Numerical discretization of a regularized inverse source problem leads to a non-symmetric saddle point linear system. Interestingly, the Schur complement of the non-symmetric saddle point system is Hermitian positive definite (HPD). Then, we propose a preconditioner matching the Schur complement (MSC). Theoretically, we show that the preconditioned conjugate gradient (PCG) method for a linear system with the preconditioned Schur complement as coefficient has a linear convergence rate independent of the matrix size and value of the regularization parameter involved in the inverse problem. Fast implementations are proposed for the matrix-vector multiplication of the preconditioned Schur complement so that the PCG solver requires only quasi-linear operations. To the best of our knowledge, this is the first solver with guarantee of linear convergence for the inversion of Schur complement arising from the discrete inverse problem. Combining the PCG solver for inversion of the Schur complement and the fast solvers for the forward problem in the literature, the discrete inverse problem (the saddle point system) is solved within a quasi-linear complexity. Numerical results are reported to show the performance of the proposed matching Schur complement (MSC) preconditioning technique.

中文翻译：

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逆源问题的匹配Schur补预处理技术

正则化逆源问题的数值离散化导致非对称鞍点线性系统。有趣的是，非对称鞍点系统的 Schur 补是 Hermitian 正定 (HPD)。然后，我们提出了一个与 Schur 补集（MSC）匹配的预处理器。理论上，我们证明了以预条件 Schur 补作为系数的线性系统的预条件共轭梯度（PCG）方法具有与反问题中涉及的矩阵大小和正则化参数值无关的线性收敛速度。提出了预条件 Schur 补集的矩阵向量乘法的快速实现，以便 PCG 求解器仅需要准线性运算。据我们所知，这是第一个保证离散反问题的 Schur 补求逆线性收敛的求解器。结合文献中用于 Schur 补数反演的 PCG 求解器和用于前向问题的快速求解器，可以在准线性复杂度内求解离散反问题（鞍点系统）。数值结果显示了所提出的匹配 Schur 补体 (MSC) 预处理技术的性能。