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How well can we hear the shape of a drum by computer algorithms?
发布时间: 2017-08-18     09:42   【返回上一页】 发布人:Zhimin Zhang



 北京师范大学数学科学学院

偏微分方程与计算数学学术报告

 

 

 

报告题目: How well can we hear the shape of a drum by computer algorithms?

 

 

报告人:Professor Zhimin Zhang (Wayne State University)

 

 

时间地点:2017821 16:00-17:00, 后主楼1220.

 

邀请人:李岩岩 教授

 

 

报告摘要:

Can we determine the shape of a domain by its Laplacian eigenvalues? The question puzzled us for many years until 1992, when three mathematicians surprised everyone by a counterexample. However, this is not the end of the story to applied mathematicians, since in most cases we are unable to obtain exact eigenvalues and the numerical approximation by computer algorithms is necessary. Naturally, another question arises: How many numerical eigenvalues can we trust? When approximating PDE eigenvalue problems by numerical methods such as finite difference and finite element, it is common knowledge that only a small portion of numerical eigenvalues are reliable. However, this knowledge is only qualitative rather than quantitative in the literature. In this talk, I will first survey some theoretical results from pure mathematics regarding eigenvalue problems. Then I will investigate the number of “trusted” eigenvalues by the finite element (and the related finite difference method results obtained from mass lumping) approximation of 2mth order elliptic PDE eigenvalue problems. Our two model problems are the Laplace and bi-harmonic operators, for which a solid knowledge regarding magnitudes of eigenvalues are available in the literature. Combining this knowledge with a priori error estimates of the finite element method, we are able to figure out roughly how many “reliable” eigenvalues can be obtained from numerical approximation under a pre-determined convergence rate.