We present a fast adaptive method for the evaluation of heat potentials, which plays a key role in the integral equation approach for the solution of the heat equation, especially in a non-stationary domain. The algorithm utilizes a sum-of-exponential based fast Gauss transform that evaluates the convolution of a Gaussian with either discrete or continuous volume distributions. The latest implementation of the algorithm allows for both periodic and free space boundary conditions. The history dependence is overcome by splitting the heat potentials into a smooth history part and a singular local part. We discuss the resolution of the history part on an adaptive volume grid in detail, providing sharp estimates that allow for the construction of an optimal grid, justifying the efficiency of the bootstrapping scheme. While the discussion in this paper is restricted to one spatial dimension, the generalization to two and three dimensions is straightforward. The performance of the algorithm is illustrated via several numerical examples.

翻译：我们提出了一种用于评估热势的快速自适应方法，该方法在热方程（特别是非稳态域）的积分方程求解中起着关键作用。该算法利用基于指数求和的快速高斯变换，可评估高斯函数与离散或连续体积分布的卷积。最新实现支持周期和自由空间边界条件。通过将热势分解为光滑历史部分和奇异局部部分，克服了历史依赖性。我们详细讨论了自适应体积网格上历史部分的分辨率，提供了构建最优网格的精确估计，论证了自举方案的高效性。虽然本文仅讨论一维空间情形，但其向二维和三维的推广是直接的。通过多个数值算例展示了该算法的性能。