A direct parallel-in-time quasi-boundary value method for inverse space-dependent source problems

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 studied an effective way for regularizing inverse problems, which was shown to achieve optimal order convergence under suitable assumptions. However, fast direct or iterative solvers resulting large-scale all-at-once linear systems have rarely literature. In this work, we propose and analyze a modified that leads diagonalization-based parallel-in-time (PinT) solver, can dramatic speedup CPU times when compared with MATLAB’s sparse solver. particular, time-discretization matrix B is be diagonalizable, condition number of its eigenvector V proven exhibit only quadratic growth, guarantees roundoff errors due diagonalization well-conditioned. Several 1D 2D examples are presented demonstrate very promising computational efficiency our proposed method, where cases speeded up by three orders magnitude.