中文版 | English
Title

A data-driven model reduction method for parabolic inverse source problems and its convergence analysis

Author
Corresponding AuthorWenlong Zhang; Zhiwen Zhang
Publication Years
2023-04
DOI
Source Title
ISSN
0021-9991
Volume487Pages:112156
Abstract

In this paper, we propose a data-driven model reduction method to solve parabolic inverse source problems with uncertain data efficiently. Our method consists of offline and online stages. In the offline stage, we explore the low-dimensional structures in the solution space of parabolic partial differential equations (PDEs) in the forward problems with a given class of source functions and construct a small number of proper orthogonal decomposition (POD) basis functions to achieve significant dimension reduction. Equipped with the POD basis functions, we can solve the forward problems extremely fast in the online stage. Thus, we develop a fast algorithm to solve the optimization problem in parabolic inverse source problems, which is referred to as the POD method. Moreover, we design an a posteriori algorithm to find the optimal regularization parameter in the optimization problem using the proposed POD method without knowing the noise level. Under a weak regularity assumption on the solution of the parabolic PDEs, we prove the convergence of the POD method in solving the forward parabolic PDEs. In addition, we obtain the error estimate of the POD method for parabolic inverse source problems. Finally, we present numerical examples to demonstrate the accuracy and efficiency of the proposed method. Numerical results show that the POD method provides considerable computational savings over the finite element method while maintaining the same accuracy.

Keywords
URL[Source Record]
Indexed By
Language
English
SUSTech Authorship
Corresponding
Publisher
ESI Research Field
PHYSICS
Data Source
人工提交
Citation statistics
Cited Times [WOS]:0
Document TypeJournal Article
Identifierhttp://kc.sustech.edu.cn/handle/2SGJ60CL/534752
DepartmentDepartment of Mathematics
Affiliation
1.Department of Statistics and CCAM, The University of Chicago, Chicago, IL 60637, USA
2.Department of Mathematics, Southern University of Science and Technology (SUSTech), 1088 Xueyuan Boulevard, University Town of Shenzhen, Xili, Nanshan, Shenzhen, Guangdong Province, PR China
3.Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong Special Administrative Region of China
Corresponding Author AffilicationDepartment of Mathematics
Recommended Citation
GB/T 7714
Zhongjian Wang,Wenlong Zhang,Zhiwen Zhang. A data-driven model reduction method for parabolic inverse source problems and its convergence analysis[J]. Journal of Computational Physics,2023,487:112156.
APA
Zhongjian Wang,Wenlong Zhang,&Zhiwen Zhang.(2023).A data-driven model reduction method for parabolic inverse source problems and its convergence analysis.Journal of Computational Physics,487,112156.
MLA
Zhongjian Wang,et al."A data-driven model reduction method for parabolic inverse source problems and its convergence analysis".Journal of Computational Physics 487(2023):112156.
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