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Inverse source problems arise often in real-world applications, such aslocalizing unknown groundwater contaminant sources. Being different fromTikhonov regularization, the quasi-boundary value method has been proposed andanalyzed as an effective way for regularizing such inverse source problems,which was shown to achieve an optimal order convergence rate under suitableassumptions. However, fast direct or iterative solvers for the resultingall-at-once large-scale linear systems have been rarely studied in theliterature. In this work, we first proposed and analyzed a modifiedquasi-boundary value method, and then developed a diagonalization-basedparallel-in-time (PinT) direct solver, which can achieve a dramatic speedup inCPU times when compared with MATLAB's sparse direct solver. In particular, thetime-discretization matrix $B$ is shown to be diagonalizable, and the conditionnumber 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 promisingcomputational efficiency of our proposed method, where the CPU times in 2Dcases can be speedup by three orders of magnitude.

A Direct Parallel-in-Time Quasi-Boundary Value Method for Inverse Space-Dependent Source Problems