Urbana Campus Research Calendar (OVCRI)

View Full Calendar

Harmonic Analysis and Differential Equations Seminar

Event Type
Seminar/Symposium
Sponsor
Department of Mathematics
Location
Altgeld Hall 143
Date
Feb 27, 2024   1:00 pm  
Speaker
Xuefeng Liu (Tokyo Woman's Christian University)
Contact
Carrie Clark
E-Mail
carriec2@illinois.edu
Views
50
Originating Calendar
Harmonic Analysis and Differential Equations Seminar (HADES)

Title: Rigorous Evaluation of the Hadamard Derivative for Shape Optimization Studies

Abstract: This presentation introduces a newly developed computational method for rigorously evaluating the Hadamard derivative of Laplacian eigenvalues, which plays an important role in studying shape optimization problems.

To evaluate the Hadamard derivative, this method employs state-of-the-art algorithms for eigenvalues and eigenfunctions via the finite element method (Liu'2013,2015; Liu-Vejchodsky'2022), effectively handling cases of repeated or closely spaced eigenvalues.

We also present a computer-assisted proof for the optimization of Laplacian eigenvalues over triangular domains (Endo-Liu'2023), demonstrating the impact of these computational advancements in spectral geometry.

References:

1) Xuefeng Liu, A framework of verified eigenvalue bounds for self-adjoint differential operators, Applied Mathematics and Computation, Vol. 267, 2015, pp. 341-355.

2) Xuefeng Liu and Tomas Vejchodský, Fully computable a posteriori error bounds for eigenfunctions, Numer. Math. 152, 183–221 (2022). 

3) Ryoki Endo, Xuefeng Liu, Shape optimization for the Laplacian eigenvalue over triangles and its application to interpolation error analysis, Journal of Differential Equations, Volume 376, 2023, pp.750-772.

link for robots only