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

Abstract

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.

A parallel implicit domain decomposition algorithm for the large eddy simulation of incompressible turbulent flows on 3D unstructured meshes

“…Here, ( f , g) Ω = ∫ Ω f · g dΩ is the standard scalar inner product in L 2 (Ω). We discretize the weak form of the LES equations (16) in space with a P 1 − P 1 stabilized finite element method. 22,23 First, we triangulate the computational domain Ω by a conformal tetrahedral mesh  h = {K} with h K the diameter of the element K ∈  h .…”

“…NKS has been successfully applied to solve different kind of nonlinear problems, for example, PDE‐constrained optimization problems, fluid‐structure interaction problems, non‐Newtonian fluid problems, inverse source problems, and elasticity problems, and has shown good parallel scalability to thousands of processors. In this work, we extend the algorithm to solve the fully implicit 3D LES problems and to investigate the performance of NKS for an industrial application.…”

Efficient time domain decomposition algorithms for parabolic PDE-constrained optimization problems

“…are the error vector functions of the state y and adjoint state p on each sub-interval. If we can show that both e k 1 (T 1 ) and w k 2 (T 1 ) converge to zero as k goes to infinity, then the uniqueness of the solution to both (25) and (26) obviously implies all error vector functions over each sub-interval also converge to zero as k goes to infinity.…”

“…In (25), it follows from multiplying from the left side of the first equation with (e k 1 ) , the transpose of (e k 1 ), and the second one with (w k 1 ) , respectively,…”

“…Although it has been pointed out in [15,17] that two-level algorithms are needed to achieve better scalable convergence rates than one-level algorithms, there is very few work discussing such two-level time domain decomposition algorithms in the setting of forward-and-backward optimality system. For instance, some two-level spacetime parallel domain decomposition preconditioners were very recently proposed in [25,11], where the full discretizations are based on first-order accurate backward Euler scheme in time and the convergence analysis of the algorithms was not given. In contrast to the aforementioned methods, our proposed time domain decomposition algorithms are based on a second-order accurate leapfrog scheme in time and are proved to be convergent as stand-alone solvers.…”