3维抛物问题时空相关扩散系数识别的一种并行区域分解方法  

A parallel domain decomposition method for identifying the space-time dependent diffusion coefficients of 3D parabolic problems

作　　者：邓小毛 江嘉华 袁敬岚 廖子菊 Xiaomao Deng;Jiahua Jiang;Jinglan Yuan;Ziju Liao

机构地区：[1]广东外语外贸大学数学与统计学院,广州510006 [2]暨南大学数学系,广州510632

出　　处：《中国科学：数学》2023年第11期1487-1508,共22页Scientia Sinica：Mathematica

基　　金：广东省自然科学基金(批准号:2020A1515010704和2021A1515012366)资助项目。

摘　　要：3维抛物方程中时空相关扩散系数的反演是一个具有严重病态性的非线性反问题.本文研究基于时空区域分解的快速并行方法.首先将该反问题转化为带有PDE(partial differential equation,偏微分方程)约束的最小二乘优化问题,然后通过添加适当的时空Tikhonov正则化项保证问题的适定性,并用先优化后离散的方法对一阶最优条件KKT(Karush-Kuhn-Tucker)系统进行空间有限元及时间有限差分离散.针对由此产生的大规模非线性系统,本文提出一种时空全耦合预条件并行算法.该算法不需要对3个PDE子系统进行交替迭代求解以及向前和向后的时间推进过程,而是将正问题、伴随问题和扩散系数方程的所有未知量耦合在一起,使用重叠型时空区域分解预条件子对每个Newton迭代步的Jacobi系统进行预处理,然后用Krylov子空间方法求解该系统.数值实验表明,本文所提出的算法对重建4维时空扩散系数具有良好的精度和鲁棒性,且对5,000个处理器核达到接近线性的加速比.The recovery of space-time dependent diffusion coefficients of three-dimensional parabolic problems is a highly nonlinear and ill-conditioned inverse problem.In this paper,we study a fast parallel space-time domain decomposition method.The inverse problem is firstly reformulated as an output least-squares minimization problem,and appropriate space and time Tikhonov regularization terms are added to ensure the well-posedness.The resulting fully coupled KKT(Karush-Kuhn-Tucker)system is discretized by finite elements in space and the Crank-Nicolson scheme in time.We introduce a parallel space-time fully coupled preconditioning algorithm which does not involve alternative iterative solutions among three PDE(partial differential equation)subsystems and sequential forward and backward time marching processes.Instead,the algorithm couples all the unknowns of the forward solution,the adjoint solution,and the diffusion coefficient in the entire space-time domain.At each Newton iteration step the Jacobi system is treated by a space-time overlapping Schwarz preconditioner and solved by a Krylov subspace method.Numerical experiments validate the efficiency and robustness of the proposed approach for reconstructing the four-dimensional space-time dependent diffusion coefficients,and achieve nearly linear speedup with 5,000 processors.

关 键 词：时空区域分解 并行计算 参数识别 反问题 

分 类 号：TP3[自动化与计算机技术—计算机科学与技术]