Abstract
In this paper, we propose a parallel space-time domain decomposition method for solving an unsteady source identification problem governed by the linear convection-diffusion equation. Traditional approaches require to solve repeatedly a forward parabolic system, an adjoint system and a system with respect to the unknowns. The three systems have to be solved one after another. These sequential steps are not desirable for large scale parallel computing. A space-time restrictive additive Schwarz method is proposed for a fully implicit space-time coupled discretization scheme to recover the time-dependent pollutant source intensity functions. We show with numerical experiments that the scheme works well with noise in the observation data. More importantly it is demonstrated that the parallel space-time Schwarz preconditioner is scalable on a supercomputer with over $10^3$ processors, thus promising for large scale applications.
Highlights
Pollutant source inversion problems have wide applications in, for example, the detection and monitoring of indoor and outdoor air pollution, underground water pollution, etc
We propose a fully coupled space-time domain decomposition method that couples the time with the space domain and decomposes the “space-time” domain into sub-domains, apply an additive Schwarz preconditioned Krylov subspace technique to solve the “space-time” problem
The subsystem is solved with a sparse LU factorization or an incomplete LU factorization (ILU) with the fill-in level denoted by ilulevel
Summary
Pollutant source inversion problems have wide applications in, for example, the detection and monitoring of indoor and outdoor air pollution, underground water pollution, etc. The lack of stability with respect to the measurement data is a major issue, which means that small noise in the data may lead to significant changes in the reconstructed source strength This problem has attracted much attention, and various methods have been developed, including both deterministic and statistical methods [27, 37]. Reduced space SQP methods decouple the system and iteratively update the state variable, the adjoint variable and the optimization variables by solving each subsystem in a sequential order. In some sense this is a block Gauss-Seidel iteration with three large blocks.
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