Efficient sum-of-exponentials approximations for the heat kernel and their applications

摘要: In this paper, we show that efficient separated sum-of-exponentials approximations can be constructed for the heat kernel in any dimension. one space dimension, admits an approximation involving a number of terms is order O(log(T?)(log(1??)+loglog(T?)))$O(\log (\frac {T}{\delta }) (\log {1}{\epsilon })+\log \log })))$ x??$x\in \mathbb R$ and ?≤t≤T, where ?? desired precision. all higher dimensions, corresponding only O(log2(T?))$O(\log ^{2}(\frac }))$ fixed accuracy ??. These used to accelerate integral equation-based methods boundary value problems governed by equation complex geometry. The resulting algorithms are nearly optimal. For NS points spatial discretization NT time steps, cost O(NSNTlog2T?)$O(N_{S} N_{T} ^{2} \frac })$ both memory CPU parallelized straightforward manner. Several numerical examples presented illustrate stability these approximations.