  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. 
Fast Parallel-in-Time Quasi-Boundary Value Methods for Backward Heat   Conduction Problems