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Time-Slicing, Rescaling & Ratio-based Parallel Time Integration

Noha Makhoul-Karam 1 
1 SAGE - Simulations and Algorithms on Grids for Environment
Inria Rennes – Bretagne Atlantique , IRISA-D1 - SYSTÈMES LARGE ÉCHELLE
Abstract : Recently, many parallel-in-time algorithms have been proposed for solving initial value problems of the form (S): dY/dt = F(Y ); Y (t0) = Y0, that could follow, for example, from the space semi-discretization of partial di erential equations. Since there is no natural parallelism across time, those algorithms are mainly meant to tackle real-time problems or to be superposed to parallelism in the space or the method directions, thus enabling a more eff ective use of a higher number of processors. In this thesis, we propose a Ratio-based Parallel Time Integration (RaPTI) algorithm for solving (S) in a time-parallel way, in the case where the behavior of the solution is known. A sliced-time computing methodology underlies this new approach. It consists of (i) a time-slicing technique that ends a slice by shooting a relevant end-of-slice condition related to the behavior of the solution and (ii) a rescaling technique that changes both the time-variable and the solution, setting them to 0 at the beginning of each time-slice. Thus, solving (S) becomes equivalent to solving a sequence of initial value "shooting" problems, in which one seeks, on each time-slice, both the solution and the end-of-slice time. RaPTI algorithm uses this methodology, and some resulting similarity properties, for generating a coarse grid and providing ratio-based predictions of the starting values at the onset of every time-slice. The correction procedure is performed on a fi ne grid and in parallel, yielding some gaps on the coarse grid. Then, the predictions are updated and the process is iterated, until all the gaps are within a given tolerance. The originality of RaPTI algorithm lies in the fact that the predictions it provides, and their update at each iteration, do not require any integration on the coarse grid, unlike other parallel-in-time schemes. Moreover, it does not start with the choice of a coarse grid, it rather starts by choosing an end-of-slice condition that will generate a coarse grid that suits the behavior of the solution. RaPTI algorithm is applied, in this thesis, to three problems: a membrane problem, a reaction-diff usion problem and a satellite trajectory in a J2-perturbed motion. In some rare cases of invariance, it yields a perfect parallelism. In the more general cases of asymptotic and weak similarity, it yields good speed-ups.
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Submitted on : Thursday, October 18, 2012 - 11:14:16 AM
Last modification on : Thursday, January 20, 2022 - 4:19:51 PM
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  • HAL Id : tel-00743132, version 1


Noha Makhoul-Karam. Time-Slicing, Rescaling & Ratio-based Parallel Time Integration. Numerical Analysis [cs.NA]. Université Rennes 1, 2010. English. ⟨NNT : ⟩. ⟨tel-00743132⟩



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