  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.
