An Adaptive Parareal Algorithm
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In this paper, we consider the problem of accelerating the numerical simulation of time dependent problems by time domain decomposition. The available algorithms enabling such decompositions present severe efficiency limitations and are an obstacle for the solution of large scale and high dimensional problems. Our main contribution is the improvement of the parallel efficiency of the parareal in time method. The parareal method is based on combining predictions made by a numerically inexpensive solver (with coarse physics and/or coarse resolution) with corrections coming from an expensive solver (with high-fidelity physics and high resolution). At convergence, the parareal algorithm provides a solution that has the fine solver's high-fidelity physics and high resolution In the classical version of parareal, the fine solver has a fixed high accuracy which is the major obstacle to achieve a competitive parallel efficiency. In this paper, we develop an adaptive variant of the algorithm that overcomes this obstacle. Thanks to this, the only remaining factor impacting performance becomes the cost of the coarse solver. We show both theoretically and in a numerical example that the parallel efficiency becomes very competitive when the cost of the coarse solver is small.
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