【Academic Seminar】Regularized Weighted Least Squares Approximation by Using Gauss Points - Prof. Congpei An
Topic: Regularized Weighted Least Squares Approximation by Using Gauss Points
Speaker: Prof. Congpei An, Southwestern University of Finance and Economics
Date and Time: 11:00 - 12:00, Monday, October 21, 2019
Venue: Room 208, Cheng Dao Building
We consider polynomial approximation over the interval [-1,1] by regularized weighted discrete least squares methods with ell_2- or ell_1-regularization, respectively. As the set of nodes we use Gauss quadrature points (which are zeros of orthogonal polynomials). The number of Gauss quadrature points is N+1. For 2L <= 2N+1, with the aid of Gauss quadrature, we obtain approximation polynomials of degree L in closed form without solving linear algebra or optimization problems. In fact, these approximation polynomials can be expressed in the form of the barycentric interpolation formula by Berrut et al. when an interpolation condition is satisfied.
We then study the approximation quality of the ell_2-regularized approximation polynomial in terms of Lebesgue constants, and the sparsity of the ell_1-regularized approximation polynomial. Finally, we give numerical examples to illustrate these theoretical results and show that a well-chosen regularization parameter can lead to good performance, with or without contaminated data.
Dr. Congpei An is currently an associate professor in the School of Economic Mathematics at the Southwestern University of Finance and Economics,Chengdu. He received his Bachelor of Science from the Central South University in 2006, the Master of Science in Computational Mathematics from the Central South University in 2008, under the supervision of Prof. Shuhuang Xiang, and the PhD in Applied Mathematics from The Hong Kong Polytechnic University in 2011, under the supervision of Prof. Xiaojun Chen and Prof. Ian Sloan. He was a faculty member of the department of Mathematics of the Jinan University, during Oct 2011-April 2019. He was the founder of the Institute of Computation Science at the Jinan University. He is an executive member of the council of the society of computational mathematics of the Guangdong province. Congpei An made fundamental contributions on spherical t-designs and their applications in three dimensional spaces. His current research interests include theory and algorithms for approximation, and their various applications in real world. Congpei An has been invited to visit many famous universities and academic institutes.