Abstract
In this paper, we show that efficient separated sum-of-exponentials approximations can be constructed for the heat kernel in any dimension. In one space dimension, the heat kernel admits an approximation involving a number of terms that is of the order \(O(\log (\frac {T}{\delta }) (\log (\frac {1}{\epsilon })+\log \log (\frac {T}{\delta })))\) for any \(x\in \mathbb R\) and δ≤t≤T, where 𝜖 is the desired precision. In all higher dimensions, the corresponding heat kernel admits an approximation involving only \(O(\log ^{2}(\frac {T}{\delta }))\) terms for fixed accuracy 𝜖. These approximations can be used to accelerate integral equation-based methods for boundary value problems governed by the heat equation in complex geometry. The resulting algorithms are nearly optimal. For N S points in the spatial discretization and N T time steps, the cost is \(O(N_{S} N_{T} \log ^{2} \frac {T}{\delta })\) in terms of both memory and CPU time for fixed accuracy 𝜖. The algorithms can be parallelized in a straightforward manner. Several numerical examples are presented to illustrate the accuracy and stability of these approximations.
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Communicated by: Zydrunas Gimbutas
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Jiang, S., Greengard, L. & Wang, S. Efficient sum-of-exponentials approximations for the heat kernel and their applications. Adv Comput Math 41, 529–551 (2015). https://doi.org/10.1007/s10444-014-9372-1
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DOI: https://doi.org/10.1007/s10444-014-9372-1