Overlapping Domain Decomposition Methods Based on Tensor Format for Solving High-Dimensional Partial Differential Equations

doi:10.1007/s42967-024-00429-3

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (3): 987-1001.doi: 10.1007/s42967-024-00429-3

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Overlapping Domain Decomposition Methods Based on Tensor Format for Solving High-Dimensional Partial Differential Equations

Yu-Han Chen^1,2, Chen-Liang Li²

1 School of Electronic Engineering and Automation, Guilin University of Electronic Technology, Guilin 541004, Guangxi, China;
2 School of Mathematics and Computing Science, Center for Applied Mathematics of Guangxi (GUET), Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin 541004, Guangxi, China

Received:2023-11-14 Revised:2024-05-20 Accepted:2024-05-21 Online:2025-09-20 Published:2025-05-23
Supported by:
This work was supported by the National Natural Science Foundation of China (12161027), the Guangxi Natural Science Foundation of China (2020GXNSFAA159143), and partially supported by the Science and Technology Project of Guangxi of China (Guike AD23023002).

Abstract

Abstract: Based on the equivalence between the Sylvester tensor equation and the linear equation obtained by discretization of partial differential equations (PDEs), an overlapping Schwarz alternative method based on the tensor format and an overlapping parallel Schwarz method based on the tensor format for solving high-dimensional PDEs are proposed. The complexity of the new algorithms is discussed. Finally, the feasibility and effectiveness of the new methods are verified by some numerical examples.

Key words: High-dimensional partial differential equations (PDEs), Sylvester tensor equation, Overlapping Schwarz alternative method based on tensor format, Overlapping parallel Schwarz method based on tensor format

CLC Number:

Yu-Han Chen, Chen-Liang Li. Overlapping Domain Decomposition Methods Based on Tensor Format for Solving High-Dimensional Partial Differential Equations[J]. Communications on Applied Mathematics and Computation, 2025, 7(3): 987-1001.

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Overlapping Domain Decomposition Methods Based on Tensor Format for Solving High-Dimensional Partial Differential Equations

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