Fast Parallel-in-Time Quasi-Boundary Value Methods for Backward Heat Conduction Problems

Abstract

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 ω-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.

A Vanka-type multigrid solver for complex-shifted Laplacian systems from diagonalization-based parallel-in-time algorithms

“…In our second example, we test a PinT quasi-boundary value method [20] for backward heat conduction problem, where the time discretization matrix (with a time step size = 1/ ) reads…”

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

“…Such a PinT direct solver can greatly speed up the quasi-boundary value methods while achieving a comparable reconstruction accuracy. Recently, such an interesting approach of integrating PinT direct solver with regularization was applied to backward heat conduction problems [27,33,34], where a block -circulant structure was exploited for developing a fast FFT-based direct PinT solver [24].…”