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.

The Discrete Structure of the KKT System

The KKT system (8)–(9) is formulated as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} \left( \partial _{\tau } C_h^n, v_h\right) + \left( a\nabla \bar{C}_h^n, \nabla v_h\right) + \left( \nabla \cdot \left( \mathbf {v}\bar{C}_h^n\right) ,v_h\right) = \left( \bar{f}^n_h, v_h\right) +\langle \bar{q}^n,v_h\rangle _{\varGamma _2}, ~~\forall \,v_h\in \mathring{V}^h\,\\ -\left( \partial _{\tau } G_h^n, w_h\right) + \left( a\nabla \bar{G}_h^n, \nabla w_h\right) + \left( \nabla \cdot \left( \mathbf {v}w_h\right) ,\bar{G}_h^n\right) \\ =-\left( A(\mathbf{x})\left( \bar{C}_h^{n}(\mathbf{x})-\bar{C}^{\varepsilon ,n}(\mathbf{x})\right) , w_h\right) , ~~\forall w_h\in \mathring{V}^h \,\\ - \left( G_h^n, g_h^{\tau }\right) +\beta _1\left( \partial _{\tau } f_h^n,\partial _{\tau } g_h^{\tau }\right) + \beta _2 \left( \nabla f_h^n, \nabla g_h^{\tau }\right) =0, ~~\forall \,g_h^{\tau }\in W_h^{\tau }. \end{array}\right. } \end{aligned}$$

(16)

To better understand the discrete structure of (16), we denote the identity and zero matrices as I and $\mathbf {0}$ respectively, and the basis functions of the finite element spaces $V^h$ and $W_h^{\tau }$ by $\phi =(\phi _i)^T,\,i=1,\,\ldots ,\,N$ and $g_j^n,\,j=1,\,\ldots ,\,N,\,n=0,\,\ldots ,\,M$, respectively, let

$$\begin{aligned}&A=(a_{ij})_{i,j=1,\ldots ,N},\quad a_{ij}=(a \nabla \phi _i, \nabla \phi _j)\\&B=(b_{ij})_{i,j=1,\ldots ,N},\quad b_{ij}=( \phi _i, \phi _j)\\&E=(e_{ij})_{i,j=1,\ldots ,N},\quad e_{ij}=(\nabla \cdot (\mathbf {v} \phi _i), \phi _j)\\&L^{mn}=(l_{ij}^{mn})_{i,j=1,\ldots ,N, 0\le m, n\le M},\quad l_{ij}^{mn}=\left( \displaystyle \frac{\partial g_i^m}{\partial t}, \displaystyle \frac{\partial g_j^n}{\partial t}\right) \\&K^{mn}=(k_{ij}^{mn})_{i,j=1,\ldots ,N, 0\le m, n\le M},\quad k_{ij}^{mn}=\big ( \nabla g_i^m, \nabla g_j^n\big )\\&D^{mn}= (d_{ij}^{mn})_{i,j=1,\ldots ,N, 0\le m, n\le M},\quad d_{ij}^{mn}=\big ( g_i^m, g_j^n\big ), \end{aligned}$$

and based on these element matrices we define

$$\begin{aligned}&A_1=B+\displaystyle \frac{\tau }{2}(A+E),\quad A_2=-B+\displaystyle \frac{\tau }{2}(A+E) \\&B_1=B+\displaystyle \frac{\tau }{2}\big (A+E^T\big ),\quad B_2 =-B+\displaystyle \frac{\tau }{2}\big (A+E^T\big )\\&B_3= \text{ zeros } \text{ except } \text{1 } \text{ at } \text{ the } \text{ measurement } \text{ locations }\quad \quad \quad \\&W^{mn}=\beta _1 L^{mn}+ \beta _2 K^{mn}, \end{aligned}$$

Then the system (16) takes the following form

$$\begin{aligned} \left( \begin{array}{ccc}BC&BG&Bf\end{array}\right) \left( \begin{array}{c}C^0\\ C^1\\ \vdots \\ C^{M-2}\\ C^{M-1}\\ C^{M}\\ G^0\\ G^1\\ G^2\\ \vdots \\ G^{M-2}\\ G^{M-1}\\ G^{M}\\ f^0\\ f^1\\ f^2\\ \vdots \\ f^{M-2}\\ f^{M-1}\\ f^{M}\\ \end{array}\right) =\left( \begin{array}{c}C^0\\ \langle \bar{q}^1,\phi \rangle _{\varGamma _2} \\ \vdots \\ \langle \bar{q}^{M-1},\phi \rangle _{\varGamma _2} \\ \langle \bar{q}^{M},\phi \rangle _{\varGamma _2}\\ \tau /2 B_3 \big (C^{\varepsilon ,0}+ C^{\varepsilon ,1}\big )\\ \vdots \\ \tau /2 B_3 \big (C^{\varepsilon ,M-2}+ C^{\varepsilon ,M-1}\big )\\ \tau /2 B_3 \big (C^{\varepsilon ,M-1}+ C^{\varepsilon ,M}\big )\\ G^M \\ 0\\ 0\\ \vdots \\ 0\\ 0\end{array}\right) , \end{aligned}$$

