Direct numerical simulations of multiscale problems are typically extremely computationally costly. Traditional multiscale methods as, for example, multigrid reduce the computational complexity to linear in the number of unknowns at the expense of increased global communication.
This has been tackled in early papers by Tony Chan. We will here discuss this type of tradeoff for the fast multipole method and the heterogeneous multiscale method in the context of modern computer architecture. Future systems are expected to have communication as a bottleneck even more than the systems of today.
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