SPADE：基于离散估计的序贯聚类粒子湮灭算法
SPADE: Sequential-clustering Particle Annihilation via Discrepancy Estimation
摘要：
对于经验有符号测度$μ= \frac{1}{N} \left(\sum_{i=1}^P δ_{x_i} - \sum_{i=1}^M δ_{y_i}\right)$，粒子湮灭（PA）同时从$\{x_i\}_{i=1}^P$和$\{y_i\}_{i=1}^M$两者中移除$N_A$粒子，从而产生另一经验有符号测度$ν$，使得$\int f d ν$在对于合适的测试函数$f$的可接受精度内近似于$\int f d μ$。这种携带相反重要性权重的粒子的湮灭已经被广泛用于减轻粒子模拟中的数值符号问题。在本文中，我们提出了一种算法PA在高维欧氏空间的基础上混合聚类和匹配，称为顺序聚类粒子湮灭通过离散估计（SPADE）。它包括两个步骤：通过控制粒子的数论差异来实现粒子的自适应聚类，并在每个簇中的正负粒子之间进行独立的随机匹配。证明了Koksma-Hlawka不等式的确定性误差界和浓度不等式的非渐近随机误差界都受两个因素的影响。一个因子度量点分布的不规则性并反映其离散性。另一种依赖于测试函数的变异性，受测试函数连续性的影响。只有后者隐含地依赖于维度$d$，这意味着SPADE可以免疫的灾难的维数为广泛的测试功能。直到$d=1080$的数值实验验证了我们的理论发现。
英文摘要：
For an empirical signed measure $μ= \frac{1}{N} \left(\sum_{i=1}^P δ_{x_i} - \sum_{i=1}^M δ_{y_i}\right)$, particle annihilation (PA) removes $N_A$ particles from both $\{x_i\}_{i=1}^P$ and $\{y_i\}_{i=1}^M$ simultaneously, yielding another empirical signed measure $ν$ such that $\int f d ν$ approximates to $\int f d μ$ within an acceptable accuracy for suitable test functions $f$. Such annihilation of particles carrying opposite importance weights has been extensively utilized for alleviating the numerical sign problem in particle simulations. In this paper, we propose an algorithm for PA in high-dimensional Euclidean space based on hybrid of clustering and matching, dubbed the Sequential-clustering Particle Annihilation via Discrepancy Estimation (SPADE). It consists of two steps: Adaptive clustering of particles via controlling their number-theoretic discrepancies, and independent random matching among positive and negative particles in each cluster. Both deterministic error bounds by the Koksma-Hlawka inequality and non-asymptotic random error bounds by concentration inequalities are proved to be affected by two factors. One factor measures the irregularity of point distributions and reflects their discrete nature. The other relies on the variation of test function and is influenced by the continuity. Only the latter implicitly depends on dimensionality $d$, implying that SPADE can be immune to the curse of dimensionality for a wide class of test functions. Numerical experiments up to $d=1080$ validate our theoretical discoveries.
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