走向现代数学学术报告 – 庞宏奎教授（No. 623）

报告题目：All-at-once method for variable-order time fractional diffusion equations

报告人：庞宏奎 教授（江苏师范大学）

报告时间：2023年5月29日 15:30

报告地点：学术报告厅(工西416)

摘要：We propose a fast solver for the variable-order (VO) time-fractional diffusion equation. Due to the impact of the time-dependent VO function, the resulting coefficient matrix of the large linear system assembling discrete equations of all time levels is a block lower triangular matrix without the block Toeplitz structure. Here, we approximate the off-diagonal blocks by low-rank matrices based on the polynomial interpolation, which can be constructed in OMlog2M operations with the same number storage requirement, where M is the number of time steps. Furthermore, a divide-and-conquer method is developed to fast solve the approximated linear system. The proposed solver can be implemented in ONMlog2M complexity with N being the degree of freedom in space. The accuracy of approximation is theoretically studied, and the stability and convergence of the proposed fast method are also investigated. Numerical experiments are carried out to exemplify the accuracy and efficiency of the proposed method.We propose a fast solver for the variable-order (VO) time-fractional diffusion equation. Due to the impact of the time-dependent VO function, the resulting coefficient matrix of the large linear system assembling discrete equations of all time levels is a block lower triangular matrix without the block Toeplitz structure. Here, we approximate the off-diagonal blocks by low-rank matrices based on the polynomial interpolation, which can be constructed in OMlog2M operations with the same number storage requirement, where M is the number of time steps. Furthermore, a divide-and-conquer method is developed to fast solve the approximated linear system. The proposed solver can be implemented in ONMlog2M complexity with N being the degree of freedom in space. The accuracy of approximation is theoretically studied, and the stability and convergence of the proposed fast method are also investigated. Numerical experiments are carried out to exemplify the accuracy and efficiency of the proposed method.

个人简介：庞宏奎，江苏师范大学数学与统计学院教授，硕士生导师，主要研究方向为大规模科学与工程计算、数值代数。主持国家自然科学基金面上项目、国家自然科学基金青年项目、江苏省自然科学基金面上项目等省部级以上课题5项，入选江苏省青蓝工程优秀青年骨干教师培养对象。在SIAM J. Sci. Comput.、SIAM J. Matrix Anal. Appl.、J. Sci. Comput.、J. Comput. Phys.、Numer. Linear Algebra Appl.、Numer. Algor.、Linear Algebra Appl.等学术期刊上发表论文多篇，在高等教育出版社出版译著1部。