Dr.  Tao  Liu
Nanyang Technological University,  Singapore

Title: Multigrid-Homotopy Method for PDE-Based Inverse Problems

Abstract:

The present study develops a novel method combining the multigrid idea and the homotopy technique for nonlinear inverse problems, in which the forward problems are modeled by some forms of partial differential equations. The method first attempts to use the multigrid method to decompose the original inverse problem into a sequence of sub-inverse problems which depend on the grid variables and are solved in proper order according to the grid size from the coarsest to the finest, and then carries out the inversion on the coarsest grid by the homotopy method. The strategy may give a rapidly and globally convergent method. As a practical application, this method is used to solve the nonlinear inverse problem of a nonlinear convection-diffusion equation, which is the saturation equation within the two-phase porous media flow. The numerical results show the effectiveness of the multigrid-homotopy method in the aspects of stability, global convergence, and fast computation. Most important of all, the comparative numerical examples show that the multigrid-homotopy method not only has a larger convergence region than the multigrid method and regularized Gauss-Newton method, but also requires less computational cost than the homotopy method and regularized Gauss-Newton method.

Biography:

Dr. Tao Liu obtained his PhD in Mathematics from Harbin Institute of Technology. Currently, he is working in School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore; School of Mathematics and Statistics, Northeastern University at Qinhuangdao, China. He served as the reviewer for Mathematical Reviews of American Mathematical Society, International Technical Committee Member of CMBDA 2022, ICARCE 2022, CPRA 2022, AMMCS 2022, AMICS 2022, AMDS 2022, and reviewer for many SCI journals. His research interests includes: Deep Learning; Reinforcement Learning; Multiscale Methods (Multigrid and Wavelet); Homotopy Method; Inverse and Ill-posed Problems; Parameter Reconstruction. In related subjects, he has published nearly 30 peer-reviewed SCI journal papers, including seven top SCI journal papers. He has been awarded as the leader of the Outstanding Project of Nature Science Foundation of Hebei Province of China in 2019.