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22 changes: 11 additions & 11 deletions _bibliography/pint.bib
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Expand Up @@ -6328,15 +6328,6 @@ @unpublished{BoschEtAl2023
year = {2023},
}

@unpublished{BossuytEtAl2023,
abstract = {We propose a micro-macro parallel-in-time Parareal method for scalar McKean-Vlasov stochastic differential equations (SDEs). In the algorithm, the fine Parareal propagator is a Monte Carlo simulation of an ensemble of particles, while an approximate ordinary differential equation (ODE) description of the mean and the variance of the particle distribution is used as a coarse Parareal propagator to achieve speedup. We analyse the convergence behaviour of our method for a linear problem and provide numerical experiments indicating the parallel weak scaling of the algorithm on a set of examples. We show that convergence typically takes place in a low number of iterations, depending on the quality of the ODE predictor. For bimodal SDEs, we avoid quality deterioration of the coarse predictor (compared to unimodal SDEs) through the usage of multiple ODEs, each describing the mean and variance of the particle distribution in locally unimodal regions of the phase space. The benefit of the proposed algorithm can be viewed through two lenses: (i) through the parallel-in-time lens, speedup is obtained through the use of a very cheap coarse integrator (an ODE moment model), and (ii) through the moment models lens, accuracy is iteratively gained through the use of parallel machinery as a corrector. In contrast to the isolated use of a moment model, the proposed method (iteratively) converges to the true distribution generated by the SDE.},
author = {Ignace Bossuyt and Stefan Vandewalle and Giovanni Samaey},
howpublished = {arXiv:2310.11365v1 [math.NA]},
title = {Monte-Carlo/Moments micro-macro Parareal method for unimodal and bimodal scalar McKean-Vlasov SDEs},
url = {http://arxiv.org/abs/2310.11365v1},
year = {2023},
}

@unpublished{BouillonEtAl2023,
abstract = {The ParaDiag family of algorithms solves differential equations by using preconditioners that can be inverted in parallel through diagonalization. In the context of optimal control of linear parabolic PDEs, the state-of-the-art ParaDiag method is limited to solving self-adjoint problems with a tracking objective. We propose three improvements to the ParaDiag method: the use of alpha-circulant matrices to construct an alternative preconditioner, a generalization of the algorithm for solving non-self-adjoint equations, and the formulation of an algorithm for terminal-cost objectives. We present novel analytic results about the eigenvalues of the preconditioned systems for all discussed ParaDiag algorithms in the case of self-adjoint equations, which proves the favorable properties the alpha-circulant preconditioner. We use these results to perform a theoretical parallel-scaling analysis of ParaDiag for self-adjoint problems. Numerical tests confirm our findings and suggest that the self-adjoint behavior, which is backed by theory, generalizes to the non-self-adjoint case. We provide a sequential, open-source reference solver in Matlab for all discussed algorithms.},
author = {Arne Bouillon and Giovanni Samaey and Karl Meerbergen},
Expand Down Expand Up @@ -6933,9 +6924,18 @@ @unpublished{BetckeEtAl2024
@unpublished{BossuytEtAl2024,
abstract = {In this paper, we are concerned with the micro-macro Parareal algorithm for the simulation of initial-value problems. In this algorithm, a coarse (fast) solver is applied sequentially over the time domain, and a fine (time-consuming) solver is applied as a corrector in parallel over smaller chunks of the time interval. Moreover, the coarse solver acts on a reduced state variable, which is coupled to the fine state variable through appropriate coupling operators. We first provide a contribution to the convergence analysis of the micro-macro Parareal method for multiscale linear ordinary differential equations (ODEs). Then, we extend a variant of the micro-macro Parareal algorithm for scalar stochastic differential equations (SDEs) to higher-dimensional SDEs.},
author = {Ignace Bossuyt and Stefan Vandewalle and Giovanni Samaey},
howpublished = {arXiv:2401.01798v1 [math.NA]},
howpublished = {arXiv:2401.01798v2 [math.NA]},
title = {Micro-macro Parareal, from ODEs to SDEs and back again},
url = {http://arxiv.org/abs/2401.01798v1},
url = {http://arxiv.org/abs/2401.01798v2},
year = {2024},
}

@unpublished{BossuytEtAl2024,
abstract = {We propose a micro-macro parallel-in-time Parareal method for scalar McKean-Vlasov stochastic differential equations (SDEs). In the algorithm, the fine Parareal propagator is a Monte Carlo simulation of an ensemble of particles, while an approximate ordinary differential equation (ODE) description of the mean and the variance of the particle distribution is used as a coarse Parareal propagator to achieve speedup. We analyse the convergence behaviour of our method for a linear problem and provide numerical experiments indicating the parallel weak scaling of the algorithm on a set of examples. We show that convergence typically takes place in a low number of iterations, depending on the quality of the ODE predictor. For bimodal SDEs, we avoid quality deterioration of the coarse predictor (compared to unimodal SDEs) through the usage of multiple ODEs, each describing the mean and variance of the particle distribution in locally unimodal regions of the phase space. The benefit of the proposed algorithm can be viewed through two lenses: (i) through the parallel-in-time lens, speedup is obtained through the use of a very cheap coarse integrator (an ODE moment model), and (ii) through the moment models lens, accuracy is iteratively gained through the use of parallel machinery as a corrector. In contrast to the isolated use of a moment model, the proposed method (iteratively) converges to the true distribution generated by the SDE.},
author = {Ignace Bossuyt and Stefan Vandewalle and Giovanni Samaey},
howpublished = {arXiv:2310.11365v2 [math.NA]},
title = {Monte-Carlo/Moments micro-macro Parareal method for unimodal and bimodal scalar McKean-Vlasov SDEs},
url = {http://arxiv.org/abs/2310.11365v2},
year = {2024},
}

Expand Down