A Parallel Method for Backward Parabolic Problems Based on the Laplace Transformation

Abstract

A parallel method for time discretization of backward parabolic problems is proposed. The problem is reformulated to a set of Helmholtz‐type problems with a parameter on a suitably chosen contour in the complex plane. After solving the resulting elliptic equations, which can be solved in parallel, we obtain a regularized solution with high frequency terms cut off by the inverse Laplace transforms without requiring the knowledge of the eigenfunctions of the differential operator. Since the regularized solution is obtained without artificial perturbation and high frequency components of the noise are suppressed, the quality of the solution is improved significantly compared to those obtained by other methods. Two different numerical inversions of Laplace transforms, with an arbitrary high order of accuracy and spectral accuracy, respectively, are used. Error estimates and numerical examples are presented.