Quantized-CP Approximation and Sparse Tensor Interpolation of Function Generated Data
Max-Planck-Institute for Mathematics in the Sciences（马克斯·普朗克科学促进部数学研究所）；Birla Institute of Technology and Science Pilani, Hyderabad Campus（比拉理工学院皮拉尼分校海德拉巴校区）；Westsächsische Hochschule Zwickau（茨维考西萨克森应用技术大学）
Boris N. Khoromskij, Kishore K. Naraparaju, Jan Schneider
本文考虑了计算由函数在实数区间大均匀网格上离散化生成的数据的canonical (CP)近似迭代方案。本文延续了针对函数相关数据量化图像的QTT方法[16]的研究。在QTT方法中，目标向量长度为$2^{L}$被重塑为一个$L$阶张量，每个模式有两 entries（量化表示），然后通过包含$2r^2 L$参数的QTT张量进行近似。接下来，我们考虑使用交替最小二乘法（ALS）迭代方案计算量化向量的rank-$r$ CP近似，仅需$2 r L \ll 2^L$参数存储。在先前的论文[17]中，这种表示被称为Q$_{Can}$格式，而在本文中我们简称为QCP表示。我们测试了ALS算法计算QCP近似在各种函数上的效果，并在所有情况下观察到QCP秩的指数误差衰减。主要思想是通过减少的函数样本数，仅在$O(2rL)$网格点计算，恢复离散化函数的rank-$r$ QCP格式。描述了求解出现的最小化问题的ALS方案的特殊版本。这种方法可以视为一种稀疏QCP插值方法，允许恢复rank-$r$ QCP张量的所有$2r L$表示参数。数值实验展示了QCP-ALS类型迭代的有效性，并表明了指数收敛率在$r$上。
英文摘要：
In this article we consider the iterative schemes to compute the canonical (CP) approximation of quantized data generated by a function discretized on a large uniform grid in an interval on the real line. This paper continues the research on the QTT method [16] developed for the tensor train (TT) approximation of the quantized images of function related data. In the QTT approach the target vector of length $2^{L}$ is reshaped to a $L^{th}$ order tensor with two entries in each mode (Quantized representation) and then approximated by the QTT tenor including $2r^2 L$ parameters, where $r$ is the maximal TT rank. In what follows, we consider the Alternating Least-Squares (ALS) iterative scheme to compute the rank-$r$ CP approximation of the quantized vectors, which requires only $2 r L\ll 2^L$ parameters for storage. In the earlier papers [17] such a representation was called Q$_{Can}$ format, while in this paper we abbreviate it as the QCP representation. We test the ALS algorithm to calculate the QCP approximation on various functions, and in all cases we observed the exponential error decay in the QCP rank. The main idea for recovering a discretized function in the rank-$r$ QCP format using the reduced number the functional samples, calculated only at $O(2rL)$ grid points, is presented. The special version of ALS scheme for solving the arising minimization problem is described. This approach can be viewed as the sparse QCP-interpolation method that allows to recover all $2r L$ representation parameters of the rank-$r$ QCP tensor. Numerical examples show the efficiency of the QCP-ALS type iteration and indicate the exponential convergence rate in $r$.
Institut für Numerische Simulation, Universität Bonn（波恩大学数值模拟研究所）；Fraunhofer Institute for Algorithms and Scientific Computing (SCAI)（弗劳恩霍夫算法与科学计算研究所）；Departement Mathematik und Informatik, Universität Basel（巴塞尔大学数学与信息学系）；Istituto Dalle Molle di studi sull’intelligenza artificiale, Università della Svizzera italiana（意大利瑞士大学达莱莫勒人工智能研究学院）
Nodal Coarsening and Approximate Ideal Interpolation for H(curl) in Algebraic Multigrid
The Pennsylvania State University（宾夕法尼亚州立大学）；Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）；Bergische Universität Wuppertal（伍珀塔尔大学）；University of New Mexico（新墨西哥大学）
Reconstructing sparse exponential polynomials from samples: Stirling numbers and Hermite interpolation
Lehrstuhl für Mathematik mit Schwerpunkt Digitale Bildverarbeitung & FORWISS, University of Passau（帕绍大学数字图像处理重点数学研究所与FORWISS）；University of Passau（帕绍大学）
