论文标题
逆空间源源问题的直接平行时间准实用值法
A Direct Parallel-in-Time Quasi-Boundary Value Method for Inverse Space-Dependent Source Problems
论文作者
论文摘要
逆源问题经常出现在现实世界中，例如定位未知的地下水污染物来源。与Tikhonov正则化不同，已提出并分析了准实数值方法，作为正规化此类反源问题的有效方法，该方法被证明在适当的假设下实现了最佳的订单收敛速率。但是，在文献中很少研究为产生的全面大规模线性系统的快速直接或迭代求解器。在这项工作中，我们首先提出并分析了一种修改的准实数值方法，然后开发了基于对角线的平行时间（PINT）直接求解器，与Matlab的稀疏直接求解器相比，该方法可以在CPU时获得巨大的加速。特别是，时间限制矩阵$ b $表明是可对角线的，并且证明其特征向量矩阵$ v $的状况数量已证明可以显示出二次增长，这保证了由于对角化而导致的圆形错误。提出了几个1D和2D示例，以证明我们提出的方法的非常有希望的计算效率，在2D情况下，CPU时间可以乘以三个数量级加速。
Inverse source problems arise often in real-world applications, such as localizing unknown groundwater contaminant sources. Being different from Tikhonov regularization, the quasi-boundary value method has been proposed and analyzed as an effective way for regularizing such inverse source problems, which was shown to achieve an optimal order convergence rate under suitable assumptions. However, fast direct or iterative solvers for the resulting all-at-once large-scale linear systems have been rarely studied in the literature. In this work, we first proposed and analyzed a modified quasi-boundary value method, and then developed a diagonalization-based parallel-in-time (PinT) direct solver, which can achieve a dramatic speedup in CPU times when compared with MATLAB's sparse direct solver. In particular, the time-discretization matrix $B$ is shown to be diagonalizable, and the condition number of its eigenvector matrix $V$ is proven to exhibit quadratic growth, which guarantees the roundoff errors due to diagonalization is well controlled. Several 1D and 2D examples are presented to demonstrate the very promising computational efficiency of our proposed method, where the CPU times in 2D cases can be speedup by three orders of magnitude.
