研究所计算数学中心）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
FFT-based homogenization on periodic anisotropic translation invariant spaces
University of Kaiserslautern（凯泽斯劳滕大学）
本文推导了在各向异性平移不变空间中均质化准静态弹性方程的离散形式，采用变分公式和所谓的Lippmann-Schwinger方程，并统一和扩展了截断傅里叶级数方法、常数有限元近似和各向异性晶格推导。所得Lippmann-Schwinger方程在各向异性平移不变空间中的形式首次将傅里叶方法和有限元方法统一到一个数学框架中。进一步定义并刻画了所得到的周期化Green算子。该算子在Dirichlet核对应于对角矩阵的情况下，与来自截断傅里叶级数方法和各向异性晶格推导的Galerkin投影算子相同，并且对于其他Dirichlet核也相同。此外，证明了周期化Green算子的有界性。该算子仅当平移空间由Dirichlet核生成时才构成投影。de la Vallée Poussin均值和Box样条的数值示例展示了该框架的灵活性。
Bashkir State University（巴什基尔国立大学）；Center for Micro and Nanoscale Dynamics of Dispersed Systems, Bashkir State University（巴什基尔国立大学分散系统微纳动力学中心）；Fantalgo, LLC.（Fantalgo有限责任公司）；Institute for Advanced Computer Studies（高级计算机研究所）；Department of Computer Science（计算机科学系）；University of Maryland, College Park（马里兰大学帕克分校）
