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

收稿日期:2023-11-14 修回日期:2024-05-20 接受日期:2024-05-21 出版日期:2025-09-20 发布日期:2025-05-23
通讯作者: Chen-Liang Li, chenli@guet.edu.cn;Yu-Han Chen, chen_yh0802@163.com E-mail:chenli@guet.edu.cn;chen_yh0802@163.com
基金资助:
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).

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).

摘要/Abstract

摘要： 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.

关键词: 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

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

中图分类号:

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.

参考文献

1. Bai, Z.-Z.: On Hermitian and skew-Hermitian splitting iteration methods for continuous Sylvester equations. J. Comput. Math. 29(2), 185–198 (2011)
2. Bai, Z.-Z., Pan, J.-Y.: Matrix Analysis and Computations. SIAM, Philadelphia (2021)
3. Baur, U., Benner, P.: Cross-Gramian based model reduction for data-sparse systems. Electron. Trans. Numer. Anal. 31, 256–270 (2008)
4. Beik, F.P.A., Ahmadi-Asl, S.: Residual norm steepest descent based iterative algorithms for Sylvester tensor equations. J. Math. Model. 2(2), 115–131 (2015)
5. Beik, F.P.A., Movahed, F.S., Ahmadi-Asl, S.: On the Krylov subspace methods based on tensor format for positive definite Sylvester tensor equations. Numer. Linear Algebra. Appl. 23(3), 444–466 (2016)
6. Boglaev, I.: On a domain decomposition algorithm for a singularly perturbed reaction-diffusion problem. J. Comput. Appl. Math. 98(2), 213–232 (1998)
7. Boglaev, I: An implicit-explicit domain decomposition algorithm for a singularly perturbed parabolic problem. Comput. Math. Appl. 38(5/6), 41–53 (1999)
8. Boglaev, I: Domain decomposition in boundary layer for singularly perturbed problem. Appl. Numer. Math. 34(2), 145–166 (2000)
9. Cai, X.-C., Casarin, M.A., Elliott, F.W.: Overlapping Schwarz algorithms for solving Helmholtz’s equation. In: Domain Decomposition Methods, vol. 10, pp. 391–399 (1998)
10. Chen, Y.-H., Li, C.-L.: A tensor multigrid method for solving Sylvester tensor equations. IEEE Trans. Auto. Sci. Eng. (2023). https://doi.org/10.1109/TASE.2023.3296110
11. Chen, Z., Lu, L.-Z.: A projection method and Kronecker product preconditioner for solving Sylvester tensor equations. Sci. China Math. 55(6), 1281–1292 (2012)
12. Chu, D., Tan, R.C.E.: Numerically reliable computing for the row by row decoupling problem with stability. SIAM J. Matrix Anal. Appl. 23(4), 1143–1170 (2002)
13. Datta, B.N., Lin, W.-W., Wang, J.-N.: Robust partial pole assignment for vibrating systems with aerodynamic effects. IEEE Trans. Autom. Control. 51(21), 1979–1984 (2006)
14. Dawson, C.N., Du, Q., Dupont, T.F.: A finite difference domain decomposition algorithm for numerical solution of the heat equation. Math. Comput. 57(195), 63–63 (1991)
15. Dawson, C.N., Dupont, T.F.: Explicit/implicit, conservative domain decomposition procedures for parabolic problems based on block-centered finite differences. SIAM J. Numer. Anal. 31(4), 1045–1061 (1994)
16. Gander, M.J., Halpern, L., Nataf, F.: Optimized Schwarz methods. In: Proceedings of the 12th International Conference on Domain Decomposition Methods, Japan, pp. 15–27 (2001)
17. Grasedyck, L.: Existence and computation of low Kronecker-rank approximations for large linear systems of tensor product structure. Computing 72(3/4), 247–265 (2004)
18. Heyouni, M., Saberi-Movahed, F., Tajaddini, A.: A tensor format for the generalized Hessenberg method for solving Sylvester tensor equations. J. Comput. Appl. Math. 377, 112878 (2020)
19. Kang, L.-S.: Domain Decomposition and Parallel Algorithm. Wuhan University Press, Wuhan (1987)
20. Kang, L.-S., Quan, H.-Y.: Splitting Method for Numerical Solution of High Dimensional PDE. Sci. Technology Press, Shanghai (1990)
21. Karimi, S., Dehghan, M.: Global least squares method based on tensor form to solve linear systems in Kronecker format. Trans. Inst. Meas. Control 40(7), 2378–2386 (2018)
22. Li, J.-H., Ding, F., Yang, G.-W.: Maximum likelihood least squares identification method for input nonlinear finite impulse response moving average systems. Math. Comput. Model. 55, 442–450 (2012)
23. Lions, P.L.: On Schwarz Alternating Method I: First International Symposium on Domain Decomposition Methods for Partial Differential Equations, pp. 1–42. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1988)
24. Lions, P.L.: On Schwarz Alternating Method II: Stochastic Interpretation and Order Properties, pp. 47–70. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1989)
25. Lions, P.L.: On the Schwarz Alternating Method III: a Variant for Nonoverlapping Subdomains, pp. 202–223. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1990)
26. Lu, T., Shih, T.M., Liem, C.B.: Two synchronous parallel algorithms for partial differential equations. J. Comput. Math. 9(1), 74–85 (1991)
27. Lyu, G.-X., Ma, F.-M.: Finite difference domain decomposition algorithm for two-dimensional heat equation. J. Numer. Methods Comput. Appl. 27(2), 96–105 (2006)
28. Lyu, T., Shi, J.-M., Li, Z.-B.: Domain Decomposition Methods: New Numerical Techniques for Solving PDE. Science Press, Beijing (1997)
29. McInnes, L.C., Susan-Resigna, R., Keyes, D.E.: Additive Schwarz methods with nonreflecting boundary conditions for the parallel computation of Helmholtz problems. In: Domain Decomposition Methods (Boulder, CO, 1997), vol. 10, pp. 325–333 (1998)
30. Miller, K.: Numerical analogs to the Schwarz alternating procedure. Numer. Math. 7(2), 91–103 (1965)
31. Najafi-Kalyani, M., Beik, F.P.A., Jbilou, K.: On global iterative schemes based on Hessenberg process for (ill-posed) Sylvester tensor equations. J. Comput. Appl. Math. 373, 112216 (2020)
32. Nataf, F.: Absorbing boundary conditions in block Gauss-Seidel methods for the convection problems. Math. Models Methods Appl. Sci. 6(4), 481–502 (1996)
33. Nataf, F., Rogier, F.: Factorization of the convection-diffusion operator and the Schwarz algorithm. Math. Models Methods Appl. Sci. 5(1), 67–93 (1995)
34. Schwarz, H.A.: Gesammelte Mathematiche Abhandlungen, vol. 2, pp. 133–143. Springer, Berlin (1890) (First published in Viertel jahrsschrift der Naturforschenden Gesellschaft 15(2), 272–286 (1870))
35. Tang, W.-P.: Schwarz Splitting and Template Operators. Stanford University, Stanford (1987)
36. Wang, W., Ding, F., Dai, J.-Y.: Maximum likelihood least squares identification for systems with autoregressive moving average noise. Appl. Math. Model. 36(5), 1842–1853 (2012)
37. Zhang, X.-F., Wang, Q.-W.: Developing iterative algorithms to solve Sylvester tensor equations. Appl. Math. Comput. 409, 126403 (2021)

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

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

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