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：本文提出了一种用于计算热势的快速自适应方法，该方法在积分方程法求解热方程（尤其在非稳态区域中）中起着关键作用。该算法采用基于指数和快速高斯变换的技术，可计算高斯函数与离散或连续体积分布的卷积。该算法的最新实现同时支持周期性和自由空间边界条件。通过将热势分解为平滑的历史部分和奇异的局部部分，克服了历史依赖性。我们详细讨论了在自适应体积网格上求解历史部分的方法，提供了精确的估计以构建最优网格，从而论证了自举方案的有效性。尽管本文的讨论限于一维空间，但该方法可直接推广至二维和三维情形。最后通过若干数值算例展示了算法的性能。