**Main innovation**: for the first time, a common methodology has been developed to analyze and compare most of the well known PinT algorithms. This will help in selecting the best time integration parameters for a given type of problem and inform the development of new and more efficient variants of existing PinT methods.

TIME-X researchers have developed an analysis framework that allows to estimate the convergence properties of most current PinT methods using generic error bounds. Such a tool is important when designing those PinT algorithms, as** their parallel efficiency is directly linked to the number of iterations** required to achieve a given accuracy.

A generic Python library **gfm** has been developed and was made publicly available. In the example below, the accuracy of different PinT algorithms is shown as a function of the number of iterations. We can identify which algorithm converges the fastest in this setting and which methods converge slowly.

Knowing only about convergence, however, is not enough to predict speedup and parallel efficiency of PinT methods, since the **computational cost for one iteration** is not the same across algorithms. Therefore, the next step is to combine the convergence model with a workload model to estimate performance and overheads. We are currently developing a Python software that will be able to predict the performance of different parallel-in-time methods and help to find optimal configurations to maximize efficiency.

#### Publication

- M. J. Gander, T. Lunet, D. Ruprecht, R. Speck, “A unified analysis framework for iterative parallel-in-time algorithms”, arXiv:2203.16069 [math.NA], 2022.