Inverse Problem for the Nonlinear Convection–Diffusion Equation by Using the Multigrid Method and Constraint Data

Shuai Wang, Shiyi Ling, Heyang Chao, Yunfei Qi (), Wenwen Zhang, Qiang Ma and Tao Liu ()
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Shuai Wang: Foundation Department, Changchun Guanghua University, Changchun 130033, China
Shiyi Ling: School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China
Heyang Chao: School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China
Yunfei Qi: Eighth Geological Brigade of Hebei Bureau of Geology and Mineral Resources Exploration (Hebei Center of Marine Geological Resources Survey), Qinhuangdao 066000, China
Wenwen Zhang: School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore
Qiang Ma: Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai 264209, China
Tao Liu: School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China

Mathematics, 2024, vol. 12, issue 15, 1-15

Abstract: In the article, we propose a combination method based on the multigrid method and constraint data to solve the inverse problem in the context of the nonlinear convection–diffusion equation in the multiphase porous media flow. The inverse problem consists of a data-fitting term involving the discretization of a direct problem, a constraint term concerning the incorporation of constraint data, and a regularization term dealing with the improvement of stability. A multigrid method, which is specialized for large-scale problems and works by keeping the consistence of objective functionals between different grids, is applied in the process of inversion. Based on the numerical results, the proposed combination method has the advantages of fast calculation, high precision, good stability, and strong anti-noise ability in computation. It obtains good performance under various noise levels, as well as outperforming any one method used alone.

Keywords: nonlinear convection–diffusion equation; inversion; multigrid; constraint data; porous media flow (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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