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(δT)(log(ϵ1)+loglog(δT))) for any x\in\bbR and
δ≤t≤T, where ϵ is the desired precision. In all
higher dimensions, the corresponding heat kernel admits an approximation
involving only O(log2(δT)) 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
NS points in the spatial discretization and NT time steps, the cost is
O(NSNTlog2δT) 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.Comment: 23 pages, 5 figures, 3 table