Journal of Scientific Computing

, Volume 67, Issue 3, pp 860–882 | Cite as

Two-Level Space–Time Domain Decomposition Methods for Three-Dimensional Unsteady Inverse Source Problems

  • Xiaomao Deng
  • Xiao-Chuan CaiEmail author
  • Jun Zou


As the number of processor cores on supercomputers becomes larger and larger, algorithms with high degree of parallelism attract more attention. In this work, we propose a two-level space–time domain decomposition method for solving an inverse source problem associated with the time-dependent convection–diffusion equation in three dimensions. We introduce a mixed finite element/finite difference method and a one-level and a two-level space–time parallel domain decomposition preconditioner for the Karush–Kuhn–Tucker system induced from reformulating the inverse problem as an output least-squares optimization problem in the entire space-time domain. The new full space–time approach eliminates the sequential steps in the optimization outer loop and the inner forward and backward time marching processes, thus achieves high degree of parallelism. Numerical experiments validate that this approach is effective and robust for recovering unsteady moving sources. We will present strong scalability results obtained on a supercomputer with more than 1000 processors.


Space–time method Multilevel method Domain decomposition preconditioner Unsteady inverse source problem  Parallel computing 

Mathematics Subject Classification

49K20 65F22 65F08 65F10 65M32 65M55 65Y05 90C06 



The authors would like to thank the anonymous referees for their insightful comments and suggestions that helped us improve the quality of the paper. The work was partly supported by NSFC 11501545, 91330111, Shenzhen Program JCYJ20140901003939012, KQCX20130628112914303, 201506303000093 and 863 Program 2015AA01A302. The second author was partly support by NSF CCF-1216314. The third author was substantially supported by Hong Kong RGC Grants 404611 and 405513.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Laboratory for Engineering and Scientific Computing, Shenzhen Institutes of Advanced TechnologyChinese Academy of SciencesShenzhenPeople’s Republic of China
  2. 2.Department of Computer ScienceUniversity of Colorado BoulderBoulderUSA
  3. 3.Department of MathematicsThe Chinese University of Hong KongShatinPeople’s Republic of China

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