  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.