where the block matrices BC, BG and Bf are given by

$$\begin{aligned} BC:= & {} \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} I&{} \mathbf {0}&{} \cdots &{} \mathbf {0}&{}\mathbf {0}&{} \mathbf {0}\\ A_2&{}A_1&{}\cdots &{}\mathbf {0}&{}\mathbf {0}&{} \mathbf {0}\\ \mathbf {0}&{}\ddots &{} \ddots &{} \mathbf {0} &{}\mathbf {0}&{} \mathbf {0}\\ \mathbf {0}&{}\mathbf {0} &{} \ddots &{} A_2 &{} A_1&{} \mathbf {0}\\ \mathbf {0}&{}\mathbf {0}&{}\cdots &{} \mathbf {0}&{} A_2 &{} A_1\\ \frac{\tau }{2} B_3&{}\frac{\tau }{2} B_3&{}\cdots &{}\mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ \mathbf {0}&{}\ddots &{} \ddots &{}\mathbf {0} &{} \mathbf {0}&{}\mathbf {0}\\ \mathbf {0}&{}\mathbf {0}&{}\cdots &{} \frac{\tau }{2} B_3 &{} \frac{\tau }{2} B_3&{}\mathbf {0}\\ \mathbf {0}&{}\mathbf {0}&{}\cdots &{}\mathbf {0}&{} \frac{\tau }{2} B_3 &{} \frac{\tau }{2} B_3\\ \mathbf {0}&{}\mathbf {0}&{}\cdots &{}\mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ \mathbf {0}&{}\mathbf {0}&{}\cdots &{}\mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ \mathbf {0}&{}\mathbf {0}&{}\cdots &{}\mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ \mathbf {0}&{}\mathbf {0}&{}\cdots &{}\mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ \mathbf {0}&{}\mathbf {0}&{}\cdots &{}\mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ \mathbf {0}&{}\mathbf {0}&{}\cdots &{}\mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\end{array},\right) \\ BG:= & {} \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \mathbf {0}&{}\mathbf {0}&{} \mathbf {0}&{} \cdots &{} \mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ \mathbf {0}&{} \mathbf {0}&{} \mathbf {0}&{} \cdots &{}\mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ \mathbf {0}&{} \mathbf {0}&{} \mathbf {0}&{}\cdots &{}\mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ \mathbf {0}&{} \mathbf {0}&{} \mathbf {0}&{}\cdots &{}\mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ \mathbf {0}&{} \mathbf {0}&{} \mathbf {0}&{}\cdots &{}\mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ B_1&{}B_2&{}\mathbf {0}&{} \cdots &{}\mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ \mathbf {0}&{}\ddots &{} \ddots &{}\cdots &{}\mathbf {0}&{} \mathbf {0}&{}\mathbf {0}\\ \mathbf {0}&{} \mathbf {0}&{} \mathbf {0}&{}\cdots &{}B_1&{}B_2&{}\mathbf {0}\\ \mathbf {0}&{} \mathbf {0}&{} \mathbf {0}&{}\cdots &{}\mathbf {0}&{}B_1&{}B_2\\ \mathbf {0}&{} \mathbf {0}&{} \mathbf {0}&{}\cdots &{}\mathbf {0}&{}\mathbf {0}&{}I\\ -D^{00}&{} -D^{01}&{} \mathbf {0}&{}\cdots &{}\mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ -D^{10}&{} -D^{11}&{} -D^{12}&{}\cdots &{}\mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ \mathbf {0}&{}\ddots &{} \ddots &{} \ddots &{} \mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ \mathbf {0}&{} \mathbf {0}&{} \mathbf {0}&{}\cdots &{}-D^{M-1,M-2}&{}-D^{M-1,M-1}&{}-D^{M-1,M}\\ \mathbf {0}&{} \mathbf {0}&{} \mathbf {0}&{}\cdots &{}\mathbf {0}&{}-D^{M,M-1}&{}-D^{MM}\end{array}\right) \end{aligned}$$

$$\begin{aligned} Bf := \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \mathbf {0}&{}\mathbf {0}&{} \mathbf {0}&{} \cdots &{} \mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ -\frac{\tau }{2} B&{}-\frac{\tau }{2} B&{}\mathbf {0}&{}\cdots &{} \mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ \mathbf {0}&{} \ddots &{} \ddots &{}\cdots &{}\mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ \mathbf {0}&{} \mathbf {0} &{} \ddots &{}\cdots &{}-\frac{\tau }{2} B&{}-\frac{\tau }{2} B&{}\mathbf {0}\\ \mathbf {0}&{}\mathbf {0}&{}\mathbf {0}&{}\cdots &{}\mathbf {0}&{}-\frac{\tau }{2} B&{}-\frac{\tau }{2} B\\ \mathbf {0}&{}\mathbf {0}&{}\mathbf {0}&{}\cdots &{}\mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ \mathbf {0}&{}\mathbf {0}&{}\mathbf {0}&{}\cdots &{} \mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ \mathbf {0}&{}\mathbf {0}&{}\mathbf {0}&{}\cdots &{}\mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ \mathbf {0}&{}\mathbf {0}&{} \mathbf {0}&{}\cdots &{}\mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ \mathbf {0}&{}\mathbf {0}&{} \mathbf {0}&{}\cdots &{}\mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ W^{00}&{} W^{01}&{} \mathbf {0}&{}\cdots &{}\mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ W^{10}&{} W^{11}&{} W^{12}&{}\cdots &{}\mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ \mathbf {0}&{}\ddots &{} \ddots &{} \ddots &{}\mathbf {0}&{}\mathbf {0}&{}\mathbf {0}\\ \mathbf {0}&{}\mathbf {0}&{} \mathbf {0}&{} \cdots &{}W^{M-1,M-2}&{}W^{M-1,M-1}&{}W^{M-1,M}\\ \mathbf {0}&{}\mathbf {0}&{}\mathbf {0}&{}\cdots &{}\mathbf {0} &{}W^{M,M-1}&{}W^{MM}\end{array}\right) . \end{aligned}$$