发表机构
Texas A&M University（德克萨斯农工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出符号切换配对策略，通过成对运行带符号梯度粒子模拟并按位置匹配粒子，显著降低一维对流扩散方程求解的采样方差，在Burgers测试中以4次模拟达到目标，优于独立采样和反射配对。
AI 中文摘要
梯度随机游走方法通过带符号粒子表示解的空间导数，并通过求和粒子质量来恢复解。我们通过将一维对流扩散方程的模拟成对运行，并按位置顺序匹配粒子，来减少此类模拟的采样误差。反射配对是标准的反对偶选择，它使匹配的粒子获得相反的位移。在符号切换配对中，同号匹配粒子获得相反位移，而异号匹配粒子获得相同位移。每个模拟保持求解器的概率分布，因此配对平均值具有单次模拟的期望和偏差。对于固定匹配，我们证明符号切换配对在最后扩散步骤中最小化每个匹配对的方差贡献。我们还推导了相对于反射配对它所消除方差的精确公式。对于热方程，这两个结果扩展到每一步，并且我们数值测量了累积方差减少。在Burgers测试中，样本量选自预运行并在新运行中检查，符号切换配对以4次模拟达到均方误差目标，而独立采样需要65次，反射配对需要34次。与仅匹配同号粒子的同号配对相比，在8192个粒子下其方差降低约26%。在2048个粒子下，它在大约少40%的时间内达到第二个目标。该增益在更多剖面、更晚时间、其他粘度和三次通量下持续存在，支持符号切换配对作为一种廉价的方法来改进异号粒子相遇的重复场估计。
英文摘要
Gradient random-walk methods represent the spatial derivative of a solution by signed particles and recover the solution by summing their masses. We reduce the sampling error of such simulations for one-dimensional convection-diffusion equations by running them in pairs with particles matched in order of position. Reflected pairing, the standard antithetic choice, gives matched particles opposite displacements. In sign-switched pairing, matched particles of equal sign receive opposite displacements and those of opposite sign the same displacement. Each simulation keeps the probability distribution of the solver, so the pair average has the expectation and bias of one simulation. For a fixed matching, we prove that sign-switched pairing minimizes the variance contribution of each matched pair at the last diffusion step. We also derive an exact formula for the variance it removes relative to reflected pairing. For the heat equation both results extend to every step, and we measure the accumulated variance reduction numerically. In Burgers tests, with sample sizes chosen from pilot runs and checked on fresh runs, sign-switched pairing meets a mean-squared-error target with 4 simulations, compared with 65 for independent sampling and 34 for reflected pairing. Against within-sign pairing, which matches only particles of equal sign, it has about 26% lower variance with 8192 particles. With 2048 particles it meets a second target in about 40% less time. The gain persists for further profiles, later times, other viscosities, and a cubic flux, supporting sign-switched pairing as an inexpensive way to improve repeated field estimates where particles of opposite sign meet.
Comments46 pages, 6 figures, 8 tables. Code and data: https://doi.org/10.5281/zenodo.23055907
