Necessary Conditions and Tight Two-level Convergence Bounds for Parareal and Multigrid Reduction in Time

Parareal and multigrid reduction in time (MGRiT) are two of the most popular parallel-in-time methods. The idea is to treat time integration in a parallel context by using a multigrid method in time. If $\Phi$ is a (fine-grid) time-stepping scheme, let $\Psi$ denote a "coarse-grid" time-stepping scheme chosen to approximate $k$ steps of $\Phi$, $k\geq 1$. In particular, $\Psi$ defines the coarse-grid correction, and evaluating $\Psi$ should be (significantly) cheaper than evaluating $\Phi^k$. A number of papers have studied the convergence of Parareal and MGRiT. However, there have yet to be general conditions developed on the convergence of Parareal or MGRiT that answer simple questions such as, (i) for a given $\Phi$ and $k$, what is the best $\Psi$, or (ii) can Parareal/MGRiT converge for my problem? This work derives necessary and sufficient conditions for the convergence of Parareal and MGRiT applied to linear problems, along with tight two-level convergence bounds. Results rest on the introduction of a "temporal approximation property" (TAP) that indicates how $\Phi^k$ must approximate the action of $\Psi$ on different vectors. Loosely, for unitarily diagonalizable operators, the TAP indicates that fine-grid and coarse-grid time integration schemes must integrate geometrically smooth spatial components similarly, and less so for geometrically high frequency. In the (non-unitarily) diagonalizable setting, the conditioning of each eigenvector, $\mathbf{v}_i$, must also be reflected in how well $\Psi\mathbf{v}_i \sim\Phi^k\mathbf{v}_i$. In general, worst-case convergence bounds are exactly given by $\min \varphi < 1$ such that an inequality along the lines of $\|(\Psi-\Phi^k)\mathbf{v}\| \leq\varphi \|(I - \Psi)\mathbf{v}\|$ holds for all $\mathbf{v}$. Such inequalities are formalized as different realizations of the TAP, and form the basis for convergence of MGRiT and Parareal.