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

Wang, Zhongjian1,2; Zhang, Wenlong3; Zhang, Zhiwen4

2023-08-15

10.1016/j.jcp.2023.112156

JOURNAL OF COMPUTATIONAL PHYSICS   Impact Factor & Quartile

0021-9991

1090-2716

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.(c) 2023 Elsevier Inc. All rights reserved.

Parabolic inverse source problems Regularization method Data -driven model reduction method Proper orthogonal decomposition (POD) Stochastic error estimate Optimal regularization parameter

SCI

English

Corresponding

Shenzhen Sci-Tech Fund[17300318] ; Hong Kong RGC["17307921","11901282"] ; null[RCBS20200714114941241] ; null[12171406]

Computer Science ; Physics

Computer Science, Interdisciplinary Applications ; Physics, Mathematical

WOS:000990548800001

ACADEMIC PRESS INC ELSEVIER SCIENCE

PHYSICS

Web of Science

1.Univ Chicago, Dept Stat, Chicago, IL 60637 USA
2.Univ Chicago, CCAM, Chicago, IL 60637 USA
3.Southern Univ Sci & Technol SUSTech, Dept Math, 1088 Xueyuan Blvd, Shenzhen, Guangdong, Peoples R China
4.Univ Hong Kong, Dept Math, Pokfulam Rd, Hong Kong, Peoples R China

Wang, Zhongjian,Zhang, Wenlong,Zhang, Zhiwen. A data-driven model reduction method for parabolic inverse source problems and its convergence analysis[J]. JOURNAL OF COMPUTATIONAL PHYSICS,2023,487.

Wang, Zhongjian,Zhang, Wenlong,&Zhang, Zhiwen.(2023).A data-driven model reduction method for parabolic inverse source problems and its convergence analysis.JOURNAL OF COMPUTATIONAL PHYSICS,487.

Wang, Zhongjian,et al."A data-driven model reduction method for parabolic inverse source problems and its convergence analysis".JOURNAL OF COMPUTATIONAL PHYSICS 487(2023).