报告名称：Numerical Approaches for Computing C-Eigenvalues of a Piezoelectric-Type Tensor

主讲人：杨宇宁 教授

邀请人：莫长鑫 讲师

时间：2022年5月24日   16:00

地点：腾讯会议（ID：840 427 857）

主办单位：数学科学学院

报告摘要

A piezoelectric-type tensor is of order three which is symmetric with respect to its last two indices. The largest C-eigenvalue of a piezoelectric-type tensor determines the highest piezoelectric coupling constant. We introduce two numerical approaches for computing C-eigenvalues: an iterative approach and a convex relaxation approach.

For the first approach, we first introduce partial maximizers and strongly partial maximizers, which are subsets of C-eigenvectors. Then a shifted eigenvalue decomposition method is proposed, which globally converges without any assumption. When the shifted parameter is small enough, the convergent limit is a partial maximizer, and for a special class of piezoelectric-type tensors, the limit is a strongly partial maximizer. Linear convergence is then established under reasonable assumptions. In addition, we introduce an approximation algorithm and provide its worst-case lower bound. Numerical experiments show the efficiency of the proposed method.

For the second approach, we first show that how to derive the convex relaxations via a newly introduce equivalence property. Such relaxations define tigher norms than usually used ones. Several insights are provided for the tightness issues of the convex relaxations. The spectral property of the dual variable, in particular, determines the tightness. When the convex relaxations are not tight, a theoretical guaranteed approximation algorithm is proposed to extract a feasible approximation solution. We provide several types of tensors to justify the tightness of the convex relaxations. In case that the relaxations are not tight, their optimal values, serving as upper bounds, are still tighter than those in the literature.

专家简介

杨宇宁，2003至2013年本硕博就读及毕业于南开大学数学科学学院。2013 至 2017 年于比利时鲁汶大学从事博士后研究。2017 年入职广西大学数学与信息科学学院。2018 年入选国家海外高层次人才青年项目，同年任教授。研究领域为张量计算和优化。发表 SCI 论文 30 余篇，发表期刊包括 SIAM J. Optim., SIAM J. Matrix Anal. Appl., J. Mach. Learn. Res., IEEE Trans. Neural Netw. Learn. Syst.等。著专著一部。主持国家自然科学基金面上基金、青年基金（已结题）各一项，主持教育部霍英东青年教师基金一项。现担任中国运筹学会数学规划分会青年理事，中国工业与应用数学学会理事，广西运筹学会理事，广西大学学术委员会委员。