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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(T¿)(log(1¿¿)+loglog(T¿)))$O(\\log (\\frac {T}{\\delta }) (\\log (\\frac {1}{\\epsilon })+\\log \\log (\\frac {T}{\\delta })))$ for any x¿¿$x\\in \\mathbb R$ and ¿≤t≤T, where ¿¿ is the desired precision. In all higher dimensions, the corresponding heat kernel admits an approximation involving only O(log2(T¿))$O(\\log ^{2}(\\frac {T}{\\delta }))$ terms for fixed accuracy ¿¿. 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 NS points in the spatial discretization and NT time steps, the cost is O(NSNTlog2T¿)$O(N_{S} N_{T} \\log ^{2} \\frac {T}{\\delta })$ in terms of both memory and CPU time for fixed accuracy ¿¿. The algorithms can be parallelized in a straightforward manner. Several numerical examples are presented to illustrate the accuracy and stability of these approximations.

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