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 $\omega$-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.

翻译：在本文中,我们提出了两种新的准边界值方法,用于规范不测的后向热导问题。由于在空间和时间上存在标准的有限差异,获得的全非对称稀散线性系统具有理想的块块($\omega$-circurculant)结构,可用于设计高效的平行实时直接解决器,它以基于时间离散矩阵的基于FFFT的明显分解法为根据。提出了趋同分析,以证明最佳选择正规化参数的合理性。报告的数字实例证实了我们的分析,并说明了我们提议的Pintt方法的优劣计算效率。