热传导反问题的快速时间离散拟边值方法
Fast Parallel-in-Time Quasi-Boundary Value Methods for Backward Heat Conduction Problems
摘要：
本文提出了两种新的拟边值方法来正则化不适定的向后热传导问题。在空间和时间的标准有限差分离散，所获得的所有在一次非对称稀疏线性系具有所需的块$ω$循环结构，它可以用来设计一个高效的并行时间（PinT）直接求解器，建立在一个显式的基于FFT的对角化的时间离散矩阵。收敛性分析证明了正则化参数的最佳选择。数值算例验证了我们的分析，并说明了优越的计算效率，我们提出的PinT方法。
英文摘要：
In this paper we proposed two new quasi-boundary value methods for regularizing the ill-posed backward heat conduction problems. With a standard finite difference discretization in space and time, the obtained all-at-once nonsymmetric sparse linear systems have the desired block $ω$-circulant structure, which can be utilized to design an efficient parallel-in-time (PinT) direct solver that built upon an explicit FFT-based diagonalization of the time discretization matrix. Convergence analysis is presented to justify the optimal choice of the regularization parameter. Numerical examples are reported to validate our analysis and illustrate the superior computational efficiency of our proposed PinT methods.
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