通过自适应粒子湮灭克服 Wigner 动力学中的数值符号问题
Overcoming the numerical sign problem in the Wigner dynamics via adaptive particle annihilation
AI总结：
针对高维 Wigner 随机模拟的符号问题，提出自适应粒子湮灭算法 SPADE，并结合稳相方差降低技术实现 6 维与 12 维质子–电子耦合模拟。
摘要：
众所周知的数值符号问题对高维相空间中基于粒子的随机 Wigner 模拟构成了根本性障碍。尽管现有的基于均匀网格的粒子湮灭方法在维度 D ≤ 4 时能够显著缓解符号问题，但当 D ≥ 6 时，由于维数灾难，网格规模急剧增长，从而使湮灭过程非常低效。本文提出一种自适应粒子湮灭算法，称为基于偏差估计的序列聚类粒子湮灭（Sequential-clustering Particle Annihilation via Discrepancy Estimation，SPADE），以克服符号问题。SPADE 遵循分治策略：通过控制粒子的数论偏差对粒子进行自适应聚类，并在每个组内进行独立随机匹配；它还可以学习能够准确捕捉 Wigner 函数非经典性所需的最少粒子数量。将 SPADE 与基于稳相近似的方差降低技术相结合，我们尝试模拟 6 维和 12 维相空间中的质子–电子耦合。本文利用特征–谱混合格式在 73^3 × 80^3 均匀网格下生成的 6 维相空间参考解，对 SPADE 进行了全面性能基准测试；该参考解充分探索了基于网格的确定性 Wigner 求解器的极限。
英文摘要：
The infamous numerical sign problem poses a fundamental obstacle to particle-based stochastic Wigner simulations in high dimensional phase space. Although the existing particle annihilation via uniform mesh significantly alleviates the sign problem when dimensionality D $\le$ 4, the mesh size grows dramatically when D $\ge$ 6 due to the curse of dimensionality and consequently makes the annihilation very inefficient. In this paper, we propose an adaptive particle annihilation algorithm, termed Sequential-clustering Particle Annihilation via Discrepancy Estimation (SPADE), to overcome the sign problem. SPADE follows a divide-and-conquer strategy: Adaptive clustering of particles via controlling their number-theoretic discrepancies and independent random matching in each group, and it may learn the minimal amount of particles that can accurately capture the non-classicality of the Wigner function. Combining SPADE with the variance reduction technique based on the stationary phase approximation, we attempt to simulate the proton-electron couplings in 6-D and 12-D phase space. A thorough performance benchmark of SPADE is provided with the reference solutions in 6-D phase space produced by a characteristic-spectral-mixed scheme under a $73^3 \times 80^3$ uniform grid, which fully explores the limit of grid-based deterministic Wigner solvers.
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